https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Ckim&feedformat=atomMath - User contributions [en]2019-07-21T19:34:24ZUser contributionsMediaWiki 1.28.3https://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17350Fall 2019-Spring 20202019-04-19T21:44:00Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Jin Woo Jang (Postech)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17349Fall 2019-Spring 20202019-04-19T21:43:30Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| J Jang (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17171PDE Geometric Analysis seminar2019-03-15T15:36:55Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
<br />
===Jiaxin Jin===<br />
<br />
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.<br />
<br />
Abstract: We first analyze a three-species system with boundary equilibria in some stoichiometric classes and study the rate of convergence to the complex balanced equilibrium. Then we prove similar results on the convergence to the positive equilibrium for a fairly general two-species reversible reaction-diffusion network with boundary equilibria.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17170PDE Geometric Analysis seminar2019-03-15T15:36:28Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17169Fall 2019-Spring 20202019-03-15T14:46:10Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ # | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17168Fall 2019-Spring 20202019-03-15T14:42:47Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[ | ]]<br />
| <br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17167Fall 2019-Spring 20202019-03-15T14:41:40Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 14-15<br />
| AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17166Fall 2019-Spring 20202019-03-15T13:51:19Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|- <br />
|- <br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17165Fall 2019-Spring 20202019-03-14T23:53:11Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 10<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17164Fall 2019-Spring 20202019-03-14T23:52:37Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 10<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17163Fall 2019-Spring 20202019-03-14T23:51:37Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Jan 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 3<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 10<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 17<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Feb 24<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 16 <br />
| No seminar (spring break)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|March 30<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 6<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 13<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 20<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|April 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17162Fall 2019-Spring 20202019-03-14T23:48:07Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
| <br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Jan 27<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17114PDE Geometric Analysis seminar2019-03-05T23:26:56Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17113PDE Geometric Analysis seminar2019-03-05T23:26:35Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
===Jonathan Luk===<br />
<br />
Title: Stability of vacuum for the Landau equation with moderately soft potentials<br />
<br />
Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17029Colloquia2019-02-22T14:59:08Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday, VV 911'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 6 '''Wednesday, room 911'''<br />
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)<br />
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs ]]<br />
| Anderson<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[#Angelica Cueto (The Ohio State University)| Lines on cubic surfaces in the tropics ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) <br />
|[[# Vladimir Sverak (Minnesota) | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Eviatar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 22 '''Monday'''<br />
| [https://justinh.su Justin Hsu] (Madison)<br />
|[[# TBA| TBA ]]<br />
| Lempp<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Angelica Cueto (The Ohio State University)===<br />
Title: Lines on cubic surfaces in the tropics<br />
<br />
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement <i>any smooth surface of degree three in P^3 contains exactly 27 lines</i> is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.<br />
<br />
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Li-Cheng Tsai (Columbia University)===<br />
<br />
Title: When particle systems meet PDEs<br />
<br />
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
<br />
===Vladimir Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17028Colloquia2019-02-22T14:58:36Z<p>Ckim: /* Spring 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday, VV 911'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 6 '''Wednesday, room 911'''<br />
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)<br />
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs ]]<br />
| Anderson<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[#Angelica Cueto (The Ohio State University)| Lines on cubic surfaces in the tropics ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) <br />
|[[# Vladimir Sverak (Minnesota) | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Eviatar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 22 '''Monday'''<br />
| [https://justinh.su Justin Hsu] (Madison)<br />
|[[# TBA| TBA ]]<br />
| Lempp<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Angelica Cueto (The Ohio State University)===<br />
Title: Lines on cubic surfaces in the tropics<br />
<br />
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement <i>any smooth surface of degree three in P^3 contains exactly 27 lines</i> is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.<br />
<br />
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Li-Cheng Tsai (Columbia University)===<br />
<br />
Title: When particle systems meet PDEs<br />
<br />
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17027Colloquia2019-02-22T14:58:19Z<p>Ckim: /* Spring 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday, VV 911'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 6 '''Wednesday, room 911'''<br />
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)<br />
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs ]]<br />
| Anderson<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[#Angelica Cueto (The Ohio State University)| Lines on cubic surfaces in the tropics ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture<br />
|[[# Vladimir Sverak (Minnesota) | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Eviatar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 22 '''Monday'''<br />
| [https://justinh.su Justin Hsu] (Madison)<br />
|[[# TBA| TBA ]]<br />
| Lempp<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Angelica Cueto (The Ohio State University)===<br />
Title: Lines on cubic surfaces in the tropics<br />
<br />
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement <i>any smooth surface of degree three in P^3 contains exactly 27 lines</i> is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.<br />
<br />
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Li-Cheng Tsai (Columbia University)===<br />
<br />
Title: When particle systems meet PDEs<br />
<br />
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17026PDE Geometric Analysis seminar2019-02-22T14:57:07Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17025PDE Geometric Analysis seminar2019-02-22T14:56:34Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | PDE aspects of the Navier-Stokes equations and simpler models (Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17024PDE Geometric Analysis seminar2019-02-22T14:56:14Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
===Sverak===<br />
<br />
Title: PDE aspects of the Navier-Stokes equations and simpler models<br />
<br />
Abstract: Does the Navier-Stokes equation give a reasonably complete description of fluid motion? There seems to be no empirical evidence which would suggest a negative answer (in regimes which are not extreme), but from the purely mathematical point of view, the answer may not be so clear. In the lecture, I will discuss some of the possible scenarios and open problems for both the full equations and simplified models.<br />
<br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17023Fall 2019-Spring 20202019-02-22T07:10:06Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17015PDE Geometric Analysis seminar2019-02-21T17:38:34Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''time:4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=17014PDE Geometric Analysis seminar2019-02-21T17:38:12Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 '''4PM-5PM, Room: VV B239'''<br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17005Fall 2019-Spring 20202019-02-20T18:24:33Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 25<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17004Fall 2019-Spring 20202019-02-20T18:23:29Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|- <br />
|Oct 7<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 14<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 21<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Oct 28<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 4<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 11<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 18<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Nov 25<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Dec 2<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17003Fall 2019-Spring 20202019-02-20T18:04:10Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Sep 9<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 16<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 23<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Sep 30<br />
| Michael Loss (Georgia tech)<br />
|[[#Michael Loss | TBA ]]<br />
| Kim<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=17002Fall 2019-Spring 20202019-02-20T18:02:21Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2019-Spring 2020 */</p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
|- <br />
|Date<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|- <br />
|Date<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16992PDE Geometric Analysis seminar2019-02-19T12:12:42Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25<br />
| Jiaxin Jin<br />
|[[# Jiaxin Jin |TBA ]]<br />
| local speaker<br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16991PDE Geometric Analysis seminar2019-02-19T03:14:51Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng (Stony Brook) <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16990PDE Geometric Analysis seminar2019-02-19T03:13:49Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Jingrui Cheng <br />
|[[#Jingrui Cheng | TBA ]]<br />
| Feldman<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Daniel Tataru===<br />
<br />
Title: A Morawetz inequality for water waves.<br />
<br />
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.<br />
<br />
Abstract: We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit. <br />
<br />
<br />
===Wenjia Jing===<br />
<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Xiaoqin Guo===<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison). <br />
<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16905PDE Geometric Analysis seminar2019-02-12T15:46:36Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''3:30PM, Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19, '''Time: 4-5PM, Room: VV B139'''<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
<br />
===Wenjia Jing===<br />
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof<br />
<br />
Abstract: In this talk, we present a unified proof to establish periodic homogenization for the Dirichlet problems associated to the Laplace operator in perforated domains; here the uniformity is with respect to the ratio between scaling factors of the perforation holes and the periodicity. Our method recovers, for critical scaling of the hole-cell ratio, the “strange term coming from nowhere” found by Cioranescu and Murat, and it works at the same time for other settings of hole-cell ratios. Moreover, the method is naturally based on analysis of rescaled cell problems and hence reveals the intrinsic connections among the apparently different homogenization behaviors in those different settings. We also show how to quantify the approach to get error estimates and corrector results.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16792PDE Geometric Analysis seminar2019-01-31T15:44:01Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16777PDE Geometric Analysis seminar2019-01-29T21:39:54Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February 13 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16774PDE Geometric Analysis seminar2019-01-29T19:35:34Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[#Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|-<br />
| February '''PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[#Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
<br />
===Ru-Yu Lai===<br />
Title: Inverse transport theory and related applications.<br />
<br />
Abstract: The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk we will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be discussed.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16716PDE Geometric Analysis seminar2019-01-25T00:22:32Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| Jan 31 '''4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[# Dean Baskin | Radiation fields for wave equations]]<br />
| Colloquium<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16715PDE Geometric Analysis seminar2019-01-25T00:21:19Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| Jan 31 '''Colloquium 4PM''',<br />
| Dean Baskin (Texas A&M)<br />
|[[# Dean Baskin | Radiation fields for wave equations]]<br />
| <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16693PDE Geometric Analysis seminar2019-01-23T20:23:00Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25 (open)<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 (open)<br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16692PDE Geometric Analysis seminar2019-01-23T20:09:10Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|-<br />
|March 25,<br />
| Open <br />
|[[# Open |Open ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|- <br />
|April 8 <br />
| Open <br />
|[[#Open | Open ]]<br />
| <br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16690PDE Geometric Analysis seminar2019-01-23T19:59:51Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 12, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16654PDE Geometric Analysis seminar2019-01-20T22:25:51Z<p>Ckim: /* Abstracts */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
|January 29, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Seokbae Yun===<br />
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
<br />
Abstract: In this talk, we consider the propagation of the uniform upper bounds<br />
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two<br />
types of estimates for the the gain part of the collision operator: namely, a potential<br />
type estimate and a relativistic hyper-surface integral estimate. We then combine them<br />
using the relativistic counter-part of the Carlemann representation to derive a uniform<br />
control of the gain part, which gives the desired propagation of the uniform bounds of<br />
the solution. Some applications of the results are also considered. This is a joint work <br />
with Jin Woo Jang and Robert M. Strain.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16653PDE Geometric Analysis seminar2019-01-20T22:24:52Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
|January 29, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16652PDE Geometric Analysis seminar2019-01-20T22:24:27Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
|January 29, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation<br />
]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[#Beomjun Choi | Evolution of non-compact hypersurfaces by inverse mean curvature]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.<br />
<br />
===Beomjun Choi===<br />
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed. <br />
<br />
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16638PDE Geometric Analysis seminar2019-01-16T17:43:01Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
|January 29, '''4:00 p.m. in VV B139'''<br />
| Trevor Leslie (UW-Madison)<br />
|[[# Trevor Leslie| TBA ]]<br />
| Analysis seminar<br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| Beomjun Choi (Columbia)<br />
|[[# | TBA ]]<br />
| Angenent<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16589PDE Geometric Analysis seminar2019-01-02T14:01:44Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16588PDE Geometric Analysis seminar2019-01-02T14:01:22Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar (several relevant colloquiums Jan/30-Feb/8)]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16587PDE Geometric Analysis seminar2019-01-02T12:02:05Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
|<br />
|[[# | No seminar]]<br />
| <br />
|-<br />
| February 11,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
| February 19,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|February 25,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=Fall_2019-Spring_2020&diff=16575Fall 2019-Spring 20202018-12-21T03:17:30Z<p>Ckim: </p>
<hr />
<div>== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
<br />
|- <br />
|Date<br />
| Speaker (Institute)<br />
|[[#Speaker | TBA ]]<br />
| Host<br />
|}</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16571PDE Geometric Analysis seminar2018-12-20T14:02:28Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, '''Room: VV B239'''<br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
|Time: TBD in February,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Time: TBD in February,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16546PDE Geometric Analysis seminar2018-12-10T15:05:46Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, <br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
|Time: TBD in February,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Time: TBD in February,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Ckimhttps://www.math.wisc.edu/wiki/index.php?title=PDE_Geometric_Analysis_seminar&diff=16545PDE Geometric Analysis seminar2018-12-10T15:01:22Z<p>Ckim: /* PDE GA Seminar Schedule Fall 2018-Spring 2019 */</p>
<hr />
<div>The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.<br />
<br />
===[[Previous PDE/GA seminars]]===<br />
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===<br />
<br />
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==<br />
<br />
<br />
{| cellpadding="8"<br />
!style="width:20%" align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!style="width:20%" align="left" | host(s)<br />
<br />
|- <br />
|August 31 (FRIDAY),<br />
| Julian Lopez-Gomez (Complutense University of Madrid)<br />
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]<br />
| Rabinowitz<br />
<br />
|- <br />
|September 10,<br />
| Hiroyoshi Mitake (University of Tokyo)<br />
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]<br />
| Tran<br />
|- <br />
|September 12 and September 14,<br />
| Gunther Uhlmann (UWash)<br />
|[[#Gunther Uhlmann | TBA ]]<br />
| Li<br />
|- <br />
|September 17,<br />
| Changyou Wang (Purdue)<br />
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]<br />
| Tran<br />
|-<br />
|Sep 28, Colloquium<br />
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)<br />
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]<br />
| Thiffeault<br />
|- <br />
|October 1,<br />
| Matthew Schrecker (UW)<br />
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]<br />
| Kim and Tran<br />
|- <br />
|October 8,<br />
| Anna Mazzucato (PSU)<br />
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]<br />
| Li and Kim<br />
|- <br />
|October 15,<br />
| Lei Wu (Lehigh)<br />
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]<br />
| Kim<br />
|- <br />
|October 22,<br />
| Annalaura Stingo (UCD)<br />
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]<br />
| Mihaela Ifrim<br />
|- <br />
|October 29,<br />
| Yeon-Eung Kim (UW)<br />
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]<br />
| Kim and Tran<br />
|- <br />
|November 5,<br />
| Albert Ai (UC Berkeley)<br />
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]<br />
| Mihaela Ifrim<br />
|- <br />
|Nov 7 (Wednesday), Colloquium<br />
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)<br />
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems ]]<br />
| Feldman<br />
|-<br />
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''<br />
| Trevor Leslie (UW)<br />
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]<br />
| Kim and Tran <br />
|-<br />
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''<br />
|Serena Federico (MIT)<br />
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]<br />
| Mihaela Ifrim <br />
|- <br />
|December 10, Colloquium, '''Time: 4:00''' <br />
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)<br />
|[[# Max Engelstein| The role of Energy in Regularity ]]<br />
| Feldman<br />
|- <br />
|January 28,<br />
| Ru-Yu Lai (Minnesota)<br />
|[[# Ru-Yu Lai | TBA ]]<br />
| Li and Kim <br />
|-<br />
| February 4,<br />
| Seokbae Yun (SKKU, long term visitor of UW-Madison)<br />
|[[# Seokbae Yun | TBA ]]<br />
| Kim <br />
|- <br />
| February 18, ''' Room: Van Vleck B239''' <br />
| Daniel Tataru (Berkeley)<br />
|[[# Daniel Tataru | TBA ]]<br />
| Ifrim <br />
|- <br />
|Time: TBD in February,<br />
| Xiaoqin Guo (UW)<br />
|[[#Xiaoqin Guo | TBA ]]<br />
| Kim and Tran<br />
|-<br />
|Time: TBD in February,<br />
| Wenjia Jing (Tsinghua University)<br />
|[[#Wenjia Jing | TBA ]]<br />
| Tran<br />
|- <br />
|March 4 <br />
| Vladimir Sverak (Minnesota)<br />
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]<br />
| Kim<br />
|- <br />
|March 11 <br />
| Jonathan Luk (Stanford)<br />
|[[#Jonathan Luk | TBA ]]<br />
| Kim<br />
|-<br />
|March 18,<br />
| Spring recess (Mar 16-24, 2019)<br />
|[[# | ]]<br />
| <br />
|- <br />
|April 1 <br />
| Zaher Hani (Michigan)<br />
|[[#Zaher Hani | TBA ]]<br />
| Ifrim<br />
|-<br />
|April 15,<br />
| Yao Yao (Gatech)<br />
|[[#Yao Yao | TBA ]]<br />
| Tran<br />
|- <br />
|April 22,<br />
| Jessica Lin (McGill University)<br />
|[[#Jessica Lin | TBA ]]<br />
| Tran<br />
|- <br />
|April 29,<br />
| ( )<br />
|[[# | TBA ]]<br />
| <br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Julian Lopez-Gomez===<br />
<br />
Title: The theorem of characterization of the Strong Maximum Principle<br />
<br />
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.<br />
<br />
===Hiroyoshi Mitake===<br />
Title: On approximation of time-fractional fully nonlinear equations<br />
<br />
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.) <br />
<br />
<br />
<br />
===Changyou Wang===<br />
<br />
Title: Some recent results on mathematical analysis of Ericksen-Leslie System<br />
<br />
Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.<br />
<br />
===Matthew Schrecker===<br />
<br />
Title: Finite energy methods for the 1D isentropic Euler equations<br />
<br />
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.<br />
<br />
===Anna Mazzucato===<br />
<br />
Title: On the vanishing viscosity limit in incompressible flows<br />
<br />
Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.<br />
<br />
===Lei Wu===<br />
<br />
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects<br />
<br />
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.<br />
<br />
<br />
===Annalaura Stingo===<br />
<br />
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D<br />
<br />
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » .<br />
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.<br />
<br />
===Yeon-Eung Kim===<br />
<br />
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties<br />
<br />
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.<br />
<br />
===Albert Ai===<br />
<br />
Title: Low Regularity Solutions for Gravity Water Waves<br />
<br />
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.<br />
<br />
===Trevor Leslie===<br />
<br />
Title: Flocking Models with Singular Interaction Kernels<br />
<br />
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system. In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon. Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents. We focus on the recent line of research that treats the case where the interaction kernel is singular. In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.<br />
<br />
===Serena Federico===<br />
<br />
Title: Sufficient conditions for local solvability of some degenerate partial differential operators <br />
<br />
Abstract: In this talk we will give sufficient conditions for the local solvability of a class of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol. By giving suitable conditions on the subprincipal part and using the technique of a priori estimates, we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.<br />
<br />
===Max Engelstein===<br />
<br />
Title: The role of Energy in Regularity<br />
<br />
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.<br />
<br />
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.<br />
<br />
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.</div>Ckim