Difference between revisions of "PDE Geometric Analysis seminar"

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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===
+
===[[Fall 2020-Spring 2021 | Tentative schedule for Fall 2020-Spring 2021]]===
  
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==
+
== PDE GA Seminar Schedule Fall 2019-Spring 2020 ==
  
  
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!align="left" | title
 
!align="left" | title
 
!style="width:20%" align="left" | host(s)
 
!style="width:20%" align="left" | host(s)
 
 
|-   
 
|-   
|August 31 (FRIDAY),
+
|Sep 9
| Julian Lopez-Gomez (Complutense University of Madrid)
+
| Scott Smith (UW Madison)
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]
+
|[[#Scott Smith | Recent progress on singular, quasi-linear stochastic PDE ]]
| Rabinowitz
+
| Kim and Tran
 
 
 
|-   
 
|-   
|September 10,
+
|Sep 14-15
| Hiroyoshi Mitake (University of Tokyo)
+
|  
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]
+
|[[ # |AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html   ]]
| Tran
+
|
|- 
 
|September 12 and September 14,
 
| Gunther Uhlmann (UWash)
 
|[[#Gunther Uhlmann | TBA ]]
 
| Li
 
|-
 
|September 17,
 
| Changyou Wang (Purdue)
 
|[[#Changyou Wang |  Some recent results on mathematical analysis of Ericksen-Leslie System ]]
 
| Tran
 
|-
 
|Sep 28, Colloquium
 
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
 
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
 
| Thiffeault
 
 
|-   
 
|-   
|October 1,
+
|Sep 23
| Matthew Schrecker (UW)
+
| Son Tu (UW Madison)
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]
+
|[[#Son Tu | State-Constraint static Hamilton-Jacobi equations in nested domains ]]
 
| Kim and Tran
 
| Kim and Tran
 
|-   
 
|-   
|October 8,
+
|Sep 28-29, VV901
| Anna Mazzucato (PSU)
+
| https://www.ki-net.umd.edu/content/conf?event_id=993
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]
+
|   | Recent progress in analytical aspects of kinetic equations and related fluid models 
| Li and Kim
+
|
 
|-   
 
|-   
|October 15,
+
|Oct 7
| Lei Wu (Lehigh)
+
| Jin Woo Jang (Postech)
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]
+
|[[#Jin Woo Jang| On a Cauchy problem for the Landau-Boltzmann equation ]]
 
| Kim
 
| Kim
 
|-   
 
|-   
|October 22,
+
|Oct 14
| Annalaura Stingo (UCD)
+
| Stefania Patrizi (UT Austin)
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]
+
|[[#Stefania Patrizi | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity ]]
| Mihaela Ifrim
+
| Tran
 
|-   
 
|-   
|October 29,
+
|Oct 21
| Yeon-Eung Kim (UW)
+
| Claude Bardos (Université Paris Denis Diderot, France)
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]
+
|[[#Claude Bardos | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture ]]
| Kim and Tran
+
| Li
 
|-   
 
|-   
|November 5,
+
|Oct 25-27, VV901
| Albert Ai (UC Berkeley)
+
| https://www.ki-net.umd.edu/content/conf?event_id=1015
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]
+
||  Forward and Inverse Problems in Kinetic Theory
| Mihaela Ifrim
+
| Li
 +
|-
 +
|Oct 28
 +
| Albert Ai (UW Madison)
 +
|[[#Albert Ai | Two dimensional gravity waves at low regularity: Energy estimates  ]]
 +
| Ifrim
 
|-   
 
|-   
|Nov 7 (Wednesday), Colloquium
+
|Nov 4
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
+
| Yunbai Cao (UW Madison)
|[[#Nov 7: Luca Spolaor (MIT) | (Log)-Epiperimetric Inequality and the Regularity of Variational Problems  ]]
+
|[[#Yunbai Cao | Vlasov-Poisson-Boltzmann system in Bounded Domains]]
| Feldman
+
| Kim and Tran
 +
|- 
 +
|Nov 18
 +
| Ilyas Khan (UW Madison)
 +
|[[#Ilyas Khan | The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension ]]
 +
| Kim and Tran
 
|-
 
|-
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''
+
|Nov 25
| Trevor Leslie (UW)
+
| Mathew Langford (UT Knoxville)
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]
+
|[[#Mathew Langford | Concavity of the arrival time ]]
| Kim and Tran
+
| Angenent
|-
+
|-  
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''
+
|Dec 9 - Colloquium (4-5PM)
|Serena Federico (MIT)
+
| Hui Yu (Columbia)
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]
+
|[[#Hui Yu | TBA ]]
| Mihaela Ifrim
+
| Tran
 
|-  
 
|-  
|December 10, Colloquium, '''Time: 4:00'''
+
|Feb. 3
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)
+
| Philippe LeFloch (Sorbonne Université)
|[[#Max Engelstein| The role of Energy in Regularity ]]
+
|[[#Speaker | TBA ]]
 
| Feldman
 
| Feldman
 
|-  
 
|-  
|January 28,
+
|Feb. 10
| Ru-Yu Lai (Minnesota)
+
| Joonhyun La (Stanford)
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]
+
|[[#Joonhyun La | TBA ]]
| Li and Kim
+
| Kim
|-
 
| February 4,
 
|
 
|[[# | No seminar (several relevant colloquiums in Feb/5 and Feb/8)]]
 
|
 
|-
 
| February 11,
 
| Seokbae Yun (SKKU, long term visitor of UW-Madison)
 
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]
 
| Kim  
 
|-
 
| February 13 '''4PM''',
 
| Dean Baskin (Texas A&M)
 
|[[#Dean Baskin | Radiation fields for wave equations]]
 
| Colloquium
 
 
|-   
 
|-   
| February 18,  '''3:30PM, Room: VV B239'''
+
|Feb 17
| Daniel Tataru (Berkeley)
+
| Yannick Sire (JHU)
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]
+
|[[#Yannick Sire (JHU) | TBA ]]
| Ifrim
 
|-                                                                                                                                                         
 
| February 19, '''Time: 4-5PM, Room: VV B139'''
 
| Wenjia Jing (Tsinghua University)
 
|[[#Wenjia Jing | Periodic homogenization of Dirichlet problems in perforated domains: a unified proof ]]
 
 
| Tran
 
| Tran
|-  
+
|-
|February 25,
+
|Feb 24
| Xiaoqin Guo (UW)
+
| Matthew Schrecker (UW Madison)
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]
+
|[[#Matthew Schrecker | TBA ]]
| Kim and Tran
+
| Feldman
|-
+
|-
|March 4 '''time:4PM-5PM, Room: VV B239'''
+
|March 2
| Vladimir Sverak (Minnesota)
+
| Theodora Bourni (UT Knoxville)
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]
+
|[[#Speaker | TBA ]]
 +
| Angenent
 +
|-
 +
|March 9
 +
| Ian Tice (CMU)
 +
|[[#Ian Tice| TBA ]]
 
| Kim
 
| Kim
|-  
+
|-
|March 11
+
|March 16
| Jonathan Luk (Stanford)
+
| No seminar (spring break)
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials ]]
+
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|March 23
 +
| Jared Speck (Vanderbilt)
 +
|[[#Jared Speck | TBA ]]
 +
| SCHRECKER
 +
|- 
 +
|March 30
 +
| Huy Nguyen (Brown)
 +
|[[#Huy Nguyen | TBA ]]
 +
| organizer
 +
|-  
 +
|April 6
 +
| Speaker (Institute)
 +
|[[#Speaker | TBA ]]
 +
| Host
 +
|- 
 +
|April 13
 +
| Hyunju Kwon (IAS)
 +
|[[#Hyunju Kwon | TBA ]]
 
| Kim
 
| Kim
|-
+
|-
|March 12, '''4:00 p.m. in VV B139'''
+
|April 20
| Trevor Leslie (UW-Madison)
+
| Speaker (Institute)
|[[# Trevor Leslie| TBA ]]
+
|[[#Speaker | TBA ]]
| Analysis seminar
+
| Host
|-
+
|-   
|March 18,
+
|April 27
| Spring recess (Mar 16-24, 2019)
+
| Speaker (Institute)
|[[#  |  ]]
+
|[[#Speaker | TBA ]]
+
| Host
|-
+
|-   
|March 25
+
|May 18-21
| Jiaxin Jin
+
| Madison Workshop in PDE 2020
|[[# Jiaxin Jin  |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. ]]
+
|[[#Speaker | TBA ]]
| local speaker
 
|-   
 
|April 1
 
| Zaher Hani (Michigan)
 
|[[#Zaher Hani | TBA ]]
 
| Ifrim
 
|-  
 
|April 8  
 
| Jingrui Cheng (Stony Brook)
 
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]
 
| Feldman
 
|-
 
|April 15,
 
| Yao Yao (Gatech)
 
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]
 
 
| Tran
 
| Tran
|-   
 
|April 22,
 
| Jessica Lin (McGill University)
 
|[[#Jessica Lin | TBA ]]
 
| Tran
 
|- 
 
|April 29,
 
| Beomjun Choi (Columbia)
 
|[[#Beomjun Choi  | Evolution of non-compact hypersurfaces by inverse mean curvature]]
 
|  Angenent
 
 
|}
 
|}
  
 
== Abstracts ==
 
== Abstracts ==
  
===Julian Lopez-Gomez===
+
===Scott Smith===
 
 
Title: The theorem of characterization of the Strong Maximum Principle
 
 
 
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.
 
 
 
===Hiroyoshi Mitake===
 
Title: On approximation of time-fractional fully nonlinear equations
 
  
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.)
+
Title: Recent progress on singular, quasi-linear stochastic PDE
  
 +
Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing .  These equations are ill-posed in the traditional sense of distribution theory.  They require flexibility in the notion of solution as well as new a priori bounds.  Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory.  This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.
  
  
===Changyou Wang===
+
===Son Tu===
  
Title: Some recent results on mathematical analysis of Ericksen-Leslie System
+
Title: State-Constraint static Hamilton-Jacobi equations in nested domains
  
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.
+
Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).
  
===Matthew Schrecker===
 
  
Title: Finite energy methods for the 1D isentropic Euler equations
+
===Jin Woo Jang===
  
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.
+
Title: On a Cauchy problem for the Landau-Boltzmann equation
  
===Anna Mazzucato===
+
Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.
  
Title: On the vanishing viscosity limit in incompressible flows
 
  
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.
+
===Stefania Patrizi===
  
===Lei Wu===
+
Title:
 +
Dislocations dynamics: from microscopic models to macroscopic crystal plasticity
  
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects
+
Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic  limit of  solutions of  Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is  joint work with E. Valdinoci.
  
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.
 
  
 +
===Claude Bardos===
 +
Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture
  
===Annalaura Stingo===
+
Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.
  
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D
+
Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.
  
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 »  .
+
Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.
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.
 
 
 
===Yeon-Eung Kim===
 
 
 
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties
 
 
 
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.
 
  
 
===Albert Ai===
 
===Albert Ai===
 +
Title: Two dimensional gravity waves at low regularity: Energy estimates
  
Title: Low Regularity Solutions for Gravity Water Waves
+
Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.
 
 
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.
 
 
 
===Trevor Leslie===
 
 
 
Title: Flocking Models with Singular Interaction Kernels
 
 
 
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.
 
 
 
===Serena Federico===
 
 
 
Title: Sufficient conditions for local solvability of some degenerate partial differential operators
 
 
 
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.
 
 
 
===Max Engelstein===
 
 
 
Title: The role of Energy in Regularity
 
 
 
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.
 
 
 
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.
 
 
 
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.
 
 
 
 
 
===Ru-Yu Lai===
 
Title: Inverse transport theory and related applications.
 
 
 
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.
 
 
 
===Seokbae Yun===
 
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation
 
 
 
Abstract: In this talk, we consider the propagation of the uniform upper bounds
 
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two
 
types of estimates for the the gain part of the collision operator: namely, a potential
 
type estimate and a relativistic hyper-surface integral estimate. We then combine them
 
using the relativistic counter-part of the Carlemann representation to derive a uniform
 
control of the gain part, which gives the desired propagation of the uniform bounds of
 
the solution. Some applications of the results are also considered. This is a joint work
 
with Jin Woo Jang and Robert M. Strain.
 
 
 
 
 
 
 
===Daniel Tataru===
 
 
 
Title: A Morawetz inequality for water waves.
 
 
 
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.
 
 
 
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.
 
 
 
 
 
===Wenjia Jing===
 
 
 
Title: Periodic homogenization of Dirichlet problems in perforated domains: a unified proof
 
 
 
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.
 
 
 
 
 
===Xiaoqin Guo===
 
 
 
Title: Quantitative homogenization in a balanced random environment
 
 
 
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).
 
 
 
===Sverak===
 
 
 
Title: PDE aspects of the Navier-Stokes equations and simpler models
 
 
 
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.
 
 
 
===Jonathan Luk===
 
 
 
Title: Stability of vacuum for the Landau equation with moderately soft potentials
 
 
 
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.
 
 
 
 
 
===Jiaxin Jin===
 
 
 
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.
 
 
 
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.
 
 
 
===Jingrui Cheng===
 
 
 
Title: Gradient estimate for complex Monge-Ampere equations
 
 
 
Abstract: We consider complex Monge-Ampere equations on a compact Kahler manifold. Previous gradient estimates of the solution all require some derivative bound of the right hand side. I will talk about how to get gradient estimate in $L^p$ and $L^{\infty}$, depending only on the continuity of the right hand side.
 
 
 
 
 
===Yao Yao===
 
 
 
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations
 
 
 
Abstract: In this talk, I will discuss some recent work on radial symmetry property for stationary or uniformly-rotating solutions for 2D Euler and SQG equation, where we aim to answer the question whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. This is a joint work with Javier Gómez-Serrano, Jaemin Park and Jia Shi.
 
  
 +
===Ilyas Khan===
 +
Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.
  
 +
Abstract: In this talk, we will consider self-shrinking solitons of the mean curvature flow that are smoothly asymptotic to a Riemannian cone in $\mathbb{R}^n$. In 2011, L. Wang proved the uniqueness of self-shrinking ends asymptotic to a cone $C$ in the case of hypersurfaces (codimension 1) by using a backwards uniqueness result for the heat equation due to Escauriaza, Sverak, and Seregin. Later, J. Bernstein proved the same fact using purely elliptic methods. We consider the case of self-shrinkers in high codimension, and outline how to prove the same uniqueness result in this significantly more general case, by using geometric arguments and extending Bernstein’s result.
  
===Beomjun Choi===
+
===Mathew Langford===
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. 
+
Title: Concavity of the arrival time
  
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.
+
Abstract:  We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a novel concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential Harnack inequality for mean curvature flow and, more generally, Andrews’ differential Harnack inequalities for certain “$\alpha$-inverse-concave” flows.

Latest revision as of 09:12, 9 December 2019

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Previous PDE/GA seminars

Tentative schedule for Fall 2020-Spring 2021

PDE GA Seminar Schedule Fall 2019-Spring 2020

date speaker title host(s)
Sep 9 Scott Smith (UW Madison) Recent progress on singular, quasi-linear stochastic PDE Kim and Tran
Sep 14-15 AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html
Sep 23 Son Tu (UW Madison) State-Constraint static Hamilton-Jacobi equations in nested domains Kim and Tran
Sep 28-29, VV901 https://www.ki-net.umd.edu/content/conf?event_id=993 Recent progress in analytical aspects of kinetic equations and related fluid models
Oct 7 Jin Woo Jang (Postech) On a Cauchy problem for the Landau-Boltzmann equation Kim
Oct 14 Stefania Patrizi (UT Austin) Dislocations dynamics: from microscopic models to macroscopic crystal plasticity Tran
Oct 21 Claude Bardos (Université Paris Denis Diderot, France) From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture Li
Oct 25-27, VV901 https://www.ki-net.umd.edu/content/conf?event_id=1015 Forward and Inverse Problems in Kinetic Theory Li
Oct 28 Albert Ai (UW Madison) Two dimensional gravity waves at low regularity: Energy estimates Ifrim
Nov 4 Yunbai Cao (UW Madison) Vlasov-Poisson-Boltzmann system in Bounded Domains Kim and Tran
Nov 18 Ilyas Khan (UW Madison) The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension Kim and Tran
Nov 25 Mathew Langford (UT Knoxville) Concavity of the arrival time Angenent
Dec 9 - Colloquium (4-5PM) Hui Yu (Columbia) TBA Tran
Feb. 3 Philippe LeFloch (Sorbonne Université) TBA Feldman
Feb. 10 Joonhyun La (Stanford) TBA Kim
Feb 17 Yannick Sire (JHU) TBA Tran
Feb 24 Matthew Schrecker (UW Madison) TBA Feldman
March 2 Theodora Bourni (UT Knoxville) TBA Angenent
March 9 Ian Tice (CMU) TBA Kim
March 16 No seminar (spring break) TBA Host
March 23 Jared Speck (Vanderbilt) TBA SCHRECKER
March 30 Huy Nguyen (Brown) TBA organizer
April 6 Speaker (Institute) TBA Host
April 13 Hyunju Kwon (IAS) TBA Kim
April 20 Speaker (Institute) TBA Host
April 27 Speaker (Institute) TBA Host
May 18-21 Madison Workshop in PDE 2020 TBA Tran

Abstracts

Scott Smith

Title: Recent progress on singular, quasi-linear stochastic PDE

Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.


Son Tu

Title: State-Constraint static Hamilton-Jacobi equations in nested domains

Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).


Jin Woo Jang

Title: On a Cauchy problem for the Landau-Boltzmann equation

Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.


Stefania Patrizi

Title: Dislocations dynamics: from microscopic models to macroscopic crystal plasticity

Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.


Claude Bardos

Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture

Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.

Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.

Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.

Albert Ai

Title: Two dimensional gravity waves at low regularity: Energy estimates

Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.

Ilyas Khan

Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.

Abstract: In this talk, we will consider self-shrinking solitons of the mean curvature flow that are smoothly asymptotic to a Riemannian cone in $\mathbb{R}^n$. In 2011, L. Wang proved the uniqueness of self-shrinking ends asymptotic to a cone $C$ in the case of hypersurfaces (codimension 1) by using a backwards uniqueness result for the heat equation due to Escauriaza, Sverak, and Seregin. Later, J. Bernstein proved the same fact using purely elliptic methods. We consider the case of self-shrinkers in high codimension, and outline how to prove the same uniqueness result in this significantly more general case, by using geometric arguments and extending Bernstein’s result.

Mathew Langford

Title: Concavity of the arrival time

Abstract: We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a novel concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential Harnack inequality for mean curvature flow and, more generally, Andrews’ differential Harnack inequalities for certain “$\alpha$-inverse-concave” flows.