Difference between revisions of "PDE Geometric Analysis seminar"

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(PDE GA Seminar Schedule Fall 2018-Spring 2019)
 
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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Spring 2018 | Tentative schedule for Spring 2018]]===
+
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==
 +
 
  
== PDE GA Seminar Schedule Fall 2017 ==
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!style="width:20%" align="left" | date   
 
!style="width:20%" align="left" | date   
Line 10: Line 12:
 
!align="left" | title
 
!align="left" | title
 
!style="width:20%" align="left" | host(s)
 
!style="width:20%" align="left" | host(s)
|-  
+
 
|September 11
+
|-
|Mihaela Ifrim (UW)
+
|August 31 (FRIDAY),
|[[#Mihaela Ifrim|  Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]
+
| Julian Lopez-Gomez (Complutense University of Madrid)
| Kim & Tran
+
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]
|-  
+
| Rabinowitz
|September 18
+
 
|Longjie Zhang (University of Tokyo)  
+
|-
|[[#Longjie Zhang | On curvature flow with driving force starting as singular initial curve in the plane]]
+
|September 10,
| Angenent
+
| Hiroyoshi Mitake (University of Tokyo)
|-  
+
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]
|September 22,
 
VV 9th floor hall, 4:00pm
 
|Jaeyoung Byeon (KAIST)  
 
|[[#Jaeyoung Byeon| Colloquium: Patterns formation for elliptic systems with large interaction forces]]
 
|  Rabinowitz
 
|-
 
|September 25
 
| Tuoc Phan (UTK)
 
|[[#Tuoc Phan |  Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]
 
 
| Tran
 
| Tran
|-  
+
|-
|September 26,  
+
|September 12 and September 14,
VV B139 4:00pm
+
| Gunther Uhlmann (UWash)
| Hiroyoshi Mitake (Hiroshima University)
+
|[[#Gunther Uhlmann | TBA ]]
|[[#Hiroyoshi Mitake Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]
+
| Li
 +
|- 
 +
|September 17,
 +
| Changyou Wang (Purdue)
 +
|[[#Changyou Wang Some recent results on mathematical analysis of Ericksen-Leslie System ]]
 
| Tran
 
| 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,
 +
| Matthew Schrecker (UW)
 +
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]
 +
| Kim and Tran
 +
|- 
 +
|October 8,
 +
| Anna Mazzucato (PSU)
 +
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]
 +
| Li and Kim
 +
|- 
 +
|October 15,
 +
| Lei Wu (Lehigh)
 +
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]
 +
| Kim
 +
|- 
 +
|October 22,
 +
| Annalaura Stingo (UCD)
 +
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]
 +
| Mihaela Ifrim
 +
|- 
 +
|October 29,
 +
| Yeon-Eung Kim (UW)
 +
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]
 +
| Kim and Tran
 +
|- 
 +
|November 5,
 +
| Albert Ai (UC Berkeley)
 +
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]
 +
| Mihaela Ifrim
 +
|- 
 +
|Nov 7 (Wednesday), Colloquium
 +
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
 +
|[[#Nov 7: Luca Spolaor (MIT) |  (Log)-Epiperimetric Inequality and the Regularity of Variational Problems  ]]
 +
| Feldman
 +
|-
 +
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''
 +
| Trevor Leslie (UW)
 +
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]
 +
| Kim and Tran
 +
|-
 +
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''
 +
|Serena Federico (MIT)
 +
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]
 +
| Mihaela Ifrim
 
|-  
 
|-  
|September 29,
+
|December 10, Colloquium, '''Time: 4:00'''
VV901 2:25pm
+
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)
| Dongnam Ko (CMU & SNU)
+
|[[#Max Engelstein| The role of Energy in Regularity ]]
|[[#Dongnam Ko | a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]
+
| Feldman
| Shi Jin & Kim
 
 
|-  
 
|-  
|October 2
+
|January 28,
| No seminar due to a KI-Net conference
+
| Ru-Yu Lai (Minnesota)
|
+
|[[#Ru-Yu Lai | Inverse transport theory and related applications ]]
 +
| Li and 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'''
 +
| Daniel Tataru (Berkeley)
 +
|[[#Daniel Tataru | A Morawetz inequality for water waves ]]
 +
| 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
 
|-  
 
|-  
|October 9
+
|February 25,
| Sameer Iyer (Brown University)
+
| Xiaoqin Guo (UW)
|[[#Sameer Iyer | Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
+
|[[#Xiaoqin Guo | Quantitative homogenization in a balanced random environment ]]
 +
| Kim and Tran
 +
|-
 +
|March 4 '''time:4PM-5PM, Room: VV B239'''
 +
| Vladimir Sverak (Minnesota)
 +
|[[#Vladimir Sverak | Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" ]]
 +
| Kim
 +
|-  
 +
|March 11
 +
| Jonathan Luk (Stanford)
 +
|[[#Jonathan Luk | Stability of vacuum for the Landau equation with moderately soft potentials  ]]
 
| Kim
 
| Kim
|-  
+
|-
|October 16
+
|March 12, '''4:00 p.m. in VV B139'''
| Jingrui Cheng (UW)
+
| Trevor Leslie (UW-Madison)
|[[#Jingrui Cheng | A 1-D semigeostrophic model with moist convection ]]
+
|[[# Trevor Leslie| TBA ]]
| Kim & Tran
+
| Analysis seminar
|-  
+
|-
|October 23
+
|March 18,
| Donghyun Lee (UW)
+
| Spring recess (Mar 16-24, 2019)
|[[#Donghyun Lee The Vlasov-Poisson-Boltzmann system in bounded domains ]]
+
|[[# |  ]]
| Kim & Tran
+
|   
|-  
+
|-
|October 30
+
|March 25
| Myoungjean Bae (POSTECH)
+
| Jiaxin Jin
|[[#Myoungjean Bae 3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system ]]
+
|[[# Jiaxin Jin  |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.  ]]
| Feldman
+
| local speaker
|-  
+
|-  
|November 6
+
|April 1
| Jingchen Hu (USTC and UW)
+
| Zaher Hani (Michigan)
|[[#Jingchen Hu | Shock Reflection and Diffraction Problem with Potential Flow Equation ]]
+
|[[#Zaher Hani | TBA ]]
| Kim & Tran
+
| Ifrim
|-  
+
|-  
|November 27
+
|April 8 
| Ru-Yu Lai (Minnesota)
+
| Jingrui Cheng (Stony Brook)  
|[[#Ru-Yu Lai |  TBD ]]
+
|[[#Jingrui Cheng | Gradient estimate for complex Monge-Ampere equations ]]
| Li
+
| Feldman
|-  
+
|-
|December 4
+
|April 15,
| Norbert Pozar (Kanazawa University)
+
| Yao Yao (Gatech)
|[[#Norbert Pozar TBD ]]
+
|[[#Yao Yao | Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations ]]
 +
| Tran
 +
|-  
 +
|April 22,
 +
| Jessica Lin (McGill University)
 +
|[[#Jessica Lin Speeds and Homogenization for Reaction-Diffusion Equations in Random Media ]]
 
| Tran
 
| Tran
 +
|- 
 +
|April 29,
 +
| Beomjun Choi (Columbia)
 +
|[[#Beomjun Choi  | Evolution of non-compact hypersurfaces by inverse mean curvature]]
 +
|  Angenent
 
|}
 
|}
  
==Abstracts==
+
== Abstracts ==
 +
 
 +
===Julian Lopez-Gomez===
 +
 
 +
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.)
 +
 
 +
 
 +
 
 +
===Changyou Wang===
 +
 
 +
Title: Some recent results on mathematical analysis of Ericksen-Leslie System
 +
 
 +
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.
  
===Mihaela Ifrim===
+
===Matthew Schrecker===
  
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation
+
Title: Finite energy methods for the 1D isentropic Euler equations
  
Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.
+
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.
  
===Longjie Zhang===
+
===Anna Mazzucato===
  
On curvature flow with driving force starting as singular initial curve in the plane
+
Title: On the vanishing viscosity limit in incompressible flows
  
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.
+
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.
  
===Jaeyoung Byeon===
+
===Lei Wu===
  
Title: Patterns formation for elliptic systems with large interaction forces
+
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects
  
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.   The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.
+
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.
  
  
===Tuoc Phan===
+
===Annalaura Stingo===
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.
 
  
Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our
+
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D
mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.
 
  
===Hiroyoshi Mitake===
+
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 »  .
Derivation of multi-layered interface system and its application
+
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===
 +
 
 +
Title: Low Regularity Solutions for Gravity Water Waves
 +
 
 +
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===
  
Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of  the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.
+
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.
  
===Dongnam Ko===
 
On the emergence of local flocking phenomena in Cucker-Smale ensembles
 
  
Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.
+
===Jiaxin Jin===
  
===Sameer Iyer===
+
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.
 
  
Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.
+
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===
 
===Jingrui Cheng===
  
A 1-D semigeostrophic model with moist convection.
+
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===
  
We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.
+
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations
  
===Donghyun Lee===
+
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.
  
We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.
+
===Jessica Lin===
  
===Myoungjean Bae===
+
Title: Speeds and Homogenization for Reaction-Diffusion Equations in Random Media
  
3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system.
+
Abstract:
 +
The study of spreadings speeds, front speeds, and homogenization for reaction-diffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, most existing arguments rely on the construction of special solutions or linearization techniques. In this talk, I will present some new approaches for their analysis which do not utilize either of these. This talk is based on joint work with Andrej Zlatos.
  
I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) component.  This talk is based on a joint work with S. Weng (Wuhan University, China).
 
  
===Jingchen Hu===
 
  
Shock Reflection and Diffraction Problem with Potential Flow Equation
+
===Beomjun Choi===
 +
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. 
  
In this talk, we will present our work on nonsymmetric shock reflection and diffraction problem, the equation concerned is potential flow equation, which is a simplification of Euler System, mainly based on the assumption that flow has no vortex. We showed in both nonsymmetric reflection case and diffraction case, that physically admissible solution does not exist. This implies that the formation of vortex is essential to maintain the structural stability of shock reflection and diffraction.
+
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.

Latest revision as of 16:54, 10 April 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 2019-Spring 2020

PDE GA Seminar Schedule Fall 2018-Spring 2019

date speaker title host(s)
August 31 (FRIDAY), Julian Lopez-Gomez (Complutense University of Madrid) The theorem of characterization of the Strong Maximum Principle Rabinowitz
September 10, Hiroyoshi Mitake (University of Tokyo) On approximation of time-fractional fully nonlinear equations Tran
September 12 and September 14, Gunther Uhlmann (UWash) TBA Li
September 17, Changyou Wang (Purdue) Some recent results on mathematical analysis of Ericksen-Leslie System Tran
Sep 28, Colloquium Gautam Iyer (CMU) Stirring and Mixing Thiffeault
October 1, Matthew Schrecker (UW) Finite energy methods for the 1D isentropic Euler equations Kim and Tran
October 8, Anna Mazzucato (PSU) On the vanishing viscosity limit in incompressible flows Li and Kim
October 15, Lei Wu (Lehigh) Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects Kim
October 22, Annalaura Stingo (UCD) Global existence of small solutions to a model wave-Klein-Gordon system in 2D Mihaela Ifrim
October 29, Yeon-Eung Kim (UW) Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties Kim and Tran
November 5, Albert Ai (UC Berkeley) Low Regularity Solutions for Gravity Water Waves Mihaela Ifrim
Nov 7 (Wednesday), Colloquium Luca Spolaor (MIT) (Log)-Epiperimetric Inequality and the Regularity of Variational Problems Feldman
December 3, Time: 3:00, Room: B223 Van Vleck Trevor Leslie (UW) Flocking Models with Singular Interaction Kernels Kim and Tran
December 10, Time: 2:25, Room: B223 Van Vleck Serena Federico (MIT) Sufficient conditions for local solvability of some degenerate partial differential operators Mihaela Ifrim
December 10, Colloquium, Time: 4:00 Max Engelstein (MIT) The role of Energy in Regularity Feldman
January 28, Ru-Yu Lai (Minnesota) Inverse transport theory and related applications Li and 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) The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation Kim
February 13 4PM, Dean Baskin (Texas A&M) Radiation fields for wave equations Colloquium
February 18, 3:30PM, Room: VV B239 Daniel Tataru (Berkeley) A Morawetz inequality for water waves Ifrim
February 19, Time: 4-5PM, Room: VV B139 Wenjia Jing (Tsinghua University) Periodic homogenization of Dirichlet problems in perforated domains: a unified proof Tran
February 25, Xiaoqin Guo (UW) Quantitative homogenization in a balanced random environment Kim and Tran
March 4 time:4PM-5PM, Room: VV B239 Vladimir Sverak (Minnesota) Wasow lecture "PDE aspects of the Navier-Stokes equations and simpler models" Kim
March 11 Jonathan Luk (Stanford) Stability of vacuum for the Landau equation with moderately soft potentials Kim
March 12, 4:00 p.m. in VV B139 Trevor Leslie (UW-Madison) TBA Analysis seminar
March 18, Spring recess (Mar 16-24, 2019)
March 25 Jiaxin Jin Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria. local speaker
April 1 Zaher Hani (Michigan) TBA Ifrim
April 8 Jingrui Cheng (Stony Brook) Gradient estimate for complex Monge-Ampere equations Feldman
April 15, Yao Yao (Gatech) Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations Tran
April 22, Jessica Lin (McGill University) Speeds and Homogenization for Reaction-Diffusion Equations in Random Media Tran
April 29, Beomjun Choi (Columbia) Evolution of non-compact hypersurfaces by inverse mean curvature Angenent

Abstracts

Julian Lopez-Gomez

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.)


Changyou Wang

Title: Some recent results on mathematical analysis of Ericksen-Leslie System

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.

Matthew Schrecker

Title: Finite energy methods for the 1D isentropic Euler equations

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.

Anna Mazzucato

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.

Lei Wu

Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects

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.


Annalaura Stingo

Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D

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 » . 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

Title: Low Regularity Solutions for Gravity Water Waves

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.

Jessica Lin

Title: Speeds and Homogenization for Reaction-Diffusion Equations in Random Media

Abstract: The study of spreadings speeds, front speeds, and homogenization for reaction-diffusion equations in random heterogeneous media is of interest for many applications to mathematical modelling. However, most existing arguments rely on the construction of special solutions or linearization techniques. In this talk, I will present some new approaches for their analysis which do not utilize either of these. This talk is based on joint work with Andrej Zlatos.


Beomjun Choi

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.

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.