Difference between revisions of "PDE Geometric Analysis seminar"

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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2017 | Tentative schedule for Fall 2017]]===
+
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==
  
= PDE GA Seminar Schedule Spring 2017 =
 
  
 
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{| cellpadding="8"
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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),
 +
| Julian Lopez-Gomez (Complutense University of Madrid)
 +
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]
 +
| Rabinowitz
 +
 +
|- 
 +
|September 10,
 +
| Hiroyoshi Mitake (University of Tokyo)
 +
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]
 +
| 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
 
|-
 
|-
|January 23<br>Special time and location:<br> 3-3:50pm, B325 Van Vleck
+
|Sep 28, Colloquium
| Sigurd Angenent (UW)
+
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
|[[#Sigurd Angenent | Ancient convex solutions to Mean Curvature Flow]]
+
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
| Kim & Tran
+
| 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
 +
|-
 +
|December 10, Colloquium, '''Time: 4:00'''
 +
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)
 +
|[[#Max Engelstein| The role of Energy in Regularity ]]
 +
| Feldman
 
|-  
 
|-  
 
+
|January 28,
 +
| 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
 
|-
 
|-
|January 30
+
| February 13 '''4PM''',
| Serguei Denissov (UW)
+
| Dean Baskin (Texas A&M)
|[[#Serguei Denissov | Instability in 2D Euler equation of incompressible inviscid fluid]]
+
|[[#Dean Baskin | Radiation fields for wave equations]]
| Kim & Tran
+
| 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
 
|-  
 
|-  
 +
|February 25,
 +
| Xiaoqin Guo (UW)
 +
|[[#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
 +
|-
 +
|March 12, '''4:00 p.m. in VV B139'''
 +
| Trevor Leslie (UW-Madison)
 +
|[[# Trevor Leslie| TBA ]]
 +
| Analysis seminar
 +
|-
 +
|March 18,
 +
| Spring recess (Mar 16-24, 2019)
 +
|[[#  |  ]]
 +
 +
|-
 +
|March 25
 +
| Jiaxin Jin
 +
|[[# Jiaxin Jin  |Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.  ]]
 +
| 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
 +
|-   
 +
|April 22,
 +
| Jessica Lin (McGill University)
 +
|[[#Jessica Lin |  Speeds and Homogenization for Reaction-Diffusion Equations in Random Media ]]
 +
| Tran
 +
|- 
 +
|April 29,
 +
| Beomjun Choi (Columbia)
 +
|[[#Beomjun Choi  | 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.
  
|-
+
 
|February 6 - Wasow lecture
+
===Annalaura Stingo===
| Benoit Perthame (University of Paris VI)
+
 
|[[#| ]]
+
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D
| Jin
+
 
|-  
+
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===
|February 13
+
Title: Inverse transport theory and related applications.
| Bing Wang (UW)
 
|[[#Bing Wang | The extension problem of the mean curvature flow]]
 
| Kim & Tran
 
|-
 
  
|-
+
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.
|February 20
 
| Eric Baer (UW)
 
|[[#Eric Baer | Isoperimetric sets inside almost-convex cones]]
 
| Kim & Tran
 
|-
 
  
|-
+
===Seokbae Yun===
|February 27
+
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation
| Ben Seeger (University of Chicago)
 
|[[#Ben Seeger | Homogenization of pathwise Hamilton-Jacobi equations ]]
 
| Tran
 
|-
 
  
|-
+
Abstract: In this talk, we consider the propagation of the uniform upper bounds
|March 7 - Mathematics Department Distinguished Lecture
+
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two
| Roger Temam (Indiana University) 
+
types of estimates for the the gain part of the collision operator: namely, a potential
|[[#Roger Temam | On the mathematical modeling of the humid atmosphere]]
+
type estimate and a relativistic hyper-surface integral estimate. We then combine them
| Smith 
+
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.
  
  
|-
 
|March 8 - Analysis/Applied math/PDE seminar
 
| Roger Temam (Indiana University) 
 
|[[#Roger Temam | Weak solutions of the Shigesada-Kawasaki-Teramoto system ]]
 
| Smith
 
|-
 
  
|-
+
===Daniel Tataru===
|March 13
 
| Sona Akopian (UT-Austin)
 
|[[#Sona Akopian | Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.]]
 
| Kim
 
  
|-
+
Title: A Morawetz inequality for water waves.
|March 27 - Analysis/PDE seminar
 
| Sylvia Serfaty (Courant)
 
|[[#Sylvia Serfaty | Mean-Field Limits for Ginzburg-Landau vortices ]]
 
| Tran
 
  
|-
+
Authors: Thomas Alazard, Mihaela Ifrim, Daniel Tataru.
|March 29 - Wasow lecture
 
| Sylvia Serfaty (Courant)
 
|[[#Sylvia Serfaty | Microscopic description of Coulomb-type systems ]]
 
|
 
  
 +
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.
  
|-
 
|March 30 <br>Special day (Thursday) and location:<br>  B139 Van Vleck
 
| Gui-Qiang Chen (Oxford)
 
|[[#Gui-Qiang Chen  | Supersonic Flow onto Solid Wedges,
 
Multidimensional Shock Waves and Free Boundary Problems ]]
 
| Feldman
 
  
 +
===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.
|April 3
 
| Zhenfu Wang (Maryland)
 
|[[#Zhenfu Wang | Mean field limit for stochastic particle systems with singular forces]]
 
| Kim
 
  
|-
 
|April 10
 
| Andrei Tarfulea (Chicago)
 
|[[#Andrei Tarfulea | Improved estimates for thermal fluid equations]]
 
| Baer
 
  
|-
+
===Xiaoqin Guo===
|April 17
 
| Siao-Hao Guo (Rutgers)
 
|[[# Siao-Hao Guo | Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]]
 
| Lu Wang
 
  
 +
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).
|April 24
 
| Jianfeng Lu
 
|[[#Jianfeng Lu | TBA]]
 
| Li
 
  
|-
+
===Sverak===
|April 25- joint Analysis/PDE seminar
 
| Chris Henderson (Chicago)
 
|[[#Chris Henderson | TBA]]
 
| Lin
 
  
|-
+
Title: PDE aspects of the Navier-Stokes equations and simpler models
|May 1st
 
| Jeffrey Streets (UC-Irvine)
 
|[[#Jeffrey Streets | Generalized Kahler Ricci flow and a generalized Calabi conjecture]]
 
| Bing Wang
 
|}
 
  
=Abstracts=
+
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.
  
===Sigurd Angenent===
+
===Jonathan Luk===
The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so.  Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such &ldquo;Ancient Solutions.&rdquo;  In doing so one finds that there is interesting dynamics associated to ancient solutions.  I will discuss what is currently known about these solutions.  Some of the talk is based on joint work with Sesum and Daskalopoulos.
 
  
===Serguei Denissov===
+
Title: Stability of vacuum for the Landau equation with moderately soft potentials
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.
 
  
 +
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.
  
===Bing Wang===
 
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.
 
  
===Eric Baer===
+
===Jiaxin Jin===
We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones.  Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere.  The work we describe is joint with A. Figalli.
 
  
===Ben Seeger===
+
Title: Convergence to the complex balanced equilibrium for some reaction-diffusion systems with boundary equilibria.
I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.
 
  
===Sona Akopian===
+
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.
Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.
 
  
We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation.
+
===Jingrui Cheng===
  
===Sylvia Serfaty===
+
Title: Gradient estimate for complex Monge-Ampere equations
Mean-Field Limits for Ginzburg-Landau vortices
 
  
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.
+
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.
  
  
===Gui-Qiang Chen===
+
===Yao Yao===
Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems
 
  
When an upstream steady uniform supersonic flow, governed by the Euler equations,
+
Title: Radial symmetry of stationary and uniformly-rotating solutions in 2D incompressible fluid equations
impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock
 
configurations if the wedge angle is less than the detachment angle -- the steady weak shock
 
with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger
 
than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which
 
satisfy the entropy conditions.
 
The fundamental issue -- whether one or both of the steady weak and strong shocks are physically
 
admissible solutions -- has been vigorously debated over the past eight decades.
 
In this talk, we discuss some of the most recent developments on the stability analysis
 
of the steady shock solutions in both the steady and dynamic regimes.
 
The corresponding stability problems can be formulated as free boundary problems
 
for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose
 
solutions are fundamental for multidimensional hyperbolic conservation laws.
 
Some further developments, open problems, and mathematical challenges in this direction
 
are also addressed.
 
  
===Zhenfu Wang===
+
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.
  
Title: Mean field limit for stochastic particle systems with singular forces
+
===Jessica Lin===
  
Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution
+
Title: Speeds and Homogenization for Reaction-Diffusion Equations in Random Media
and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos
 
through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.
 
  
===Andrei Tarfulea===
+
Abstract:
We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and
+
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.
dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.
 
  
===Siao-hao Guo===
 
Analysis of Velázquez's solution to the mean curvature flow with a type II singularity
 
  
Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.
 
  
===Jeffrey Streets===
+
===Beomjun Choi===
Generalized Kahler Ricci flow and a generalized Calabi conjecture
+
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. 
  
Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.'  In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.
+
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.