Difference between revisions of "PDE Geometric Analysis seminar"

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(PDE GA Seminar Schedule Spring 2018)
(PDE GA Seminar Schedule Fall 2018-Spring 2019)
 
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===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2018 | Tentative schedule for Fall 2018]]===
+
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===
  
 
+
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==
 
+
== PDE GA Seminar Schedule Spring 2018 ==
+
  
  
Line 16: Line 14:
  
 
|-   
 
|-   
|January 29, '''3-3:50PM, B341 VV.'''
+
|August 31 (FRIDAY),
| Dan Knopf (UT Austin)
+
| Julian Lopez-Gomez (Complutense University of Madrid)
|[[#Dan Knopf | Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons]]
+
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]
| Angenent
+
| Rabinowitz
|-  
+
 
|February 5, '''3-3:50PM, B341 VV.'''
+
|-   
| Andreas Seeger (UW)
+
|September 10,
|[[#Andreas Seeger Singular integrals and  a problem on mixing flows ]]
+
| Hiroyoshi Mitake (University of Tokyo)
| Kim & Tran
+
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]
|-  
+
| Tran
|February 12
+
|-
| Sam Krupa (UT-Austin)
+
|September 12 and September 14,
|[[#Sam Krupa Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case ]]
+
| Gunther Uhlmann (UWash)
| Lee
+
|[[#Gunther Uhlmann | TBA ]]
|-  
+
| Li
|February 19
+
|-   
| Maja Taskovic (UPenn)
+
|September 17,
|[[#Maja Taskovic TBD ]]
+
| Changyou Wang (Purdue)
 +
|[[#Changyou Wang Some recent results on mathematical analysis of Ericksen-Leslie System ]]
 +
| Tran
 +
|-
 +
|Sep 28, Colloquium
 +
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
 +
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
 +
| Thiffeault
 +
|-  
 +
|October 1,
 +
| 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
 
| 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
 
|-  
 
|-  
|March 5
+
|December 10, Colloquium, '''Time: 4:00'''
| Khai Nguyen (NCSU)
+
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)
|[[#Khai Nguyen | TBD ]]
+
|[[# Max Engelstein| The role of Energy in Regularity ]]
| Tran
+
| Feldman
 
|-  
 
|-  
|March 12
+
|January 28,
| Hongwei Gao (UCLA)
+
| Ru-Yu Lai (Minnesota)
|[[#Hongwei Gao |  TBD ]]
+
|[[# Ru-Yu Lai | TBA ]]
 +
| Li and Kim
 +
|-
 +
| February 4,
 +
| Seokbae Yun (SKKU, long term visitor of UW-Madison)
 +
|[[# Seokbae Yun | TBA ]]
 +
| Kim
 +
|-  
 +
| February 18, 
 +
| Daniel Tataru (Berkeley)
 +
|[[# Daniel Tataru | TBA ]]
 +
| Ifrim
 +
|-                                                                                                                                                         
 +
|Time: TBD in February,
 +
| Xiaoqin Guo (UW)
 +
|[[#Xiaoqin Guo | TBA ]]
 +
| Kim and Tran
 +
|-
 +
|Time: TBD in February,
 +
| Wenjia Jing (Tsinghua University)
 +
|[[#Wenjia Jing | TBA ]]
 
| Tran
 
| Tran
 
|-  
 
|-  
|March 19
+
|March 4
| Huy Nguyen (Princeton)
+
| Vladimir Sverak (Minnesota)
|[[#Huy Nguyen TBD ]]
+
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]
| Lee
+
| Kim
 +
|-   
 +
|March 11
 +
| Jonathan Luk (Stanford)
 +
|[[#Jonathan Luk | TBA ]]
 +
| Kim
 
|-
 
|-
|March 24 (Mar 24-Apr 1, 2018)
+
|March 18,
| Spring recess
+
| Spring recess (Mar 16-24, 2019)
|[[#  |  TBD ]]
+
|[[#  |  ]]
 
|   
 
|   
 +
|-   
 +
|April 1
 +
| Zaher Hani (Michigan)
 +
|[[#Zaher Hani | TBA  ]]
 +
| Ifrim
 
|-
 
|-
|April 9
+
|April 15,
| reserved
+
| Yao Yao (Gatech)
|[[# | TBD ]]
+
|[[#Yao Yao | TBA ]]
 
| Tran
 
| Tran
|-  
+
|-  
|April 21-22 (Saturday-Sunday)
+
|April 22,
| Midwest PDE seminar
+
| Jessica Lin (McGill University)
|[[#Midwest PDE seminar | ]]
+
|[[#Jessica Lin | TBA ]]
| Angenent, Feldman, Kim, Tran.
+
| Tran
|-  
+
|-  
|April 25 (Wednesday)
+
|April 29,
| Hitoshi Ishii (Wasow lecture)
+
|   ( )
|[[#Hitoshi Ishii | TBD]]
+
|[[# | TBA ]]
| Tran.
+
|
 
|}
 
|}
  
 
== Abstracts ==
 
== Abstracts ==
  
===Dan Knopf===
+
===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.
  
Title: Non-Kähler Ricci flow singularities that converge to Kähler-Ricci solitons
+
===Serena Federico===
  
Abstract: We describe Riemannian (non-Kähler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kähler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered by Feldman, Ilmanen, and Knopf in 2003. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kähler solutions of Ricci flow that become asymptotically Kähler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kähler metrics under Ricci flow.
+
Title: Sufficient conditions for local solvability of some degenerate partial differential operators
  
===Andreas Seeger===
+
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.
  
Title: Singular integrals and a problem on mixing flows
+
===Max Engelstein===
  
Abstract: The talk will be about  results related to Bressan's mixing problem. We present  an inequality for the change of a  Bianchini semi-norm of characteristic functions under the  flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator  for which one proves bounds  on Hardy spaces. This is joint work with Mahir Hadžić,  Charles Smart and    Brian Street.
+
Title: The role of Energy in Regularity
  
===Sam Krupa===
+
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.
  
Title: Proving Uniqueness of Solutions for Burgers Equation Entropic for a Single Entropy, with Eye Towards Systems Case
+
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.
  
Abstract: For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (Panov). This single entropy result was proven again by De Lellis, Otto and Westdickenberg in 2004. These two proofs both rely on the special connection between Hamilton--Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In our new work, we prove the single entropy result for scalar conservation laws without using Hamilton--Jacobi.  Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. This is joint work with A. Vasseur.
+
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.

Latest revision as of 09:05, 10 December 2018

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) TBA Li and Kim
February 4, Seokbae Yun (SKKU, long term visitor of UW-Madison) TBA Kim
February 18, Daniel Tataru (Berkeley) TBA Ifrim
Time: TBD in February, Xiaoqin Guo (UW) TBA Kim and Tran
Time: TBD in February, Wenjia Jing (Tsinghua University) TBA Tran
March 4 Vladimir Sverak (Minnesota) TBA(Wasow lecture) Kim
March 11 Jonathan Luk (Stanford) TBA Kim
March 18, Spring recess (Mar 16-24, 2019)
April 1 Zaher Hani (Michigan) TBA Ifrim
April 15, Yao Yao (Gatech) TBA Tran
April 22, Jessica Lin (McGill University) TBA Tran
April 29, ( ) TBA

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.