PDE Geometric Analysis seminar: Difference between revisions

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===[[Fall 2015 | Tentative schedule for Fall 2015]]===
===[[Fall 2015 | Tentative schedule for Fall 2015]]===


= Seminar Schedule Spring 2015 =
= Seminar Schedule Fall 2015 =
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|-
|-
|January 21 (Departmental Colloquium: 4PM, B239)  
|September 7 (Labor Day)
|Jun Kitagawa (Toronto) 
|  
|[[#Jun Kitagawa (Toronto)  | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics  ]]
|[[# | ]]
|Feldman
|
|-
|-
|February 9
|September 14
|Jessica Lin (Madison)
|
|[[#Jessica Lin (Madison) | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]
|[[#  | ]]
|Kim
|  
|-
|-
|February 17 (Tuesday) (joint with Analysis Seminar: 4PM, B139)
|September 21
|Chanwoo Kim (Madison)  
|Luis Silvestre (Chicago)
|[[#Chanwoo Kim (Madison)  | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]
||[[# Luis Silvestre | TBA ]]
|Seeger
| Kim
|-
|-
|February 23 (special time*, '''3PM, B119''')
|September 28
| Yaguang Wang (Shanghai Jiao Tong)
|  
|[[ #Yaguang Wang | Stability of Three-dimensional Prandtl Boundary Layers ]]
|[[# | ]]
|Jin
|
|-
|-
|March 2 
|October 5
|Benoit Pausader (Princeton)
|Hyung-Ju Hwang (Postech)
|[[#Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions]]
|[[# Hyung-Ju Hwang | TBA  ]]
|Kim
| Kim
|-
|-
|March 9
|October 12
|Haozhao Li (University of Science and Technology of China)
|  
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]
|[[# | ]]
|Wang
|  
|-
|-
|March 16
|October 19
| Jennifer Beichman (Madison) 
|  
|[[#Jennifer Beichman (Madison) |Nonstandard dispersive estimates and linearized water waves ]]
||[[#  |  ]]
| Kim
|  
|-
|-
|March 23
|October 26
| Ben Fehrman (University of Chicago)
|  
|[[#Ben Fehrman (University of Chicago) | On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments ]]
|[[#  | ]]
| Lin
|
|-
|-
|March 30
|November 2
| Spring recess Mar 28-Apr 5 (S-N)
|  
|[[#  |  ]]
|[[#  |  ]]
|
|-
|November 9
| Yifeng Yu (UC Irvine)
|[[# Yifeng Yu | TBA ]]
| Tran
|-
|November 16
|
||[[#  |  ]]
|  
|  
|-
|-
|April 13
|November 23
| Sung-Jin Oh (Berkeley)
|
|[[# Berkeley | Global well-posedness of the energy critical Maxwell-Klein-Gordon equation ]]
|[[#  |  ]]
| Kim
|
|-
|November 30
|
|[[# | ]]
|
|-
|-
|April 20
|December 7
|Yuan Lou (Ohio State)
|  
|[[#Yuan Lou (Ohio State) | Asymptotic behavior of the smallest eigenvalue of an elliptic operator and its applications to evolution of dispersal]]
|[[# | ]]
|Zlatos
|
|-
|-
|'''April 28''' (a joint seminar with analysis, '''4:00 p.m B139''')
|December 14
| Diego Córdoba (ICMAT, Madrid)
| reserved
|[[# Diego Córdoba |Global existence solutions and geometric properties of the SQG sharp front ]]
|[[# |  ]]
| Zlatos
| Zlatos
|-
|May 4 
| Vera Hur (UIUC) 
|[[# Vera Hur (UIUC) |Instabilities in nonlinear dispersive waves  ]]
| Yao
|-
|}
== Abstracts ==
===Jun Kitagawa (Toronto)===
Regularity theory for generated Jacobian equations: from optimal transport to geometric optics
Equations of Monge-Ampere type arise in numerous contexts, and solutions often exhibit very subtle qualitative and quantitative properties; this is owing to the highly nonlinear nature of the equation, and its degeneracy (in the sense of ellipticity). Motivated by an example from geometric optics, I will talk about the class of Generated Jacobian Equations; recently introduced by Trudinger, this class also encompasses, for example, optimal transport, the Minkowski problem, and the classical Monge-Ampere equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field reflector problems (of interest from a physical and practical point of view). This talk is based on joint works with N. Guillen.
===Jessica Lin (Madison)===
Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations
We establish error estimates for the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. Based on the approach of Armstrong and Smart in the elliptic setting, we construct a quantity which captures the geometric behavior of solutions to parabolic equations. The error estimates are shown to be of algebraic order. This talk is based on joint work with Charles Smart.
===Yaguang Wang (Shanghai Jiao Tong)===
Stability of Three-dimensional Prandtl Boundary Layers
In this talk, we shall study the stability of the Prandtl boundary layer
equations in three space variables. First, we obtain a well-posedness
result of the three-dimensional Prandtl equations under some constraint on
its flow structure. It reveals that the classical Burgers equation plays an
important role in determining this type of flow with special structure,
that avoids the appearance of the complicated secondary flow in the
three-dimensional Prandtl boundary layers. Second, we give an instability
criterion for the Prandtl equations in three space variables. Both of
linear and nonlinear stability are considered. This criterion shows that
the monotonic shear flow is linearly stable for the three dimensional
Prandtl equations if and only if the tangential velocity field direction is
invariant with respect to the normal variable, which is an exact complement
to the above well-posedness result for a special flow. This is a joint work
with Chengjie Liu and Tong Yang.
===Benoit Pausader (Princeton)===
Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions
It is well known that pure compressible fluids tend to develop shocks, even from small perturbation. We study how self consistent electromagnetic fields can stabilize these fluids. In a joint work with A. Ionescu and Y. Deng, we consider a compressible fluid of electrons in 2D, subject to its own electromagnetic field and to a field created by a uniform background of positively charged ions. We show that small smooth and irrotational perturbations of a uniform background at rest lead to solutions that remain globally smooth, in contrast with neutral fluids. This amounts to proving small data global existence for a system of quasilinear Klein-Gordon equations with different speeds.
===Haozhao Li (University of Science and Technology of China)===
Regularity scales and convergence of the Calabi flow
We define regularity scales to study the behavior of the Calabi flow.
Based on estimates of the regularity scales, we obtain convergence theorems
of the Calabi flow on extremal K\"ahler surfaces, under the assumption of global existence
of the Calabi flow solutions. Our results partially confirm Donaldson’s conjectural picture for
the Calabi flow in complex dimension 2. Similar results hold in high dimension with an extra
assumption that the scalar curvature is uniformly bounded.
===Jennifer Beichman (UW-Madison)===
Nonstandard dispersive estimates and linearized water waves
In this talk, we focus on understanding the relationship between the decay of a solution to the linearized water wave problem and its initial data. We obtain decay bounds for a class of 1D dispersive equations that includes the linearized water wave. These decay bounds display a surprising growth factor, which we show is sharp. A further exploration leads to a result relating singularities of the initial data at the origin in Fourier frequency to the regularity of the solution.
===Ben Fehrman (University of Chicago)===
On The Existence of an Invariant Measure for Isotropic Diffusions in Random Environments
I will discuss the existence of a unique mutually absolutely continuous invariant measure for isotropic diffusions in random environment, of dimension at least three, which are small perturbations of Brownian motion satisfying a finite range dependence.  This framework was first considered in the continuous setting by Sznitman and Zeitouni and in the discrete setting by Bricmont and Kupiainen.  The results of this talk should be seen as an extension of their work.
I will furthermore mention applications of this analysis to the stochastic homogenization of the related elliptic and parabolic equations with random oscillatory boundary data and, explain how the existence of an invariant measure can be used to prove a Liouville property for the environment.  In the latter case, the methods were motivated by work in the discrete setting by Benjamini, Duminil-Copin, Kozma and Yadin.
===Vera Hur===
Instabilities in nonlinear dispersive waves
I will speak on the wave breaking and the modulational instability of nonlinear wave trains in dispersive media. I will begin by a gradient blowup proof for the Boussinesq-Whitham equations for water waves. I will then describe a variational approach to determine instability to long wavelength perturbations for a general class of Hamiltonian systems, allowing for nonlocal dispersion. I will discuss KdV type equations with fractional dispersion in depth. Lastly, I will explain an asymptotics approach for Whitham's equation for water waves, qualitatively reproducing the Benjamin-Feir instability of Stokes waves.
===Sung-Jin Oh===
Global well-posedness of the energy critical Maxwell-Klein-Gordon equation
The massless Maxwell-Klein-Gordon system describes the interaction between an electromagnetic field (Maxwell) and a charged massless scalar field (massless Klein-Gordon, or wave).
In this talk, I will present a recent proof, joint with D. Tataru, of global well-posedness and scattering of this system for arbitrary finite energy data in the (4+1)-dimensional Minkowski space, in which the PDE is energy critical.
===Yuan Lou===
Asymptotic behavior of the smallest eigenvalue of an elliptic operator and its applications to evolution of dispersal
We investigate the effects of diffusion and drift on the smallest eigenvalue of an elliptic operator with zero Neumann boundary condition. Various asymptotic behaviors of the smallest eigenvalue, as diffusion and drift rates approach zero or infinity, are derived. As an application, these qualitative results yield some insight into the evolution of dispersal in heterogeneous environments.
===Diego Cordoba===
Global existence solutions and geometric properties of the SQG sharp front
A particular kind of weak solutions for a 2D active scalar are the so called sharp fronts, i.e., solutions for which the scalar
is a step function. The evolution of such distribution is completely determined by the evolution of the boundary, allowing the problem
to be treated as a non-local one dimensional equation for the contour. In this setting we will present several analytical results for the surface
quasi-geostrophic equation (SQG): the existence of convex $C^{\infinity}$ global rotating solutions, elliptical shapes are not rotating solutions (as opposed to 2D Euler equations) and the existence of convex solutions that lose their convexity in finite time.

Revision as of 22:36, 4 May 2015

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 2015

Seminar Schedule Fall 2015

date speaker title host(s)
date speaker title host(s)
September 7 (Labor Day)
September 14
September 21 Luis Silvestre (Chicago) TBA Kim
September 28
October 5 Hyung-Ju Hwang (Postech) TBA Kim
October 12
October 19
October 26
November 2
November 9 Yifeng Yu (UC Irvine) TBA Tran
November 16
November 23
November 30
December 7
December 14 reserved Zlatos
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