PDE Geometric Analysis seminar: Difference between revisions

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===[[Previous PDE/GA seminars]]===
===[[Previous PDE/GA seminars]]===
===[[Fall 2015 | Tentative schedule for Fall 2015]]===
===[[Spring 2018 | Tentative schedule for Spring 2018]]===


= Seminar Schedule Spring 2015 =
== PDE GA Seminar Schedule Fall 2017 ==
{| cellpadding="8"
{| cellpadding="8"
!align="left" | date   
!style="width:20%" align="left" | date   
!align="left" | speaker
!align="left" | speaker
!align="left" | title
!align="left" | title
!align="left" | host(s)
!style="width:20%" align="left" | host(s)
|-
|-  
|January 21 (Departmental Colloquium: 4PM, B239)
|September 11
|Jun Kitagawa (Toronto)
|Mihaela Ifrim (UW)
|[[#Jun Kitagawa (Toronto)  | Regularity theory for generated Jacobian equations: from optimal transport to geometric optics  ]]
|[[#Mihaela Ifrim| Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]
|Feldman
| Kim & Tran
|-
|-  
|February 9
|September 18
|Jessica Lin (Madison)
|Longjie Zhang (University of Tokyo)  
|[[#Jessica Lin (Madison)  | Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations ]]
|[[#Longjie Zhang | On curvature flow with driving force starting as singular initial curve in the plane]]
|Kim
| Angenent
|-
|-  
|February 17 (Tuesday) (joint with Analysis Seminar: 4PM, B139)
|September 22,
|Chanwoo Kim (Madison)  
VV 9th floor hall, 4:00pm
|[[#Chanwoo Kim (Madison) | Hydrodynamic limit from the Boltzmann to the Navier-Stokes-Fourier ]]
|Jaeyoung Byeon (KAIST)  
|Seeger
|[[#Jaeyoung Byeon| Colloquium: Patterns formation for elliptic systems with large interaction forces]]
|-
| Rabinowitz
|February 23 (special time*, '''3PM, B119''')
|-  
| Yaguang Wang (Shanghai Jiao Tong)
|September 25
|[[ #Yaguang Wang | Stability of Three-dimensional Prandtl Boundary Layers ]]
| Tuoc Phan (UTK)
|Jin
|[[#Tuoc Phan | Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]
|-
| Tran
|March 2
|-  
|Benoit Pausader (Princeton)
|September 26,
|[[#Benoit Pausader (Princeton) | Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions]]
VV B139 4:00pm
|Kim
| Hiroyoshi Mitake (Hiroshima University)
|-
|[[#Hiroyoshi Mitake | Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]
|March 9
| Tran
|Haozhao Li (University of Science and Technology of China)  
|-  
|[[#Haozhao Li|Regularity scales and convergence of the Calabi flow]]
|September 29,
|Wang
VV901 2:25pm
|-
| Dongnam Ko (CMU & SNU)
|March 16
|[[#Dongnam Ko a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]
| Jennifer Beichman (Madison)
| Shi Jin & Kim
|[[#Jennifer Beichman (Madison)  |  ]]
|-  
| Kim
|October 2
|-
| No seminar due to a KI-Net conference
|March 23
|
| Ben Fehrman (University of Chicago)
|
|[[#Ben Fehrman (University of Chicago  | TBA ]]
|-  
| Lin
|October 9
|-
| Sameer Iyer (Brown University)
|March 30
|[[#Sameer Iyer Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
| Spring recess Mar 28-Apr 5 (S-N)
| Kim
|[[# |  ]]
|-  
|  
|October 16
|-
| Jingrui Cheng (UW)
|April 6
|[[#Jingrui Cheng A 1-D semigeostrophic model with moist convection ]]
| Vera Hur (UIUC)
| Kim & Tran
|[[# |  ]]
|-  
| Yao
|October 23
|-
| Donghyun Lee (UW)
|April 13
|[[#Donghyun Lee The Vlasov-Poisson-Boltzmann system in bounded domains ]]
|  
| Kim & Tran
|[[# |  ]]
|-  
|  
|October 30
|-
| Myoungjean Bae (POSTECH)
|April 20
|[[#Myoungjean Bae | TBD ]]
|Yuan Lou (Ohio State)
| Feldman
|[[#Yuan Lou (Ohio State) | TBA]]
|-  
|Zlatos
|November 6
|-
| Jingchen Hu (USTC and UW)
|April 27
|[[#Jingchen Hu TBD ]]
|  
| Kim & Tran
|[[# |  ]]
|-  
|  
|December 4
|-
| Norbert Pozar (Kanazawa University)
|May 4  
|[[#Norbert Pozar TBD ]]
|  
| Tran
|[[# |  ]]
|  
|-
|}
|}


==Abstracts==


== Abstracts ==
===Mihaela Ifrim===


===Jun Kitagawa (Toronto)===
Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation


Regularity theory for generated Jacobian equations: from optimal transport to geometric optics
Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.


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.
===Longjie Zhang===


===Jessica Lin (Madison)===
On curvature flow with driving force starting as singular initial curve in the plane


Algebraic Error Estimates for the Stochastic Homogenization of Uniformly Parabolic Equations
We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.


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.
===Jaeyoung Byeon===


Title: Patterns formation for elliptic systems with large interaction forces


===Yaguang Wang (Shanghai Jiao Tong)===
Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions.  The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.


Stability of Three-dimensional Prandtl Boundary Layers


In this talk, we shall study the stability of the Prandtl boundary layer
===Tuoc Phan===
equations in three space variables. First, we obtain a well-posedness
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.
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.


Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our
mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.


===Benoit Pausader (Princeton)===
===Hiroyoshi Mitake===
Derivation of multi-layered interface system and its application


Global smooth solutions for the Euler-Maxwell problem for electrons in 2 dimensions
Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of  the Allen-Cahn equation.  By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.


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.


===Dongnam Ko===
On the emergence of local flocking phenomena in Cucker-Smale ensembles


===Haozhao Li (University of Science and Technology of China)===
Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.


Regularity scales and convergence of the Calabi flow
===Sameer Iyer===
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.


We define regularity scales to study the behavior of the Calabi flow.  
Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.
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
===Jingrui Cheng===
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
A 1-D semigeostrophic model with moist convection.
assumption that the scalar curvature is uniformly bounded.
 
We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.
 
===Donghyun Lee===
 
We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.

Revision as of 12:37, 14 October 2017

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 Spring 2018

PDE GA Seminar Schedule Fall 2017

date speaker title host(s)
September 11 Mihaela Ifrim (UW) Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation Kim & Tran
September 18 Longjie Zhang (University of Tokyo) On curvature flow with driving force starting as singular initial curve in the plane Angenent
September 22,

VV 9th floor hall, 4:00pm

Jaeyoung Byeon (KAIST) Colloquium: Patterns formation for elliptic systems with large interaction forces Rabinowitz
September 25 Tuoc Phan (UTK) Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application Tran
September 26,

VV B139 4:00pm

Hiroyoshi Mitake (Hiroshima University) Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application Tran
September 29,

VV901 2:25pm

Dongnam Ko (CMU & SNU) a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles Shi Jin & Kim
October 2 No seminar due to a KI-Net conference
October 9 Sameer Iyer (Brown University) Global-in-x Steady Prandtl Expansion over a Moving Boundary Kim
October 16 Jingrui Cheng (UW) A 1-D semigeostrophic model with moist convection Kim & Tran
October 23 Donghyun Lee (UW) The Vlasov-Poisson-Boltzmann system in bounded domains Kim & Tran
October 30 Myoungjean Bae (POSTECH) TBD Feldman
November 6 Jingchen Hu (USTC and UW) TBD Kim & Tran
December 4 Norbert Pozar (Kanazawa University) TBD Tran

Abstracts

Mihaela Ifrim

Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation

Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.

Longjie Zhang

On curvature flow with driving force starting as singular initial curve in the plane

We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.

Jaeyoung Byeon

Title: Patterns formation for elliptic systems with large interaction forces

Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions. The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.


Tuoc Phan

Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.

Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.

Hiroyoshi Mitake

Derivation of multi-layered interface system and its application

Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.


Dongnam Ko

On the emergence of local flocking phenomena in Cucker-Smale ensembles

Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.

Sameer Iyer

Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.

Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.

Jingrui Cheng

A 1-D semigeostrophic model with moist convection.

We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.

Donghyun Lee

We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.