PDE Geometric Analysis seminar: Difference between revisions

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===[[Previous PDE/GA seminars]]===
===[[Previous PDE/GA seminars]]===
===[[Fall 2016 | Tentative schedule for Fall 2016]]===
===[[Spring 2018 | Tentative schedule for Spring 2018]]===


= Seminar Schedule Spring 2016 =
== 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 25
|September 11
||Tianling Jin (HKUST and Caltech)
|Mihaela Ifrim (UW)
|[[#Tianling Jin | Holder gradient estimates for parabolic homogeneous p-Laplacian equations  ]]
|[[#Mihaela Ifrim| Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation]]
| Zlatos
| Kim & Tran
|-
|-  
|February 1
|September 18
|Russell Schwab (Michigan State University)
|Longjie Zhang (University of Tokyo)  
|[[#Russell Schwab | Neumann homogenization via integro-differential methods ]]
|[[#Longjie Zhang On curvature flow with driving force starting as singular initial curve in the plane]]
| Lin
Angenent
|-
|-  
|February 8
|September 22,
|Jingrui Cheng (UW Madison)
VV 9th floor hall, 4:00pm
|[[#Jingrui Cheng | Semi-geostrophic system with variable Coriolis parameter ]]
|Jaeyoung Byeon (KAIST)  
| Tran & Kim
|[[#Jaeyoung Byeon| Colloquium: Patterns formation for elliptic systems with large interaction forces]]
|-
| Rabinowitz
|February 15
|-  
| Paul Rabinowitz (UW Madison)
|September 25
|[[# Paul Rabinowitz | On A Double Well Potential System ]]
| Tuoc Phan (UTK)
| Tran & Kim
|[[#Tuoc Phan Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application]]
|-
|February 22  
| Hong Zhang (Brown)
|[[# Hong Zhang | On an elliptic equation arising from composite material ]]
| Kim
|-
|February 29
|Aaron Yip (Purdue university)  
|[[# Aaron Yip TBD ]]
| Tran
| Tran
|-
|-  
|March 7
|September 26,
|Hiroyoshi Mitake (Hiroshima university)  
VV B139 4:00pm
||[[#Hiroyoshi Mitake | Selection problem for fully nonlinear equations]]
| Hiroyoshi Mitake (Hiroshima University)
|[[#Hiroyoshi Mitake | Joint Analysis/PDE seminar: Derivation of multi-layered interface system and its application]]
| Tran
| Tran
|-
|-  
|March 15
|September 29,
| Nestor Guillen (UMass Amherst)
VV901 2:25pm
|[[#Nestor Guillen | TBA  ]]
| Dongnam Ko (CMU & SNU)
| Lin
|[[#Dongnam Ko |  a joint seminar with ACMS: On the emergence of local flocking phenomena in Cucker-Smale ensembles ]]
|-
| Shi Jin & Kim
|March 21 (Spring Break)
|-  
|
|October 2
|[[# ]]
| No seminar due to a KI-Net conference
|
|-
|March 28
| Ryan Denlinger (Courant Institute)
|[[#Ryan Denlinger | The propagation of chaos for a rarefied gas of hard spheres in vacuum ]]
| Lee
|-
|April 4
|
||[[#  |  ]]
|
|-
|April 11
|
|[[#  |  ]]
|
|
|-
|April 18
|
|[[#  |  ]]
|
|
|-
|-  
|April 25
|October 9
| Moon-Jin Kang (UT-Austin)
| Sameer Iyer (Brown University)
|[[# |  ]]
|[[#Sameer Iyer Global-in-x Steady Prandtl Expansion over a Moving Boundary ]]
| Kim
| Kim
|-
|-  
|May 2
|October 16
|  
| Jingrui Cheng (UW)
|[[#  |  ]]
|[[#Jingrui Cheng |  A 1-D semigeostrophic model with moist convection ]]
|
| Kim & Tran
|-
|October 23
| Donghyun Lee (UW)
|[[#Donghyun Lee |  The Vlasov-Poisson-Boltzmann system in bounded domains ]]
| Kim & Tran
|-
|October 30
| Myoungjean Bae (POSTECH)
|[[#Myoungjean Bae |  TBD ]]
|  Feldman
|-
|November 6
| Jingchen Hu (USTC and UW)
|[[#Jingchen Hu | TBD ]]
| Kim & Tran
|-
|December 4
| Norbert Pozar (Kanazawa University)
|[[#Norbert Pozar TBD ]]
| Tran
|}
|}


=Abstracts=
==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.


===Tianling Jin===
===Jaeyoung Byeon===


Holder gradient estimates for parabolic homogeneous p-Laplacian equations
Title: Patterns formation for elliptic systems with large interaction forces


We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation
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.
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.


===Russell Schwab===


Neumann homogenization via integro-differential methods
===Tuoc Phan===
Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.


In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data. I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions. The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain.  This is joint work with Nestor Guillen.
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.


===Jingrui Cheng===
===Hiroyoshi Mitake===
Derivation of multi-layered interface system and its application


Semi-geostrophic system with variable Coriolis parameter.
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.
   
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.


===Hong Zhang===


On an elliptic equation arising from composite material
===Dongnam Ko===
On the emergence of local flocking phenomena in Cucker-Smale ensembles


I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.
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.


===Paul Rabinowitz===
===Sameer Iyer===
Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.


On A Double Well Potential System
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.


We will discuss an elliptic system of partial differential equations of the form
===Jingrui Cheng===
\begin{equation}
\label{*} \tag{*}
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}
\end{equation}
\[\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,\]
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$.
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations.
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,
i.e. solutions that are of phase transition type.


This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).
A 1-D semigeostrophic model with moist convection.


===Hiroyoshi Mitake===
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.


Selection problem for fully nonlinear equations
===Donghyun Lee===


Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.
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.