Difference between revisions of "PDE Geometric Analysis seminar"

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(PDE GA Seminar Schedule Fall 2017)
(PDE GA Seminar Schedule Fall 2017)
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==Abstracts==
 
==Abstracts==

Revision as of 08:54, 19 September 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 B239 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: TBD 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) TBD Kim & Tran
October 23 Donghyun Lee (UW) TBD 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.

Iyer Sameer

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.