Abstract:
Vortex dipoles are one of the most iconic structures in two-dimensional incompressible flows. In this talk, I will present recent results on the existence and stability of traveling wave solutions to the two-dimensional incompressible Euler equations. These solutions take the form of counter-rotating vortex dipoles symmetric across a horizontal axis. A classical example is the Chaplygin–Lamb dipole, where the two vortex regions are tightly packed near the symmetry axis, leading to intense interaction. I will describe a variational framework for constructing such solutions and discuss their dynamical properties. This is joint work with Kyudong Choi and Young-Jin Sim (UNIST).
4:00--4:50 Speed of propagating fronts in spatially periodic KPP equations
5:00--6:00 Propagation of fronts through a perforated wall
4:00--4:50
Title: Speed of propagating fronts in spatially periodic KPP equations
Abstract : In this talk I will discuss front propagation in the KPP type
reaction-diffusion equations with spatially periodic coefficients. Since
the pioneering work of Kolmogorov--Petrovsky--Piscounov and Fisher in
1937, front propagations in KPP type reaction-diffusion equations have
been studied extensively. Starting around 1950's, KPP type equations
have played an important role in mathematical ecology, in particular, in
the study of biological invasions in a given habitat. What is
particularly important is to estimate the speed of propagating fronts.
In the spatially homogeneous case, there is a simple formula for the
speed, which was given in the work of KPP and Fisher in 1937. However,
if the coefficients are spatially periodic, estimating the front speed
is much more difficult, and it involves the principal eigenvalue of a
certain operator that is not self-adjoint. In this talk, I will mainly
focus on the one-dimensional problem and give an overview of the past
research on this theme starting around 1980's. I will also present a
work of mine on KPP type equations in 2D in periodically stratified media.
5:00--6:00
Title: Propagation of fronts through a perforated wall
Abstract: In recent years, the behavior of solution fronts of reaction-diffusion
equations in the presence of obstacles has attracted attention among
many researchers. Of particular interest is the case where the equation has a bistable nonlinearity.
In this talk, I will consider the case where the obstacle is a wall of
infinite span with many holes and discuss whether the front can pass
through the wall and continue to propagate (“propagation”) or is
blocked by the wall (“blocking”). The answer depends largely on the
size and the geometric configuration of the holes.
This problem has led to a variety of interesting mathematical
questions that are far richer than we had originally anticipated. Many
questions still remain open. This is joint work with Henri Berestycki
and François Hamel.
Abstract
We consider a class of linear estimates for evolution PDEs on the Euclidean space, called Strichartz estimate. Strichartz estimates are well-established for fundamental linear PDEs, such as heat and wave equations. As a simple model of such, we consider the Schrödinger example, introducing classical Strichartz estimates with proofs.
Reference
Terence Tao, Nonlinear dispersive equations: local and global analysis, Chapter 2.3
Abstract: In this talk, we consider the second-order quasilinear degenerate elliptic equation whose dominant part has the form $(2x - au_x)u_{xx} + bu_{yy} - u_x = 0$, where $a$ and $b$ are positive constants. We first introduce the physical situation that motivates the present analysis in a very brief manner, and then discuss mathematical difficulties involved in the analysis of the problem. The main part of this talk focuses on methods to overcome those difficulties, such as vanishing viscosity approximation and parabolic scaling.
- Reference:
[1] Chen, G.-Q. and Feldman, M. (2010). Global solutions to shock reflection by large-angle wedges, Ann. of Math. 171: 1019–1134. *Main reference
[2] Bae, M., Chen, G.-Q. and Feldman, M. (2009). Regularity of solutions to regular shock reflection for potential flow, Invent. Math. 175: 505–543.
Abstract
In this talk, I will present the local existence theory for quasilinear symmetric hyperbolic systems, based on Sections 1.3 and 2.1 of [1]. I will begin by reviewing the framework of symmetric systems and then explain how it is applied to establish local-in-time existence of classical solutions.
The main focus will be on the iteration scheme, energy estimates, and convergence arguments. We aim to understand how regularity and a priori bounds are used to construct solutions from smooth initial data.
Reference:
[1] Andrew Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, 1984.