Monday PDE Meetup

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

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.

초록: In this talk, we prove that the inviscid surface quasi-geostrophic (SQG) equation is strongly ill-posed in critical Sobolev spaces: there exists an initial data $H^2(\mathbb{R}^2)$ without any solutions in $L^{\infty}_tH^2$. Then, we introduce similar ill-posedness results for $\alpha$-SQG and two-dimensional incompressible Euler equations. This talk is based on joint works with In-Jee Jeong(SNU), Young-Pil Choi(Yonsei Univ.), Jinwook Jung(Hanyang Univ.), and Min Jun Jo(Duke Univ.).

Abstract: We study support properties of solutions to stochastic heat equations $\partial_t u = \Delta u + \sigma(u) \xi$ where $\xi$ is Gaussian noise. For $\sigma(u) = u^\lambda$ with colored noise, we show the compact support property holds if and only if $\lambda \in (0, 1)$. Here, the compact support property refers to the property that if the initial function has compact support, then so does the solution for all time. For space-time white noise with general $\sigma$, we characterize when solutions maintain compact support versus become strictly positive. We also discuss how the initial function influences these support properties. This is based on joint work with Beom-Seok Han and Jaeyun Yi.

The Lyapunov-Schmidt reduction is a powerful tool to solve PDEs. This method reduces the equations, which are essentially infinite-dimensional, to finite-dimensional ones. In this talk, we illustrate the reduction by showing the existence of a positive solution to the singularly perturbed problem  in  for positive smooth  and appropriate . To show the existence, we first construct an -dimensional surface of approximate solutions. Then, we reduce the problem onto that surface by the Lyapunov-Schmidt reduction. The key to the reduction is proving the invertibility of a certain operator, which in turn, is proved by a certain uniqueness result. After the reduction, we end the proof by solving the equation on the -dimensional surface.

Reference 

Y.Y. Li, On a singularly perturbed elliptic equation, Advances in Differential Equations, Adv. Differential Equations 2(6) (1997) 955-980

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.

We briefly introduce the restriction theory in harmonic analysis and its connections with PDEs through Strichartz estimate.

We then discuss the Kakeya and multilinear Kakeya estimates, which naturally arise from restriction theory.

The main part of the talk will focus on Larry Guth’s proof of the multilinear Kakeya estimate via the induction on scales method.

References:

[1] Terence Tao, “Restriction Theory,” Lecture Notes for Math 247B, 2020.

[2] Larry Guth, 18.118 Decoupling Lecture Notes.

[3] Larry Guth, “A Short Proof of the Multilinear Kakeya Inequality,” Math. Proc. Camb. Philos. Soc. 158 (2015), 147–153.

[4] Bennett, J., Carbery, A. and Tao, T. (2006). On the Multilinear Restriction and Kakeya Conjectures., Acta Math. 196(2), 261–302. 

초록: I will discuss recent progress on the vanishing-viscosity limit of the two-dimensional Navier–Stokes equation. Our approach is Lagrangian and probabilistic:

1. We develop a stochastic counterpart of the DiPerna–Lions theory to construct and control stochastic Lagrangian flows for the viscous dynamics.
2. We also establish a large-deviation principle that quantifies convergence to the Euler dynamics.
This talk is based on joint work with Chanwoo Kim, Dohyun Kwon, and Jinsol Seo.