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.

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.).

Classical variational approach of maximizing the kinetic energy with various constraints provides vortex stability in several special cases, but in general this approach fails when the vorticity is concentrated at several points in the fluid domain. This is simply because such configurations are not local kinetic energy maximizers, even when we restrict the admissible class using all the other coercive conserved quantities of fluid motion. In this talk, we present several results on the stability of multi-vortex solutions, obtained by combining classical variational approach with dynamical bootstrapping schemes. We focus on the case of multiple Lamb dipoles weakly interacting with each other. This is based on joint works with Ken Abe, Kyudong Choi, and Yao Yao.

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.

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

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.

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.