The Korteweg-de Vries-Burgers (KdVB) equation is a fundamental model capturing the interplay of nonlinearity, viscosity (dissipation), and dispersion, with broad physical relevance. It is well known that the KdVB equation admits traveling wave solutions, called viscous-dispersive shocks. These shock profiles are monotone in the viscosity-dominant regime, while they exhibit infinitely many oscillations when dispersion dominates. In this talk, we study the stability of such viscous-dispersive shocks, focusing on an L2 contraction property under arbitrarily large perturbations, up to a time-dependent shift. We begin with viscous shocks of the viscous Burgers equation (i.e., the KdVB equation without dispersion), then treat monotone viscous-dispersive shocks and finally address oscillatory shocks. We also present detailed structural properties of the oscillatory profiles. This is joint work with Geng Chen (University of Kansas), Moon-Jin Kang (KAIST), and Yannan Shen (University of Kansas).

In this talk, we discuss the initial–boundary value problem for one-dimensional hyperbolic conservation laws on the half-line, focusing on linear systems and scalar conservation laws. We begin with a discussion of the theory of the Cauchy problem. We then turn to the half-line setting, where we introduce two formulations of boundary conditions: one based on the vanishing viscosity method and the other based on the Riemann problem. We show that these two formulations are equivalent for linear systems and scalar conservation laws. Finally, we present remarks on boundary conditions for general hyperbolic systems of conservation laws. Reference: Dubois, F., and LeFloch, P. Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71, 1 (1988), 93–122.

In this seminar, we study the Vlasov–Maxwell system, a fundamental collisionless kinetic model for plasmas, posed in a three-dimensional half-space with boundaries. We begin with a brief warm-up by revisiting the one-dimensional Vlasov–Poisson system in the absence of magnetic fields, focusing on Penrose’s classical 1960 spectral criterion for linear stability and instability. We then turn to the full Vlasov–Maxwell system and discuss the major analytical difficulties introduced by electromagnetic coupling, boundary effects, and nonlinear interactions. In particular, we highlight the role of an effective gravitational force directed toward the boundary and its interplay with boundary temperature conditions. This viewpoint naturally leads us to formulate a conjectural linear instability criterion associated with boundary-induced confinement effects. Within this framework, we construct global-in-time classical solutions to the nonlinear Vlasov–Maxwell system beyond the vacuum scattering regime. Our approach combines the construction of stationary boundary equilibria with a proof of their asymptotic stability in the $L^\infty$ setting under small perturbations. This work provides a new framework for analyzing long-time plasma dynamics in bounded domains with interacting magnetic fields. To our knowledge, it yields the first construction of asymptotically stable non-vacuum steady states for the full three-dimensional nonlinear Vlasov–Maxwell system. This is joint work with Chanwoo Kim.

In this talk, we study the non-cutoff Boltzmann equation with moderately soft potentials, a classical kinetic model. The uniqueness of large weak solutions is challenging due to the nonlinearity and limited regularity. To overcome these difficulties, we utilize dilated dyadic decompositions in phase space $(v,\xi,\eta)$ to capture hypoellipticity and reduce the fractional derivative structure $(-\Delta_v)^s$ of the Boltzmann collision operator to a zeroth-order form. Within this framework, we establish the uniqueness of large-data weak solutions under the assumption of finite $L^2$--$L^r$ energy, namely that $\|\mu^{-\frac{1}{2}}(F-\mu)\|_{L^\infty_t L^{r}_{x,v}}+\|\mu^{-\frac{1}{2}}(F-\mu)\|_{L^\infty_t L^2_{x,v}}$ is bounded for some sufficiently large $r>0$. The challenges arising from large solutions are handled via a negative-order hypoelliptic estimate, which yields additional integrability in $(t,x)$.