Abstract: In this seminar, we study the logistic diffusion equation, a reaction–diffusion model, and its equilibria. We first establish existence and regularity of positive solutions to the parabolic problem. We then use the comparison principle to show that, as time tends to infinity, the solution converges to a steady state solving the corresponding elliptic equation.
We recall why the existence of solutions to this elliptic problem is not easily obtained by standard variational methods. Finally, we discuss how stability depends on the resource term and how the solution behavior changes with the diffusion rate.
References:
[1] Cantrell, R.S., Cosner, C. Spatial ecology via reaction-diffusion equation. Wiley series in mathematical and computational biology, John Wiley & Sons Ltd (2003)
Abstract: In this talk, we discuss finite-time blow-up dynamics for the nonlinear heat equation (NLH). We explain the notion of finite-time blow-up, introduce Type I and Type II blow-ups, and discuss the difference between these two behaviors. Restricting to radially symmetric solutions, we review known blow-up results and give a heuristic explanation of when only Type I blow-up is possible and when Type II blow-up may occur. Finally, we describe possible Type II blow-up scenarios through their formal mechanisms.
Reference:
[1] Hiroshi Matano, Frank Merle. On Nonexistence of type II blowup for a supercritical nonlinear heat equation. Communications on Pure and Applied Mathematics, 2004, 57. 1494 - 1541.
[2] Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41(10): 4847-4885.
Abstract. We discuss sharp local smoothing estimates for curve averages. The proof introduces a new method for estimating oscillatory integrals based on wave packet analysis and a high–low decomposition. We outline the main ideas of the local smoothing estimates for curve averages in three dimensions, focusing on the treatment of the relevant oscillatory integrals.
Abstract: We establish the stability of a pair of Hill's spherical vortices moving away from each other in 3D incompressible axisymmetric Euler equations without swirl. Each vortex in the pair propagates away from its odd-symmetric counterpart, while keeping its vortex profile close to Hill's vortex. This is achieved by analyzing the evolution of the interaction energy of the pair and combining it with the compactness of energy-maximizing sequences in the variational problem concerning Hill's vortex. The key strategy is to confirm that, if the interaction energy is initially small enough, the kinetic energy of each vortex in the pair remains so close to that of a single Hill's vortex for all time that each vortex profile stays close to the energy maximizer: Hill's vortex. An estimate of the propagating speed of each vortex in the pair is also obtained by tracking the center of mass of each vortex. This estimate is optimal in the sense that the power exponent of the epsilon (the small perturbation measured in the "L^1+L^2+impulse" norm) appearing in the error bound cannot be improved. This talk is based on the paper [Y.-J. Sim, Nonlinearity, 2026].
Abstract: 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 $(t,x)$.
초록: 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.
Abstract: 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.
Abstract:
In this talk, we discuss the existence and stability of subsonic potential flow for the steady Euler--Poisson system in a 2-dimensional nozzle of finite length with prescribing some suitable boundary conditions. The purpose of this talk is to introduce the small-perturbation method for the steady Euler--Poisson system, which reduces to an elliptic system in our present situation. As a starting point of our discussion, we introduce the notion of background solution about which we may linearize the equations, and then point out the property of the linearized coefficients that turns out to be extremely crucial for the $$H^1$$-estimates. Next, we establish the iteration scheme and show the necessary estimates in a very brief manner, which may immediately lead us to the proof of the main existence and stability theorem. Finally, if time permits, we will take a glance at the general situation such as flows with nonzero vorticity.
Main Reference: M. Bae, B. Duan, and C. Xie, Subsonic Flow for the Multidimensional Euler–Poisson System, Arch. Rational Mech. Anal. 220 (2016), 155–191.
Abstract: 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).