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

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.

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].

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 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)