Seorabeol Research Station

Gichan Bae (Seoul Nat'l Univ.)

Title: High Reynolds number limit of 2D Boltzmann equation
Abstract: We prove the hydrodynamic limit of the 2D Boltzmann equation to the incompressible Euler equation in a periodic box. This is joint work with Chanwoo Kim.

Young-Pil Choi (Yonsei Univ.)

Title: Asymptotic analysis of kinetic models towards the compressible Euler and continuity equations

Abstract: In this talk, we discuss the analysis of asymptotic limits for kinetic models to compressible Euler equations or continuity equation. Precisely, we introduce some kinetic models relaxing to the compressible Euler-Poisson system or drift-diffusion equations.

Dingqun Deng (Postech)

Title: Nonlinear stability of the planar shock and rarefaction waves for the 3-D Boltzmann equation

Abstract: This talk focuses on the stability and long-time behavior of the 3-D Boltzmann equation in $\mathbb{R}\times\mathbb{T}^2$ near 1-D shock profiles and 1-D rarefaction waves. On the one hand, we consider the nonlinear stability of the composite wave consisting of two shock profiles of the Boltzmann equation under general perturbations without the integral zero assumption of the macroscopic quantities. On the other hand, we consider the stability of the Vlasov-Poisson-Boltzmann system with (-out) specular boundary condition near a local Maxwellian with macroscopic quantities given by the rarefaction wave solution of one-dimensional compressible Euler equations. We will show that in these two cases, the solution with small perturbations tends to the composite shock profiles or rarefaction waves as time goes to infinity.

Sonae Hadama (RIMS, Kyoto Univ.)

Title: Asymptotic stability of a wide class of stationary solutions for the Hartree and Schrödinger equations for infinitely many particles

Abstract: We consider the Hartree and Schrödinger equations describing the time evolution of wave functions of infinitely many interacting fermions in three-dimensional space. These equations can be formulated using density operators and have infinitely many stationary solutions. In this talk, we give a result and rough idea to deal with a wide class of stationary solutions. We emphasize that our result includes Fermi gas at zero temperature. This is one of the most important steady states from the physics point of view; however, its asymptotic stability has been left open after the seminal work by Lewin and Sabin which first formulated this stability problem and gave significant results.

In-Jee Jeong (Seoul Nat'l Univ.)

Title: Stability and instability of Vlasov-Poisson equation

Abstract: We consider the dynamics of electrons under the Vlasov-Poisson equation with a fixed and given ion density background. When the ion density is sufficiently well-localized, classical works give the existence and stability of steady electron distributions. We study the dynamics of perturbations of such steady distributions. Furthermore, we address the question of stability and instability when the ion density does not satisfy the localization hypothesis.

Junha Kim (KIAS)

Title: On the axially symmetric solutions to the spatially homogeneous Landau equation

Abstract: TBA

Bongsuk Kwon (UNIST)

Title: Euler-Poisson system of ion dynamics and related problems

Abstract: Various plasma phenomena are mathematically studied using a fundamental fluid model for plasmas, called the Euler-Poisson system. Among them, plasma solitary waves are of our interest, for which existence, stability, and the time-asymptotic behavior of the solitary wave will be briefly discussed. On the other hand, to study nonlinear stability, a question of existence of smooth global solution naturally arises, which is completely open, to the best of our knowledge. We introduce the finite time blow-up results for the Euler-Poisson system, and discuss the related open questions. This talk is based on joint work with Junsik Bae and Yunju Kim at UNIST.

Shota Sakamoto (Kyushu Univ.)

Title: Title: Faster decay of the microscopic part of a solution to the Boltzmann equation without angular cut-off

Abstract: We consider a Cauchy problem of the Boltzmann equation without angular cut-off near the global Maxwellian. It is widely observed that, under various settings of kinetic equations, the kinetic (microscopic) part of a solution decays faster in time than its fluid (macroscopic) counterpart. We will prove that a solution to the Boltzmann equation in the so-called low-regularity spaces also satisfies this property. The proof is based on the derivation of higher-order energy estimates and differential inequality for the microscopic part, which obtains (1+t)^{-1/2} faster decay for any order of differentiation.

Tak Kwong Wong (Univ. of Hong Kong)

Title: Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus

Abstract: This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global-in-time well-posedness of axisymmetric solutions to the relativistic Vlasov-Maxwell system in a two-dimensional annulus, provided that a huge external magnetic potential is imposed near the boundary. The external magnetic potential well that we impose remains finite within any finite time interval and from that, we prove that the plasma never touches the spatial boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. This is a joint-work with Jin Woo Jang and Robert M. Strain.

Seok-Bae Yun (Sungkyunkwan Univ.)

Title: Weak solutions for the stationary BGK model

Abstract: We consider the rarefied stationary flow emerging from the evaporation and condensation process on the two parallel condensed phases in the framework of the stationary BGK model. Under the physically minimum conditions on the inflow functions, namely the finite mass flux, energy flux and entropy flux, the existence of stationary weak solutions is derived. Due to the combined effect of the singularity near vanishing first velocity component and the highly nonlinear structure of the relaxation operator, various novel difficulties unobserved in the corresponding problem for the Boltzmann equation arise. (1) Unlike the Boltzmann equation, truncation of the relaxation operator in the vanishing velocity region in the last limit process turns out to be not feasible. (2) The limit process may send the local Maxwellian into a Dirac mass. (3) controlling the velocity distribution functions near the vanishing velocity region requires a refined understanding of the behavior of the macroscopic fields. This is a joint work with Stephane Brull (Univ. of Bordeaux)