Seorabeol Research Station

Gichan Bae (Dalian University of Technology)

Title: Global Incompressible Euler Limits of the Landau and Non-Cutoff Boltzmann Equations for Axisymmetric Flows without Swirl
Abstract: We establish the incompressible low-Mach/high-Reynolds limit for the Landau equation with Coulomb interactions and the non-cutoff Boltzmann equation with soft potentials in the three-dimensional axisymmetric setting without swirl, without recourse to any asymptotic expansion. Acoustic dispersion and the transport structure of the relative vorticity yield convergence to global axisymmetric solutions of the incompressible Euler equations without swirl, with relative vorticity in Lᵖ for 1 < p < ∞. Under additional sign and finite-impulse assumptions, we also establish conservation of the limiting kinetic energy.

Junsik Bae (Kyungpook National Univ.)

Title: Spectral stability of steady solutions in some shallow water models
Abstract: We consider some shallow water models that admit various steady solutions depending on the prescribed boundary conditions and bottom topography. In particular, discontinuous steady solutions arise when the flow undergoes a transition from a supercritical to a subcritical regime, whereas continuous transcritical solutions occur in transitions from subcritical to supercritical flow. We investigate the linear stability of these steady flows. This talk is based on the ongoing project with Miguel Rodrigues (University of Rennes) and Nina Aguillon (Sorbonne University).

Younghun Hong (Chung-Ang University)

Title: Semiclassical limit of quantum scattering states

Abstract: In this talk, we consider small-data solutions to the nonlinear Hartree equation in the density-matrix formulation. For short-range interactions, it is known that such solutions scatter as time tends to infinity. We prove that, in the semiclassical limit, their scattering states converge strongly to the scattering states of the corresponding kinetic model, namely, the Vlasov equation. Moreover, we establish the optimal convergence rate O(ℏ²). This improves the weak convergence result obtained in our previous work. This talk is based on joint work with Sonae Hadama (Osaka University).

Junyeong Jang (Chung-Ang University)

Title: Extending the time interval of validity of the long-wave approximation
Abstract: The water wave problem is a free-boundary problem describing the motion of a sea with a free surface. But the full water wave equation is difficult to use directly for the analysis of specific phenomena or for long-time numerical simulations. For this reason, one often replaces the full water wave equation by simpler asymptotic models and then rigorously justifies their validity through multi-scaling analysis. However, most existing justification results require high regularity of the initial data and are valid only on limited time intervals, which leaves a gap between the mathematical theory and the practical use of water wave models. In this talk, I will present a method for extending the temporal validity by studying the KdV limit of the BBM and Boussinesq equations. This talk is based on joint work with Prof. Younghun Hong (Chung-Ang Univ.).

In-Jee Jeong (KIAS)

Title: Stability of vortex dipoles
Abstract: We consider the existence, uniqueness, and geometric properties of vortex dipoles, which are pairs of steadily-translating and counter-rotating vortices. First, we make precise the notion that the Lamb dipole, an explicit traveling wave solution described by Lamb back in 1895, is the dipole with "maximal mass" under enstrophy and impulse constraints. Then we discuss rigidity of dipoles near the Lamb dipole. Lastly, we discuss stability of multiple Lamb dipoles, allowing for dipoles with mixed signs. This is based on joint works with Ken Abe, Kyudong Choi, Guolin Qin, Yao Yao, and Weicheng Zhan.

Jinwook Jung (Hanyang Univeristy)

Title: Quantitative light-particle limit for the Vlasov-Fokker-Planck-Navier-Stokes system
Abstract: In this talk, we investigate the hydrodynamic limit of the Vlasov-Fokker-Planck-Navier-Stokes system in the light particle regime, where the particle relaxation takes place on a singularly fast time scale. Using a relative entropy method adapted to this scaling, we develop the first quantitative convergence theory for the light particle limit. Our analysis yields explicit rates for the convergence of both the kinetic distribution and the fluid velocity, extending the qualitative compactness-based result of Goudon, Jabin, and Vasseur [Indiana Univ. Math. J., 53, (2004), 1495--1515]. Moreover, we derive refined convergence estimates for the macroscopic density and fluid velocity in negative Sobolev spaces, consistent with the formally optimal rates predicted by the Hilbert expansion. The results apply to both the torus and the whole space, providing a unified quantitative description of the light particle hydrodynamic limit. This is based on the joint work with Y.-P. Choi (Yonsei University).

Dong-Ha Kim (Seoul National University)

Title: A Lagrangian approach to uniqueness for active scalars equations in generalized mean oscillation spaces

Abstract: We prove uniqueness for a class of two-dimensional inviscid active scalar equations in generalized mean-oscillation spaces via a Lagrangian approach. Our unified framework covers the Euler equation, α-SQG equation, and SQG equation in spaces which are slightly rougher than BMO, Cᵅ, and C^{0,1}, respectively. In each case, the corresponding modulus is enlarged by a factor of log log(1/|x − y|). It also yields analogous results for the IPM equation and the logarithmically perturbed Euler equation. This work is joint with Junha Kim, Hyunjin In, and Seungjae Lee.

Junha Kim (Ajou University)

Title: Boundary effects for α-SQG equations in the half-plane

Abstract: We discuss the Cauchy problem for the inviscid α-SQG equations in the half-plane, a family of active scalar equations interpolating between the 2D Euler equations (α = 0) and the SQG equation (α = 1). We consider initial data smooth up to the boundary. When it does not vanish on the boundary, the normal derivative of the tangential velocity exhibits a singularity of order α at the boundary. This singularity determines the threshold α = 1/2. For 0 < α ≤ 1/2, we establish local well-posedness in an anisotropic Lipschitz space contained in C^{1−α}, and show that the same singularity leads to escape from C^β regularity for all β > 1 − α. For 1/2 < α < 1, it leads to nonexistence of C^α solutions. In the SQG case, we study data vanishing on the boundary. A refined analysis of the velocity shows that the second normal derivative of the tangential velocity admits a logarithmic singularity at the boundary. As a consequence, smooth data remain in C^{2,β} for any β < 1 for a short time, but in general fail to stay in C^{2,1}. This talk is based on joint works with In-Jee Jeong, Yao Yao, and Hideyuki Miura.

Junsung Kim (POSTECH)

Title: Dynamical concentration and self-field stability for the Relativistic Vlasov-Maxwell system under Maxwell-compatible forcing.
Abstract: We study the relativistic Vlasov–Maxwell system in 1Dₓ + 3Dₚ slab geometry under Maxwell-compatible external electromagnetic fields and establish concentration of the particle distribution together with global stability of the self-consistent field. The evolution is divided into an initial acceleration layer and a subsequent asymptotic regime. In the acceleration layer, direct control of the self-field yields a large transverse velocity gain, which generates the relativistic anisotropy needed for the later dynamics. In the asymptotic regime, this velocity gain suppresses longitudinal transport and drives spatial concentration, while the resulting geometry, combined with a reflection-induced cancellation in the field representation, provides global control of the self-consistent electromagnetic field. This reveals a two-stage mechanism in which acceleration prepares the phase-space structure that ultimately produces both concentration and stability. This work is joint with Jin Woo Jang, Tak Kwong Wong, and Robert M. Strain.

Joonhyun La (KIAS)

Title: On kinetic theory of oscillator chains

Abstract: In this talk, we discuss the kinetic theory of one-dimensional nonlinear oscillator chains, of which the most famous example is the Fermi-Pasta-Ulam-Tsingou equation.

Jinyeop Lee (Kyung Hee University)

Title: Derivation of the Chern-Simons-Schrödinger equation from the dynamics of an almost-bosonic-anyon gas

Abstract: We study the time evolution of an initial product state in a system of almost-bosonic-extended-anyons in the large-particle limit. We show that the dynamics of this system can be well approximated, in finite time, by a product state evolving under the effective Chern--Simons--Schrödinger equation. Furthermore, we provide a convergence rate for the approximation in terms of the radius R=(logN)^{-1/2+ε} of the extended anyons. These results establish a rigorous connection between the microscopic dynamics of almost-bosonic-anyon gases and the emergent macroscopic behavior described by the Chern--Simons--Schrödinger equation.

Sungbin Park (POSTECH)

Title: Exponential-Type Ill-Posedness for the BGK Model and Well-Posedness for the Boltzmann Equation in a Half-Space with Inflow Boundary Conditions
Abstract: The Boltzmann equation and the BGK model are widely used to describe the nonequilibrium statistical behavior of dilute gases. Although the two models share fundamental features, such as the conservation of macroscopic quantities, they exhibit different behavior when their solutions are considered in exponentially weighted spaces. Extending previous results from the whole-space setting, we study both models in a half-space with an inflow boundary condition and vacuum initial data. We establish ill-posedness for the BGK model and well-posedness for the Boltzmann equation in suitable exponentially weighted spaces. Unlike in the whole-space setting, the analysis of the half-space problems requires different exponential weights for the two models, reflecting their distinct responses to the boundary inflow. This is a joint work with Donghyun Lee, Sung-Jun Son, and Seok-Bae Yun.

Jinsol Seo (KIAS)

Title: Unique Solvability and Stability for the Landau Equation with Inflow Boundary Data

Abstract: The Landau equation is a fundamental kinetic model for collisional plasmas. In this talk, I will discuss the Landau equation in a bounded domain with inflow boundary data. We establish a well-posedness theory for both the stationary and time-dependent problems, and prove the asymptotic stability of steady states under small perturbations of the inflow and initial data. This is joint work with Chanwoo Kim.

Jinmyoung Seok (Seoul National University)

Title: Measure-Valued Bose Free Energies: Obstacle Problems and Sharp Condensation Transitions
Abstract: We study a free-energy minimization problem for a trapped semiclassical Bose gas with repulsive Coulomb self-interaction. Since the bosonic entropy is not superlinear and its lower-semicontinuous extension has zero recession cost, the natural variational problem is posed over nonnegative finite Radon measures on phase space. The possible singular component of a minimizer is interpreted as condensation within this relaxed semiclassical model. We first prove existence and uniqueness of the minimizer at every positive temperature. Its absolutely continuous part is given by a Bose–Einstein distribution in a self-consistent effective potential, while its singular part is concentrated at zero momentum and supported on the contact set where the effective potential reaches the chemical potential. When the trapping potential is locally C², this contact condition becomes an obstacle problem and yields an explicit density formula for the condensate. In radial examples, the resulting condensate may occupy either a ball or a spherical shell. We then establish sharp condensation transitions. At every fixed mass, there exists a unique finite positive critical temperature: the minimizer is condensed below this temperature and normal at and above it. At fixed temperature, there is likewise a sharp critical mass, which may be infinite. For harmonic traps, we obtain an exact confinement-strength dichotomy between a finite critical mass and the absence of condensation at every mass. This is joint work with Gi-Chan Bae.

Page updated

Google Sites

Report abuse