Plenary Lecture 

• Jinmyoung Seok (Seoul National Univ.) 

Title : Variational Methods for Steady States and Dynamical Stability in Kinetic and Fluid Equations 

Abstract : In these two lectures, I will present variational methods for constructing steady states and establishing their nonlinear dynamical stability for a range of kinetic and fluid equations. The first lecture will be devoted to foundational theory and key historical developments. In the second lecture, I will discuss several recent results, including some of my own work. 

Invited Talk

• Gi-Chan Bae (Seoul National Univ.) 

Title : Quantum kinetic equations near a global equilibrium 

Abstract : This talk considers the existence and asymptotic behavior of two quantum kinetic equations, the quantum BGK model and the relativistic quantum Boltzmann equation. More precisely, we establish the existence of unique classical solutions and their exponentially fast stabilization when the initial data starts sufficiently close to a global quantum equilibrium based on the nonlinear energy method. The two models have different difficulties. For the quantum BGK model, the difficulty is to extract dissipation from the highly nonlinear quantum local equilibrium. For the relativistic quantum Boltzmann equation, we control the singular integral on the energy-momentum 4-vector space. 

• Sangdon Jin (Chungbuk National Univ.) 

Title : Semiclassical equivalence of two white dwarf models as ground states of the relativistic Hartree-Fock and Vlasov-Poisson energies

Abstract : We study the semi-classical limit for ground states of the relativistic Hartree-Fock energies (HF) under a mass constraint, which are considered as the quantum mean-field model of white dwarfs. Fermionic ground states of the relativistic Vlasov-Poisson energy (VP) are constructed as a classical mean-field model of white dwarfs, and are shown to be equivalent to the classical Chandrasekhar model. In this talk, we prove that as the reduced Planck constant ℏ goes to the zero, the ℏ-parameter family of the ground energies and states of (HF) converges to the fermionic ground energy and state of (VP) with the same mass constraint. 

• Jinwook Jung (Hanyang Univ.) 

Title : Incompressible fluid limits from a nonlinear Vlasov-Fokker-Planck equation with constant temperature 

Abstract : In this talk, we provide the rigorous derivation of the incompressible Navier–Stokes/Euler equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian can be well established, which also yields some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system. Under the incompressible Euler scaling, we use relative entropy methods and uniform moment bounds to show that weak solutions converge to dissipative solutions of the incompressible Euler equations on the torus. This talk is based on the collaboration with Y.-P. Choi (Yonsei Univ.). 

• Jeongho Kim (Kyung Hee Univ.) 

Title : Nonextensive BGK model

Abstract : In this talk, I will introduce a nonextensive BGK model, which is a generalization of standard BGK model with Maxwellian equilibrium. Precisely, the relaxation distribution of the nonextensive BGK model is a one-parameter family of distribution with parameter q, called the Tsallis distribution (or q-Maxwellian), which is reduced to the standard Maxwellian when q=1. I will introduce several properties of the nonextensive BGK model and present results on its Cauchy problems. This talk is based on joint works with Prof. Seok-Bae Yun (Sungkyunkwan University) and Seung-Yeon Cho (Gyeongsang National University).

• Sungbin Park (POSTECH) 

Title : Well-Posedness Theory and Lower and Upper Bounds for the Boltzmann Equation with Fermi–Dirac Statistics 

Abstract :The Boltzmann-Fermi-Dirac equation, proposed by L.W. Nordheim (1928) and by E. A. Uehling and George Uhlenbeck (1933), is a quantum modification of the classical Boltzmann equation for Fermi-Dirac statistics. In this talk, we extend some results in the classical Boltzmann equation to the Boltzmann-Fermi-Dirac equation. (1) We first state the existence and uniqueness of the solution of the Boltzmann-Fermi-Dirac equation. (2) We construct a Gaussian lower bound for the solution. (3) Finally, we consider polynomial and exponential moments in the $L^1$ and $L^\infty$ settings. We will discuss the main difficulties and techniques in proving these results compared to the classical equation. 

• Yeongseok Yoo (Yonsei Univ.) 

Title : Global Weak Solutions to the Nonlinear Vlasov-Fokker-Planck Equation

Abstract : In this work, we study the Nonlinear Vlasov-Fokker–Planck equation that models collective behavior such as flocking and swarming. The equation includes a local alignment force that depends on density, velocity, and temperature. This singularity makes the mathematical analysis difficult, especially when the density may vanish. We prove the global-in-time existence of weak solutions for initial data with finite mass, energy, and entropy. The proof is based on a regularization and iteration scheme that constructs smooth approximate solutions and provides uniform bounds independent of the regularization. A key tool is the velocity averaging lemma, which allows us to obtain strong compactness of the macroscopic quantities needed to pass to the limit in nonlinear terms.