Program and Abstracts

Yan Guo, Brown University

Title: Hydrodynamic Limit of Steady Kinetic Models

Abstract: In this series of lectures, I would like to present recent progress of hydrodynamic limits of various steady kinetic models in the presence of physical boundary conditions. (1) Diffuse Limit of Neutron Transport Equation: It was discovered that classical diffusive expansion for such a basic problem for kinetic theory is invalid in L^\infty. It is established that a cutoff diffusive expansion is valid in L^2 via new boundary layer estimates. (2) Hydrodynamic Limit of Gas Subject to Temperature Variation: In this basic problem in physics, it is established that the hydrodynamic limit of a Boltzmann gas leads to the celebrated Fourier law in case the temperature variation is of the size of the mean free path. On the other hand,  it is established that the hydrodynamic limit of a Boltzmann gas leads to a fluid model with ghost effect if the temperature variation is o(1). New mathematical tools and framework will be introduced.

Jinwook Jung, Jeonbuk National University

Title: On the Vlasov- and Euler-alignment model with singular communication weights

Abstract: In this talk, we investigate the solutions to the Vlasov- and Euler-alignment model with singular communication weights $\phi(r) = r^{-\gamma}$, which are also known as the kinetic and hydrodynamic Cucker-Smale model, respectively. For the Vlasov-alignment model, we establish the local-in-time well-posedness of strong solutions in a weighted Sobolev space for $\gamma \in [d-1, d+1/4)\setminus \{d\}$. On the other hand, we present the global-in-time well-posedness result for the Euler-alignment model when $\gamma <d$ based on the method of characteristics and optimal transport techniques. This talk is based on the collaboration with Y.-P. Choi (Yonsei University).

Gi-Chan Bae, Seoul National University

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.

Myeongju Kang, KIAS

Title: Trend to equilibrium for the inertial Kuramoto-Sakaguchi equation

Abstract: In this talk, we show exponential relaxation of solutions to the inertial Kuramoto-Sakaguchi equation toward corresponding phase-homogeneous stationary states in weighted Lebesgue norm sense. In [Choi-Ha-Xiao-Zhang, SIMA, 2021], it is proved that when the noise intensity is sufficiently large, equilibrium of the inertial Kuramoto-Sakaguchi equation is asymptotically stable. For generic initial data, every solutions converges to equilibrium in weighted Sobolev norm sense. We improve this previous result by showing the convergence for a larger class of functions and by providing a simpler proof.

Donghyun Lee, POSTECH

Title: DYNAMICAL BILLIARD AND A LONG-TIME BEHAVIOR OF THE BOLTZMANN EQUATION IN GENERAL 3D TOROIDAL DOMAINS

Abstract: Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors’ chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular- bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains. This is joint work with Gyounghun Ko and Chanwoo Kim.

Ho Lee, Kyung Hee University

Title: Global existence of small solutions of the Einstein-Boltzmann system with soft potentials

Abstract: In this talk we consider the Einstein-Boltzmann system with soft potentials. For the Einstein equations we assume the Bianchi I symmetry, which describes a spatially homogeneous universe, and a positive cosmological constant, which describes an accelerated expansion of the universe. For the Boltzmann equation, we consider a certain range of soft potentials in a spatially homogeneous setting. It is well known that the global existence for the Boltzmann equation can be obtained by considering the dispersion effect of small solutions. In our case, global existence is obtained by investigating the accelerated expansion of the universe. We introduce a new weight function and will show that the weight function works nicely in the case of a spatially homogeneous universe with an accelerated expansion.

Seok-Bae Yun, Sungkyunkwan University

Title: Weak solutions to stionary BGK model in a slab

Abstract: We consider stationary flow between two condensed phases that emerges from the evaporation and condensation process on the two phases in the framework of the stationary BGK model in a slab. 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. The main difficulties are, among others, (1) the impossibility of truncation of the relaxation operator in the vanishing velocity region in the last limit processs, and  (2) the control of the velocity distribution functions near vanishing velocity region using the macroscopic fields. The first difficulty is overcome by replacing the transport operator with aa suitable regularized version, instead of introducing truncation of the relaxation operator. To treat the second difficulty, we start from the observation that the temperature is well-controlled from below when the density does not blow up, and the reamining part can be controlled to be small. This is a joint work with Stephane Brull.

Junsik Bae, UNIST

Title: Nonexistence of multi-dimensional solitary waves in unmagnetized plasma

Abstract: We study the nonexistence of multi-dimensional solitary waves for the Euler- Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler- Poisson system has solitary waves travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves for any traveling velocity and for general pressure laws. Our finding is strong evidence for the transverse stability of line solitary waves in the multi-dimensional Euler-Poisson system. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler-Poisson system for ions and electrons. This is a joint work with Daisuke Kawagoe.

Jeongho Kim, Kyung Hee University

Title: Quantified asymptotic analysis for the relativistic quantum mechanical system with electromagnetic fields

Abstract: We study asymptotic analysis of the relativistic quantum mechanical system interacting with a self-consistent electromagnetic fields. In particular, our analysis focuses on the Maxwell-Klein-Gordon (MKG) model, wherein the Klein-Gordon scalar field is coupled with the Maxwell equations for electromagnetic fields. To derive an asymptotic analysis, we consider the semi-classical and non-relativistic regime at the same time, by introducing a single scaling parameter that simultaneously parametrizes the Planck constant and the speed of light. As the scaling parameter vanishes, we derive rigorous and quantified estimates regarding the asymptotic convergence of the MKG system towards the classical Euler-Poisson system. Our analytical framework relies on the modulated energy estimate.

Joonhyun La, KIAS

Title: Hydrodynamic limit of incompressible Euler equation

Abstract: In this talk, we discuss recent progress in the hydrodynamic limit of incompressible Euler equation from the Boltzmann equation.

Dowan Koo, Yonsei University

Title: CRITICAL THRESHOLDS IN PRESSURELESS EULER–POISSON EQUATIONS WITH BACKGROUND STATES

Abstract: We investigate the critical threshold phenomena in a large class of pressureless Euler--Poisson (EP) equations in one dimension. We propose a new method based on Lyapunov functions to construct the supercritical region with finite-time breakdown and the subcritical region with global-in-time regularity of C^1 solutions for the pressureless damped EP equations with background states. We identify for the first time critical threshold curves for the pressureless damped EP equations with repulsive forces and variable background. For the supercritical initial data, the lower and upper bounds on the blow-up time are analyzed, and the large-time behavior of solutions with the subcritical initial data is also obtained. We finally apply our new method to the study of critical thresholds in the damped EP system for a cold plasma, in which the density of electrons is given by the so-called Maxwell--Boltzmann relation. In particular, our result shows the global-in-time existence and uniqueness of classical solutions to the damped EP system for a cold plasma only under the smallness assumption on the initial energy compared to the strength of damping.

Dingqun Deng, POSTECH

Title: The Non-cutoff Boltzmann Equation in Convex Domains

Abstract: The initial-boundary value problem for the inhomogeneous non-cutoff Boltzmann equation is a long-standing open problem. In this talk, we investigate the stability and long-time dynamics of the Boltzmann equation near global Maxwellian without angular cutoff assumption in a convex domain $\Omega$ with physical boundary conditions: inflow and Maxwell-reflection (including diffuse-reflection) boundary conditions. When the domain $\Omega$ is bounded, we obtain the global stability in time, which has an exponential decay rate for the inflow boundary for both hard and soft potentials, and for the Maxwell-reflection boundary for hard potentials. The crucial method is to extend the boundary problem in a convex domain to the whole space, followed by the De Giorgi iteration and the $L^2$--$L^\infty$ method. We believe that the current work will have a significant impact on the generation of robust applications for the non-cutoff Boltzmann equation in bounded domains.

Yong-Geun Oh, IBS-CGP & POSTECH

Title: NONEQUILIBRIUM THERMODYNAMICS, INFORMATION ENTROPY AND CONTACT GEOMETRY

Abstract: Starting from Carathedory and Hermann, contact geometry is proposed as the correct geometric framework of thermodynamics, especially of its equilibrium thermodynamics. It has been observed in this formulation that the state of thermodynamic equilibrium can be interpreted as a Legendrian submanifold in contact geometry. In this lecture, we will explain the origin of aforementioned contact structure in thermodynamic phase space (TPS), in particular of its odd dimensionality, as a symplecto-contact re- duction of the kinetic theory phase space (KTPS) of the statistical phase space (SPS) of many-body systems. In this reduction process, the relative information entropy associated to the probability distribution plays the role of a generating function of thermodynamic equilbria as a Legendrian submanifold. The associated thermodynamic potential is multi-valued for general thermodynamic equilibria. We then interpret the Maxwell construction in thermodynamics as the procedure of finding a continuous, not necessarily differentiable, thermodynamic potential and explain the associated phase transition by identifying the Maxwell construction with the general algorithm of finding a single-valued graph selector in symplecto-contact geometry and in the Aubry-Mather theory of dynamical system. (This is based on a joint work with my student Jinwook Lim.)

Byung-Hoon Hwang, Sangmyung University

Title: Derivation of the relativistic BGK model for gas mixtures

Abstract: In this talk, we study the derivation of the relativistic BGK model for gas mixtures in which the existence of each equilibrium coefficients in the relaxation operator is rigorously guaranteed in a way that all the essential physical properties are satisfied such as the conservation laws, the H-theorem, the capturing of the correct equilibrium state, the indifferentiability principle, and the recovery of the classical BGK model in the Newtonian limit. This is joint work with Myeong-Su Lee and Seok-Bae Yun.