Global stability of kinetic equations with large-oscillation initial data

Abstract

In this thesis, we focus on nonlinear partial differential equations of various kinetic models. In kinetic theory, we analyze a system with a large number of interacting particles. Kinetic theory is one of the major mathematical tools to study motions of interacting particle systems such as gas, fluid, plasma, etc. Because there are too many particles (∼ 10^23) in general, it is practically impossible to describe velocity and position of each particle. Instead, the motion of the system is described by partial differential equations in terms of a probability distribution function defined on time, space, and velocity with assumptions of continuum physics. There are various models regarding particle interactions. Among them, the Boltzmann equation is one of the fundamental kinetic models for collisional particle systems. Additionally, there is a relaxation model of the Boltzmann equation known as the Bhatnagar–Gross–Krook (BGK) model. The BGK model, an alternative to Boltzmann equation, is useful for the numerical simulation of various kinetic models. Although there has been significant progress in kinetic equations, especially the Boltzmann equation, some questions have remained open. My thesis has been de- voted to two questions about kinetic models. The first question concerns the Cauchy problem: it aims to prove the existence and uniqueness of a solution when initial data are given. The second question is the asymptotic behavior of the solution, whether it reaches an equilibrium state. In [27, 36, 41], boundary value problems of the Boltzmann equation have been solved under the assumption that the initial condition is very close to an equilibrium state. However, the assumption is very specific and not general. Thus, we try to relax this assumption by imposing smallness of initial relative entropy. Despite the assumption of a small initial relative entropy, there are examples where the difference between the initial data and equilibrium state could be large. We refer to these Cauchy problems as large-amplitude problems. We firstly study the large-amplitude problem of the Boltzmann equation. More- over, we consider both hard potential and soft potential cases. For the hard potential case, we are concerned the initial-boundary value problem on the Boltzmann equation with specular reflection boundary condition in a C^3 uniformly convex bounded domain. For the soft potential case, we solve the Boltzmann equation in torus domain T^3 for given initial data. Next, we study the asymptotic stability of the BGK model with large-amplitude initial data. In all cases, our aim is to prove the well-posedness of solutions and convergence to the equilibrium state when the initial data are not uniformly close to the equilibrium state but the relative entropy is sufficiently small.