波茲曼方程式的解的存在性與性質

Abstract

在本文中，我們將探討在有界域上各種邊界條件的線性與非線性波茲曼方程式的解。在第二章中，我們探討在硬勢假設下，在具有高斯正曲率的足夠小有界域上，在入射邊界條件下，我們證明穩態 H^1 解的存在性。在第三章中，我們進一步探討在特殊情況下，在硬球假設下，在入射邊界非線性下 W^{1,p} 穩態解的存在性。在第四章中我們研究在硬球假設下，鏡面和擴散反射邊界條件下的行為。我們證明無凸域假設下在任意短時間下高斯分布下界的存在性，並提供在非截斷假設下較弱的下界。

We investigate solutions of the Boltzmann equations, linearized and nonlinear ones, with various boundary conditions on bounded domains. In Chapter 2, we consider the incoming boundary problem for the linearized Boltzmann equation with the hard potential cutoff assumption. With the assumption of the C2 bounded domain with a positive Gaussian curvature boundary, we provide the H1 existence of stationary solutions provided that the diameter of the domain Ω is small. In Chapter 3, we consider the incoming boundary problem for the Boltzmann equation with the hard potential cutoff assumption. With some additional assumptions to the space domain, we establish a W 1,p stationary solution provided that the domain and the boundary data are small enough. In Chapter 4, we investigate the time evolutionary behavior of mild solutions on a bounded connected domain without convex assumption. For specular or diffusive reflection boundary condition, we prove that a Maxwellian lower bound in cut-off case and a weaker lower bound for non-cutoff case can generate instantly.