Boltzmann方程大尺度解的整体适定性

The Boltzmann equation, which was used to describe the motion of the rarefied gas in the collisional kinetic theory, is fundamental equation in the Statistical Physics. The global existence, uniqueness and large time behavior of Boltzmann solutions have been the key problems in the area of kinetic theory and nonlinear partial differential equations. As far as we know, the global solutions of Boltzmann equation with uniqueness are abtained in the perturbation framework around global Maxwellian. Indeed, the oscillations of those solutions have to be small enough due to the nonlinear fearture of Boltzmann collisions, and thus the vacuum is initially excluded. The project of this proposal is to develop the mathematical theory of the Boltzmann equation for large amplitude initial data in bounded domains with or without large external forces.

Boltzmann方程被用来描述稀薄气体的运动规律，是统计物理中的基本方程。关于Boltzmann方程整体解的存在唯一性以及大时间渐近行为的研究一直以来都是动理学理论和非线性偏微分方程的核心课题。据申请人所知，具有唯一性的Boltzmann方程的整体解都是在扰动框架下取得的，并且初始扰动必须很小, 因此初始真空是不被允许的。本项目拟主要研究：有界区域中带或不带大外力的Boltzmann方程的大扰动解的整体存在性、唯一性以及大时间渐近行为。