动理学方程的适定性理论与渐进分析

The project is devoted to study of the well-posedness and asymptotics of the kinetic equations which can be unfolded as follows. In the first part, we will try to establish the well-posedness theory for Boltzmann and Landau equations and also for the complex systems such as, Vlasov-Possion-Boltzmann and Vlasov-Maxwell-Boltzmann systems in the vacuum regime or in the strong magnetic background. The theory will cover the local solvability of the equations, the global stability theory and the long time behavior of the solutions. Based on these results, in the second part, we will consider the numerical simulation, hydrodynamical limit and the asymptotics of the equations. In particular, the asymptotics from Boltzmann equation with angular cutoff to the equation without angular cutoff and from Boltzmann equation to Landau equation, will be investigated. We believe that it will be helpful to update the knowledge on the mechanism of micro-macro scale to the fluid and promote the development of the applied technology.

本项目的研究将围绕动理学方程（Boltzmann和Landau方程）的适定性理论和渐进分析而展开。第一，建立动理学方程（Boltzmann和Landau方程）以及复杂系统，Vlasov-Possion-Boltzmann，Vlasov-Maxwell-Boltzmann方程组，在真空附近或大背景场情形下的适定性理论，包括局部可解性理论，整体稳定性理论以及解的长时间行为。第二，研究方程的数值求解，流体力学极限以及两类渐进行为: (1)Boltzmann方程从截断情形到非截断情形的渐进过程;(2)从Boltzmann方程到Landau方程的极限过程。这些研究将深化和完善动理学方程的数学理论并提高人们对流体的宏观-微观机理的认识，推动应用技术的发展。

项目研究的主要内容在于统计物理和等离子物理中出现的动理学方程的适定性以及相应的渐近分析。项目的主要成果主要分为下面几个方面：第一，研究了Boltzmann方程在Sobolev空间的局部适定性理论，并研究齐次情形下方程弱解的唯一性以及长时间行为；第二，研究了Boltzmann方程在Coulomb位势下的平擦极限问题并在扰动框架下证明了Landau本人推导Landau方程的正当性；第三，研究了量子Boltzmann方程到Landau到半经典极限问题，证明了在所谓弱耦合意义下，Landau方程是唯一的极限方程。