几类复杂的动理学方程组的整体解

Some complex kinetic equations, including the Boltzmann equation as a typical model, are the basic equations of the kinetic theory and the study of their mathematical theories has been one of the hottest topics in this field. In this project, our study will concentrate on the Vlasov-Maxwell-Boltzmann equations and the Vlasov-Poisson-Boltzmann equations, for example, how to construct the global-in-time solutions to the Vlasov-Maxwell-Boltzmann equations and the Vlasov-Poisson-Boltzmann equations near Maxwellians for soft potentials and weak singularity without angular cutoff, how to construct the global- in-time solutions near vacuum to the Vlasov-Maxwell-Boltzmann equations for hard sphere model and the Vlasov-Poisson-Boltzmann equations for very soft potentials, and whether the solutions of the Vlasov-Maxwell-Boltzmann equations converge to the solutions of the Vlasov-Poisson-Boltzmann equations when the speed of the light goes to infinity.

以Boltzmann方程为典型代表的几类复杂的动理学（kinetic）方程组是动理学理论的最基本的方程组，关于它们的数学理论研究是本领域的一个热点问题。在本项目中我们将围绕Vlasov-Maxwell-Boltzmann方程组合Vlasov-Poisson-Boltzmann方程组展开研究，例如如何构造平衡态附近，非角截断假设下软势情形和弱角奇性的Vlasov-Maxwell-Boltzmann方程组和Vlasov-Poisson-Boltzmann方程组的整体经典解、如何构造真空附近硬球模型的Vlasov-Maxwell-Boltzmann方程组和非常软势情形的Vlasov-Poisson-Boltzmann方程组整体经典解以及当光速趋于无穷时，Vlasov-Maxwell-Boltzmann方程组的解是否逼近Vlasov-Poisson-Boltzmann的解。

动理学方程组主要用来描述稀薄气体的数学理论。对于带电磁场的Boltzmann方程组或者Landau方程组一直以来都是本领域的一个研究热点。我们建立了非硬球模型的VMB系统在平衡态附近的适定性理论，对于Cutoff和Noncutoff两种情形我们均给出了正面的回答和严格的证明。