几类动理学方程的解的性态研究

Kinetic equations, which include the Boltzmann equation as a typical model, are the cornerstone of the kinetic theory of diluted gases and the study of their mathematical theories has been one of the hottest topics in the field of nonlinear partial differential equations. In this project we will focus on the following problems: global wellposedness of the Cauchy problem of some complex kinetic equations, such as one-species Vlasov-Maxwell-Boltzmann system for cutoff intermolecular interactions etc., near Maxwellians in the perturbative framework; rigorously mathematical justification of the global Hilbert and Chapman-Enskog expansions of some kinetic equations for the case when the resulting compressible Euler type equations admit non-trivial solutions; and global existence of measured-valued solutions, especially for solutions with infinite energy, to space homogeneous Boltzmann equation and the precise description of their large time behaviors.. .Because these problems not only have strong physical background, but also contain challenging mathematical difficulties, the study on them are in the frontier of the research in the field of mathematical theories on kinetic equations. We expect that progress in solving these problems will not only enrich the mathematical theories in these areas, but also shed some light on the explanations of the related physical phenomena.

以Boltzmann方程为典型特例的动理学方程是稀薄气体动理学理论的基石，关于它们数学理论的研究一直是非线性偏微分方程研究领域的一个焦点。本项目拟在项目组成员前期研究工作的基础上，主要围绕扰动框架下某些复杂的动理学方程，例如带角截断的单个粒子的Vlasov-Maxwell-Boltzmann方程组等，初值问题在一个给定的整体Maxwell分布附近的适定性理论、当相应的可压缩Euler-型的方程组的解不再是平凡解时，一些动理学方程Hilbert展开和Chapman-Enskog展开关于时间变元的整体有效性的严格数学证明和空间齐次的Boltzmann方程初值问题整体测度值解，特别是能量无限整体解的构造以及其大时间渐近行为的精细刻画等问题开展研究。这些问题是本领域国内外数学工作者所关注的焦点问题，对它们的研究将不仅丰富动理学方程的数学理论，而且有助于理解一些相关的物理现象。

本项目主要研究动理学方程及其相关宏观模型的一些数学理论，所取得的主要研究进展包括：弱角奇性情形下或带强背景磁场的Vlasov-Maxwell-Boltzmann方程组在扰动框架下的整体适定性理论、一类大初始扰动下输运系数为正常数或者依赖于温度和密度情形下一维可压缩Navier-Stokes方程组Cauchy问题粘性激波的非线性稳定性、不考虑热传导效应的一维可压缩Navier-Stokes方程Cauchy问题稀疏波的非线性稳定性、一类大初始扰动下一维可压缩Navier-Stokes方程组内流问题粘性激波的非线性稳定性以及高维Burgers方程外区域问题球对称静态解的存在性和稳定性等。