Boltzmann方程的数学理论

The Boltzmann equation is the basic equation in statistical physics and the cornerstone of the kinetic theory of diluted gases. Due to the nonlinear structure of the equations themselves and the complexities of phenomena described by them, the study on the mathematical theories of certain kinetic equations, which include the Boltzmann equation as a typical example, and related macroscopic equations, such as the Euler type equations, the Navier-Stokes type equations, etc., has provided many challenging mathematical problems to the mathematical community and is one of the hottest topic in the field of nonlinear partial differential equations. ..In this project, based on our former research on the topics mentioned above, we will focus on the following problems: Construction of measure valued solutions to the space inhomogeneous Boltzmann equation; L^∞ theory of the Boltzmann equation without angular cutoff assumption; global solvability of compressible Navier-Stokes equations with temperature dependent transport coefficients and large data; boundary layer theory of the MHD system, etc. ..The study on these problems are in the frontier of the research in the field of mathematical theories on kinetic equations and related macroscopic models because they not only have strong physical background and but also contain challenging mathematical difficulties. We expect that we can make some progress on these problems which will not only enrich the mathematical theories in these areas, but also shed some light on the explanation of the related phenomena.

Boltzmann方程是统计力学中的基本方程，稀薄气体动理学(kinetic)理论的基石。由于方程的非线性性以及所描述现象的复杂性，以其为典型特例的动理学方程以及相关的宏观模型，如Euler型方程、Navier-Stokes型方程等，的数学理论的研究给数学工作者提出了许多具有挑战性的数学问题，一直是国内外非线性偏微分方程研究领域中的一个焦点。..本项目拟在项目组成员多年来围绕这方面所开展前期研究工作的基础上，主要围绕空间非齐次Boltzmann方程测度值解的构造、不带角截断的Boltzmann方程的L^∞理论、大初始扰动下输运系数依赖于温度的可压缩Navier-Stokes型方程的整体适定性和MHD方程的边界层理论等问题开展研究。..这些问题具有很强的物理背景，数学上也有相当的难度。我们希望在这些问题的研究中取得一些有意义的进展，一方面丰富动理学方程及相关宏观模型的数学理论，另一方面还有助于理解一些相关的现象。

本项目主要研究动理学方程及相关宏观模型的数学理论，在低正则函数空间中非角截断Boltzmann方程和Landau方程的整体适定性、扰动框架下带强背景磁场的VMB方程组Cauchy问题的整体适定性及其整体VPB极限、带代数衰减初始扰动的非截断Boltzmann方程周期初边值问题的整体适定性、Boltzmann方程一致剪切流的存在性与稳定性、抛物尺度下带一般碰撞核的Boltzmann方程的不可压缩Navier-Stokes-Fourier极限、有界区域上Boltzmann方程的Chapman-Enskog展开关于时间变元的整体有效性、双粒子VMB方程的不可压缩带欧姆定律的Navier-Stokes-Maxwell方程极限、量子Boltzmann方程的Fermi-Dirac分布的流体极限、几类可压缩Navier-Stokes型方程组大初值整体解的构造和大初始扰动下基本波的非线性稳定性、相对论Euler方程的Riemann问题、双曲守恒律方程组特征间断的稳定性、理想可压缩MHD方程组自由边界问题的可解性等问题研究中取得了一系列有意义的研究进展，在Comm. Pure Appl. Math.、J. Eur. Math. Soc.、Arch. Ration. Mech. Anal.、Math. Ann.、J. Math. Pures Appl.、Rev. Mat. Iberoam.、Trans. Amer. Math. Soc.、SIAM J. Math. Anal.、J. Differential Equations、 J. Funct. Anal.、Math. Models Methods Appl. Sci.、Indiana Univ. Math. J.、Sci. China Math.等刊物上发表论文或接受发表论文58篇。