相对论Boltzmann方程解的性态研究

The relativistic Boltzmann equation is a basic model of dilute gas in kinetic theory. It describes the evolutive law of fast moving particles interacting through binary collisions. Because of its nonlinearity and complexity of physical phenomenon described, the research of its mathematical theory gives mathematicians many challenging problems. Therefore, it was a hot topic in our research area in recent years and it attracted many colleagues’ attention in mathematics around the world. By now, there have been a lot of progresses in its mathematic theory: global existence of solutions near vacuum and their L^1 stability, the construction of global solutions around given global Maxwell distribution, decay estimates, hydrodynamic limits from relativistic Boltzmann equation to relativistic Euler and so on. However, rigorous mathematic theory of many basic problems such as the nonlinear stability of fundamental waves for the relativistic Boltzmann equation and the hydrodynamic limit call for further investigation. This program will discuss the above problems. We plan to study the nonlinear stability of fundamental waves of relativistic Boltzmann equation and the relativistic Navier-Stokes approximation of the relativistic Boltzmann equation.

相对论Boltzmann方程是稀薄气体动理学理论(kinetic theory)中的一个基本模型，它描述了高速运动的稀薄气体在气体分子之间两体碰撞作用下其分布函数随时间演化的规律。该方程是近年来本领域的一个研究热点，吸引了众多国内外数学同行的关注。到目前为止，关于它的数学理论已经有很多进展：真空附近整体解的存在性和 L^1 稳定性，给定的整体Maxwell分布附近整体解的构造、衰减估计和从相对论 Boltzmann 方程到相对论 Euler 方程的流体动力学极限等等。虽然如此，许多基本的问题例如相对论Boltzmann方程基本波的非线性稳定性及其流体动力学极限问题严格数学理论等还有待进一步研究。本项目将围绕这些问题展开，拟研究相对论Boltzmann方程基本波的非线性稳定性和相对论Boltzmann方程的相对论可压 Navier-Stokes 逼近。

Boltzmann 型方程是稀薄气体动理学理论(kinetic theory)中的基本模型，该类方程是近年来本领域的一个研究热点，吸引了众多国内外数学同行的关注。我们研究了具有短程相互作用的相对论 Vlasov-Maxwell-Boltzmann 方程组、Vlasov-Poisson-Boltzmann 方程组(带角度截断和不带角度截断两种情形)经典解的存在性和解的各阶导数收敛率，系数依赖于温度和密度的可压缩 Navier-Stokes 方程稀疏波的非线性稳定性，动理学的 Cucker-Smale 方程解的存在性和大时间行为，以及动理学的 Cucker-Smale 方程通过摩擦力与可压缩/不可压缩 Navier-Stokes 方程耦合的耦合方程组大初值解的存在性。相应研究结果在 7 篇论文中给出，发表在 J. Func. Anal.，M3AS，J. Differential Equations 等杂志上。这些结果完善和发展了复杂 Boltzmann 型方程、Cucker-Smale 型动理学方程 经典解的存在性和大时间行为的新理论和方法，将Navier-Stokes 方程稀疏波波稳定性理论推广到系数依赖于密度和温度情形，具有比较重要的科学意义。