相对论Boltzmann方程解的性态研究

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。

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.