Boltzmann型方程的流体动力学极限问题的研究

As an important link between the molecular dynamics and fluid dynamics in physics, hydrodynamic limits of the Boltzmann equation has always been one of the hottest topics that many scholars are concerned about. In this project, we will focus on the following two topics: (1) Global fluid dynamical limits of the Valsov-Poisson-Boltzmann to the level of the compressible Euler-Poisson equation in the whole space or a periodic box; (2) Global hydrodynamic limits of the Boltzmann-type equation to the level of the compressible Euler-type equation with boundary condition.

Boltzmann方程作为联系物理学中分子动力学和流体动力学理论的重要桥梁，其流体动力学极限问题一直以来都是许多学者所关注的焦点问题之一。在本项目中，我们拟研究以下两个方面的内容：（1）在全空间或者周期区域中，Vlasov-Poisson-Boltzmann方程到相应的Euler-Poisson方程全局极限的问题；（2）带边界条件的动理学方程（Boltzmann型方程）到相应的流体动力学方程（可压Euler型方程）全局极限的问题。

动理学方程在稀薄气体动力学、等离子体等复杂的物理现象研究中有着广泛的应用，其定解问题的适定性研究一直是一个热点问题。在本项目中我们主要得到了两类动理学方程--非常软势情形下带角截断和摩擦外力项的Boltzmann方程及弱角奇性条件下Vlasov-Poisson-Boltzmann方程的柯西问题在平衡态附近的解的整体存在性。