Boltzmann方程初边值问题的研究

Boundary effects plays a crucial role in the dynamics of dilute gases governed by the Boltzmann equation which also brings much mathematical difficulty. Based on the recent L2-L infinity method developed by Y. Guo, we consider the application of energy method to the mathematical study of initial boundary value problem of Boltzmann equation. In this project, we focus on the following three topics: (1)The well-posedness of the Boltzmann equation with soft potentials in exterior domains, especially in the case of isothermal boundary. (2)Non-trivial profile (with boundary layer) solutions in the half space of Boltzmann equation.(3)Hydrodynamic limits of the Boltzmann equation in domains with boundary condition.

带边界区域上的Boltzmann方程的研究一直是流体动理学领域的重点和难点问题，本项目拟在最近由郭岩老师发展起来的L2-L无穷能量方法的基础上，来讨论能量方法在处理Boltzmann方程的边界层问题中的应用。本项目的研究目标主要集中在以下三个方面：（1）带软势情形下的Boltzmann方程在外区域上的适定性问题（2）Boltzmann方程在半空间上的带边界层的非平凡profile解的构造（3）带边区域上的流体动力学极限问题。

本项目的主要研究内容包括以下几个方面：（1）考虑带Navier-slip边界条件下的等熵Navier-Stokes方程的不可压极限，（2）在整体 Maxwellian 附近带角截断的单个粒子 Vlasov‐Poisson‐Boltzmann 方程组Cauchy问题整体解的存在性，（3）对一类介观的self-organized动理学模型（SOK）到其宏观的动力学模型（SOH）的流体动力学极限我们进行了数学上的严格的论证。本项目的研究成果目前发表在包括Journal of Functional Analysis，SIAM Journal on Mathematical Analysis，Kinetic and Related Models等杂志，得到国内国际同行的引用。