Boundary Problems in the Boltzmann theory

运动论及其模型，如玻尔兹曼方程和弗拉索夫方程，在理解气体动力学、等离子体物理和流体方程等问题方面发挥了重要作用。在许多物理情况下，不同模型中的粒子与边界相互作用，这种相互作用产生了一些有趣的现象，例如奇点的形成。一般来说，边界效应可能不仅停留在边界附近，而且会影响整个内部动力学。因此，边界效应非常重要，在许多情况下有着丰富的应用。然而，由于动力学模型的奇异性，边界问题在数学上具有挑战性。这个项目旨在开发新的数学工具来处理这些问题。本课题研究动力学模型和流体方程中出现的几个重要边界问题。第一个主题是关于动态和静态问题在具有几种物理边界条件的有界域上玻尔兹曼解的最优正则性。第二个主题涉及动力学模型中的边界场相互作用，如弗拉索夫-泊松-玻尔兹曼系统。第三个主题是理解稳定玻尔兹曼解与不可压缩的纳维-斯托克斯-傅立叶系统之间的关系，当平均自由程足够小时，存在边界。最后一个主题是流体-物质相互作用，如表面活性剂在粘性表面波上的动力学。该项目的结果有望提高其他科学家和工程师感兴趣的广泛问题的建模能力。

The kinetic theory and its models such as the Boltzmann equation and Vlasov equations have played an important role in the understanding of problems in gas dynamics, plasma physics, and fluid equations. In many physical situations, the particles in various models interact with boundaries, and this interaction creates several interesting phenomena such as the formation of singularities. In general, boundary effects may not stay only near the boundary but also impact on the whole interior dynamics. Therefore the boundary effects are important and have rich application in many cases. However, boundary problems in kinetic models are mathematically challenging due to their singular nature. This project aims to develop new mathematical tools to handle these problems.This research project studies several important boundary problems arising in kinetic models and fluid equations. The first topic regards optimal regularity of Boltzmann solutions in bounded domains with several physical boundary conditions for both dynamical and stationary problems. The second topic concerns the boundary-field interaction in kinetic models such as Vlasov-Poisson-Boltzmann systems. The third topic is to understand the relation between steady Boltzmann solutions and the incompressible Navier-Stokes-Fourier systems in the presence of a boundary when the mean free path is sufficiently small. The last topic concerns fluid-material interaction such as surfactant dynamics on viscous surface waves. Results of the project are expected to improve modeling capabilities for a wide range of problems of interest to other scientists and engineers.