Dynamics of compressible fluids near a free surface and collisional kinetic models

研究气体和流体动力学中的非线性偏微分方程及其相关应用。第一个主题是关于可压缩流体在重力驱动的移动边界附近的动力学。本课题考虑的模型包括牛顿引力理论中的欧拉方程和狭义相对论中的相对论欧拉方程，以及具有恒定粘度的双流体Navier-Stokes方程。重点是研究重要的物理现象，如引力坍缩和瑞利-泰勒不稳定性。第二个主题涉及碰撞动力学模型的定性行为，如玻尔兹曼方程和福克-普朗克方程。研究了在电磁场作用下，当平均自由程足够小时，以磁流体动力学为重点的玻尔兹曼方程的渐近行为。在有界域上研究了Fokker-Planck碰撞驱动的动力学。可压缩流体和气体是自然界中随处可见的常见物体。可压缩流体的运动边界问题和动力气体的渐近极限问题的研究具有重要的意义，因为它们在数学科学和工程中有着丰富的应用，以及它们背后的迷人的数学。研究自由表面附近流体的动力学和电磁场平衡中尺度碰撞的动力学，旨在推进这一基础数学领域的知识。对于物理学家、其他科学家和工程师来说，这个项目所解决的问题也非常有趣。研究结果为天体物理学、等离子体物理学、空气动力学、计算物理学、化学和生物学等其他学科提供了一些基础证据，并有望引领科技进步。

JangDMS-1212142 The investigator studies nonlinear partial differential equations arising in gas and fluid dynamics and related applications. The first topic regards the dynamics of compressible fluids near a moving boundary driven by gravity. The models considered in this topic include Euler equations in Newtonian gravity theory and relativistic Euler equations in special relativity, and two-fluid Navier-Stokes equations with constant viscosity. The focus is on the investigation of important physical phenomena such as gravitational collapse and Rayleigh-Taylor instability. The second topic concerns the qualitative behavior of collisional kinetic models such as Boltzmann equations and Fokker-Planck equations. The asymptotic behavior of Boltzmann equations in the presence of electromagnetic fields with focus on magnetohydrodynamic regimes is investigated when the mean free path is sufficiently small. Also the dynamics driven by Fokker-Planck collisions are studied on a bounded domain. Compressible fluids and gases are the common objects found everywhere in nature. The study of moving boundary problems for compressible fluids and the asymptotic limit of kinetic gases is very important because of their rich applications in mathematical sciences and engineering as well as the fascinating mathematics behind them. The investigation of the dynamics of fluids near a free surface and the dynamics of mesoscale collisions balanced with electromagnetic fields aims to advance knowledge in this fundamental area of mathematics. The problems addressed in this project are also very interesting to physicists, other scientists, and engineers. The results provide some foundational evidences to other disciplines such as astrophysics, plasma physics, aerodynamics, and computational physics, chemistry, and biology, and is expected to lead to scientific and technological advances.