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

JangDMS-1212142 研究非线性偏微分方程产生的气体和流体动力学及相关应用。 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，以及具有常粘性的双流体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，如玻尔兹曼方程和福克-普朗克方程的定性行为。 研究了在电磁场作用下，当平均自由程足够小时，Boltzmann方程组的渐近行为。 在有界区域上研究了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。 可压缩流体的动边界问题和运动气体的渐近极限问题的研究是非常重要的，因为它们在数学科学和工程中有着丰富的应用，以及它们背后迷人的数学。 对自由表面附近的流体动力学和与电磁场平衡的中尺度碰撞动力学的研究旨在推进数学这一基本领域的知识。 这个项目中解决的问题对物理学家、其他科学家和工程师来说也非常有趣。 这些结果为天体物理、等离子体物理、空气动力学、计算物理、化学和生物学等学科的发展提供了基础性依据，并有望推动科学技术的进步。

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.