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

点击翻译按钮获取中文摘要

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.