Analysis and applications of nonlinear partial differential equations in conservation laws and kinetic theories

主要研究流体力学和动力学理论中某些非线性偏微分方程组的数学问题，如可压缩相对论的Navier-Stokes方程、无磁扩散系数的磁流体动力学(MHD)方程、可压缩粘弹性流动和Vlasov-Maxwell-Boltzmann方程。这些方程模拟了各种物理过程，需要显著的新的数学方法，从数学和物理的角度来看，理解这些模型的性质都是一个根本的挑战。研究方案由四部分组成：(1)第一部分讨论相对论Navier-Stokes方程弱解的整体存在性和弱解的一些定性性质。紧致性理论和动力学公式。(2)第二部分讨论了无磁扩散的磁流体力学整体强解的存在性。第三部分考虑了可压缩粘弹性流体的不可压缩极限和二维定常可压缩粘弹性流体的弱解。(4)第四部分研究了Vlasov-Maxwell-Boltzmann方程的重整化解的整体存在性及其流体动力学极限。本课题涉及的数学问题涉及等离子体物理、弹性动力学、天体物理、流体动力学以及生物和化学反应动力学等多个学科。数学问题，如整体存在、不可压缩极限和流体动力学极限，在生物学、工程学和物理学中具有重要意义。这项研究计划的主要目标是提供基本模型的数学验证，找到这些模型之间的联系，并调查分析性质，这些性质将对流体的数学产生新的见解，对等离子体物理、弹性动力学和其他物理模型至关重要，这可以促进对相关物理现象的更好理解。

The principal investigator will study mathematical problems of certain nonlinear partial differential equations in fluid dynamics and the kinetic theory, such as the compressible relativistic Navier-Stokes equation, the magnetohydrodynamics (MHD) equations without the magnetic diffusivity, the compressible viscoelastic flows, and the Vlasov-Maxwell-Boltzmann equations. These equations model a variety of physical processes requiring significantly new mathematical approaches, and understanding of properties of these models is a fundamental challenge both from the mathematical and physical viewpoints. The research program consists of four parts:(1) The first part concerns the global existence of weak solutions of the relativistic Navier-Stokes equation and some qualitative properties of weak solutions. Compactness theory and the kinetic formulation will be explored.(2) The second part regards the global existence of strong solutions to the magnetohydrodynamics without magnetic diffusivity. The decay of the component of the magnetic field which is parallel to the equilibrium will be fully developed.(3) The third part considers the incompressible limit of the compressible viscoelastic fluids and the weak solutions to the two dimensional steady compressible viscoelastic fluids. The oscillation of the asymptotic solutions will be exploited.(4) The fourth part studies the Vlasov-Maxwell-Boltzmann equations.Topics that to be addressed include the global existence of the renormalized solution and its hydrodynamic limit.The mathematical problems to be investigated in the project arise in many scientific disciplines including plasma physics, elastodynamics, astrophysics, fluid dynamics, and the dynamics of biological and chemical reactions. The mathematical issues, such as global existence, incompressible limits, and hydrodynamic limits are of fundamental importance in biology, engineering, and physics. The primary goal of this research program is to provide mathematical verification of the fundamental models, to find a connection among these models, and to investigate analytical properties that will yield new insight into the mathematics of fluids, critical for plasma physics, elastodynamics, and other physical models, which could facilitate better understanding of the relevant physical phenomena.