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.
During the period of the award, I worked with my collaborators on different subjects as proposed in the proposal, for example the incompressible/compressible viscoelasticity, incompressible/compressible magnetohydrodynamics with zero magnetic diffusivity, and compressible liquid crystals. We primarily worked on the global existence issue for those PDE system. Results in manuscripts or publications help both mathematician and physician to understand better these PDE systems. Our main outcomes can be discribed as follows: (1) Global solutions of incompressible viscoelasticity with discontinuous initial data. In fact, this manuscript addresses the physical interesting problem when the initial data of incompressible viscoelasticity is discontinuous. Due to the hyperbolic nature of the deformation gradient, the discontinuity of solutions will develop along the flow. We find out a way to overcome difficulties arising from the possible oscillation and concentration to construct a global solution; (2) Global existence of classical solution of incompressible/compressible magnetohydrodynamics with zero magnetic diffusivity. More precisely, we find out exactly the condition which ensures a global-in-time existence of incompressible/compressible magnetohydrodynamics based on the introduction of the flow maps and the scaling structure of PDE systems. The regularity of solutions is the lowest according to the scaling; (3) Optimal decay of classical solutions to compressible viscoelasticity and incompressible liquid crystals; (4) Global existence of classical solutions to compressible liquid crystals.