An Introduction of Compressible Navier-Stokes Equation
课程摘要：The incompressible Navier-Stokes Equation (NSE) is the fundamental equation to describe the motion of the incompressible, viscous fluid from physics. It is used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. Coupled with Maxwell's equation and Fokker-Planck equation, it can be used to study magnetohydrodynamics (MHD) and complex fluid flows including viscoelastic fluids respectively. NSEs are also of great interests from mathematical viewpoints, in which many fundamental questions are yet to be understood. The outline of the course may consist of: The linear Stokes equation: Basic theory behind both steady and dynamic cases. The steady Navier-Stokes equation: Existence, regularity, and stability. The dynamic Navier-Stokes equation: Ladyzhenskya's theory in dimension two, Leray and Hopf's theory in dimension three, Serrin-Prodi's regularity and uniqueness theorem, and Caffarelli-Kohn-Nirenberg and Escauriaza-Seregin- Sverak's regularity theories.