2012-12-19       培养处

1219-21日（周三至周五），1224-28日（周一至周五）。

Outline of the course

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.

No previous knowledge of these topics will be assumed, but the participants should have some knowledge of Ordinary and Partial Differential Equations, at the level of an introductory graduate course. Open problems of various level of difficulty will be mentioned. Pending on the availability of schedule, each student may be assigned to study and present an article at the end of the mini-course.

Within this course, we aim to introduce some of the very basic issues on NSE varying from existence, uniqueness, stability to regularity. In particular, we plan to discuss the following topics:

1、The basic of fluid mechanics: Derivation and basic property of Navier-Stokes equation.

2、Model equations: Simple model equations which are more manageable than the Navier-Stokes equation can bring important insights into these problems and will be discussed at various points.

3、The linear Stokes equation: Basic theory behind both steady and dynamic cases.

4、The steady Navier-Stokes equation:  Existence, regularity, and stability.

5、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.

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