流體力學

The aim of this course is to discuss the theory of Stationary Navier-Stokes equation. More precisely, we will study the so-called obstacle problem as posted for solutions to the Stationary Navier-Stokes equation in the plane. In the obstacle problem for stationary Navier-Stokes flows in the plane, a cricular shape obstacle is given at the middle of the plane. In the far range, we prescribed the uniform velocity to which the stationary Navier-Stokes flow should approach in the far range. Under this setting, we seek for a mathematical solution the stationary Navier-Stokes equation in the exterior domain which is outside of the obstacle. The solution which we are interested in should satisfy the no-slip boundary condition along the boundary of the obstacle, yet at the same time should converge to the prescribed uniform velocity as the limiting profile. This was a difficult mathematical problem, and significant progress was made by Charles Amick in the 1980's. The first aim of this course is to give a detailed exposition of the classical paper by Amick. We will also explore some recent breakthroughs on this problem by Mikhail V. Korobkov.

A student taking this course should be familiar with basics of undergraduate real analysis, the techniques of vector calculus, basics of linear algebras. A student should know the basics of functional analysis and the theory of Lebesgue measure theory.

The course grade will be based on a final written and oral report by the student on a selected topic on his own.