The Properties Of The Boundary Layer Solution To The Shallow Water Equations In The Half Space

The shallow water equation is very important in partial differential equations. Though form is simple,it contains many valuable information. In the several years,many authors in different fields pay attention to the shallow water equation. There are some results on existence and stability of its solutions. For example:L. Sundbye has studied the Cauchy problem in the 2-D spaces,and obtained global existence and uniqueness of solutions. But there are lots of subjects to be studied on the initial-boundary problem in the half space. In this article,we construct boundary layer functions near the point x=0 and obtain that the stationary solution is convergent to the boundary layer function. In additions,we obtain the L2 local stability of the solutions with the aid of energy estimate.This article is divided into six parts .In the first part,we introduce the origin of the shallow water equations and some of its results. In the second part,we introduce the fundamental theory in the fluid mechanics. In the third part,we give the preparation knowledge. In the fourth part,we obtain the existence and boundary layer convergence of the stationary solution. In the fifth part,we obtain the local stability of the solution. In the sixth part,we sum the results in this article and give some problems to be study in the deeper step.