Existence And Uniquence Of The Global Solutions To Prandtl Equation And Properties Of The Solutions To Some Degenerate Parabolic Equation

This paper consists of two chapters. Chapter 1 is to study the existence and uniquence of solutions to the Prandtl system. In this chapter, we assume that the velocity at outer edge of the boundary layer U(t,x) = x^mU₁(t,x), m≥1. We consider the globall existence of the solution. We first translate the initial and boundary value problem of the Prandtl system to the initial and boundary value problem for a single parabolic equation (1.19) by the Crocco change. Then we regularize the problem (1.19) and to do some prior estimates to the solution of the regularized problem. Lastly, we proved the existence of the global solution and study the uniqueness of solution.The second part in this paper is to study the limit behaviour of the solution to some degenerate parabolic equation with shift element i. We prove that the limit (ε→0) of the solution to the problem with shift element is the solution to the degenerate parabolic equation without shift element. Firstly, the degenerate parabolic with shift element is regularized. Secondly, some prior estimate of solutions to the regularized problem are obtained. Lastly, the character of the limit (ε→0) of the solution is proved.The third part in this paper is to study the properties to some degenerate parabolic equation not in divergence form with strongly nonlinear soures.The first section is about the initial-boundary problem to this equation with bounded set. The second section is about some properties of the solution to the initial problem with unbounded set.