Existence Of Solutions Of The Viscous Incompressible Fluid In Bounded Domain

In this paper,in the first part,we mainly discuss the existence of the weak solutions of the viscous incompressible MHD fluid in bounded domain.In a general bounded domain with multiply connected boundary,the problem of existence of solutions of the viscous incompressible flow is among the most outstanding and still open questions in the whole theory.Usually,for flow in bounded regions,the driving mechaninsm is due either to the movement of part of the boundary,as in the flow between two rotating,concentric sphere,or to the injection and removal of liquid through the permeable part of the boundary,as in the case of a flow in a region with a finite number of "sources" and "sinks." For this kind of problem,we try to solve the problem in two directions.First,in order to use the fixed point theory,we try to construct the compression map;second,the use of blow-up proceed to obtain contradictions.Here,we study the problem from the second direction.In the second part,we will discuss the regularity problem of Navier-Stokes equations.This thesis is divided into five chapters.The first chapter is the introduction,we introduce the background of MHD and some partial regularity results of Navier-Stokes equations;the second chapter is to to prove the existence of MHD in a class of admissible domain;the third chapter is to state the results of MHD in general bounded domain;the forth chapter is to discuss some improvement on the Navier-Stokes equations,the last chapter,we give a briefly proof of Morse-Sard theorem on Sobolev space.More details in the following:In Chapter 1,we introduce the physical backgrounds,and main developments about our researches,and some notations.In Chapter 2,we mainly prove the existence of MHD in a class of admissible domain;In Chapter 3,we investigate the flux problem of MHD in bounded domain with multi-component boundary as follows.(?)Chapter 4,we come back to the regularity problem of Navier-Stokes equations.Chapter 5,we will give a brief proof of Morse-Sard theorem.