The 3D Navier-Stokes Equations With A Free Surface

As a thesis of literature reviews,the main content of this thesis is to summarize a part of representative research methods and results of the time-dependent three-dimensional Navier-Stokes equation,and to propose the solution of the equation under random conditions.The purpose is to provide some clear summaries and reviews for researchers who are interested in this field.This paper mainly refers to references[1](Bae H.,2011),[3](Beale J.T.,1984)and[5](Coutand D.,2002).Comparing the hypothesis,research methods and results,we select two research findings to introduce.On the basis,this thesis is going to propose a feasible direction to study the existence of solutions of three-dimensional Navier-Stokes equations in random time-dependent domain.This paper is divided into four chapters:the first chapter is the introduction,which introduces the problems and its background;the second chapter is concerned with a fluid in a three-dimensional ocean of infinite extent,and supposes that the initial surface and velocity field are sufficiently small in Sobolev spaces,there is a unique solution existing for all time.The main idea of the proof is to transform the problem to the fixing domain,and to prove the existence of the solution by constructing compression mapping,and the solution is in L2 in-time.Chapter 3 introduces the existence and uniqueness of the global-in-time solution of Navier-Stokes equation in the three-dimensional moving domain of finite depth under the condition of small initial value.As same as the previous chapter,the proof also include transforming the system of equations to a fixed domain,and the difference is that the solution is obtained by using the energy method rather than talking the Laplace transform in-time to tranform the problem into the stationary elliptic problem,and the solution is in L? in-time;the Chapter 4 mainly aims to propose a feasible solution of the stochastic Navier-Stokes equation in time-dependent domain,which is based on the summaries of the research method of the three-dimensional Navier-Stokes equation in time-dependent domain.