Existence And Decay Of Solution To The Two- Dimensional Quasi-geostrophic Equation

In this paper,the authors consider the existence and decay of the global solutions of the quasi-geostrophic equation.In the third part of the paper,the authors firstly consider the existence of the solutions by using the Galerkin approximation method the periodic case in a domain,and then give some energy estimates and employ the compactness theorem in Sobolev space to obtain the existence of the global solutions.In addition,the uniqueness of the solutions can be obtained using the Gronwall inequality.In the fourth part of the paper,the author use the Fourier splitting method to give decay estimates of the solutions.The decay rate is?e?10?t?^-1.This paper is divided into five chapters:The first chapter is an introduction and is divided into two sections.We introduce the research background of the issue,the research status at home and abroad and what I have done,respectively.The second chapter is preliminary knowledge.We introduce the basic symbols and function space,as well as inequality commonly used in the paper.The third chapter gives the definition of weak solutions and the existence of weak solutions in periodic region.By using Galerkin approximation method,the existence of weak solutions is obtained,and combines energy estimates and employ the compactness theorem in Sobolev space to the existence of the global solutions.In addition,the uniqueness of the solutions can be obtained by using the Growall inequality.The fourth chapter focuses on the proof of theorem to get decay estimates,and gives the general conclusion.The fifth chapter is the last part of this paper,we make a summary of the work and some shortcomings that appear in the paper and give the prospect of future research.