Global Conservative Solutions Of A Modified Two-component Camassa-holm Equation

This paper is designed to consider the global conservative solutions for the Cauchy problem of a modified two-component Camassa-Holm system. The paper consists of five chapters.The first chapter is concerned with the background of shallow water equation. Moreover, the author introduces some known results of two-component Camassa-Holm system.By using the basic theory of ordinary differential equations, two-component Camassa-Holm system can be rewritten into convolution style, and the conservation law is obtained. The main results and the preliminaries of this dissertation are given in the second chapter.By introducing some independent and dependent variables, the original system is rewritten as a semilinear system in the third chapter.The forth chapter is mainly divided into two parts. The local well-posedness for the semilinear system is obtained as fixed points of a contractive transformation in the first half part. Then local solutions can be extended to global solutions by using the estimates which established in the second half part.In the fifth chapter, returning to the original variables, the author obtains the global conservative solutions of original Cauchy problem. The solutions depends continuously on the initial data.