Global Well-posedness For The Two-component Camassa-Holm Equation With Fractional Dissipation

In this paper,we investigate the two-component Camassa-Holm equation with fract.ional dissipation where ?? denotes the fractional dissipative operator which is defined by the Fourier transform F(??u)(?)=|?|?F(u)(?).In this paper,we use some inequalit,ies norm estimation as a tool,such as Holder inequality,Gronwall's inequality,and embed-ding inequality to estimate norm.To prove the local-posedness of this equation via the Littlewood-Paley theory and the suitable iterative scheme.Furthermore,un-der appropriate discussions,so we can give the global well-posedness of the above equation.In chapter 1,we introduce the origin and research status of relevant research work,and summarize the main work of this paper.In chapter 2,we list the preliminaries and the five important lemmas of this paper.In chapter 3,we present the local well-posedness of the equation via the fol-lowing six steps:constructing approximate solution,proving that uniform bounds,convergence,checking that the limit satisfies our equation,presenting the uniqueness and regularity of the solution.In chapter 4,we give the global well-posedness result of the two-component fractional Camassa-Holm equation with dissipative term by using the method of contradiction.In chapter 5,we summarize the main conclusions and methods used in this paper.