The Research On The Regularity Of Solution To The Camassa-Holm Equation

The research on the regularity of the solutions of the nonlinear partial differential equations is always the important study fields and directions of partial differential equations. In this paper, the regularity of the seconder-order Camassa-Holm equations and the two-component Camassa-Holm equations is researched by using Holder inequality,Gronwall inequality and a series of a prior estimates.The research work of this paper are mainly two parts. First part, we study the regularity of the second-order Camassa-Holm equations and prove it. All these are in chapter 3. This chapter is that the Cauchy problem of the second-order Camassa-Holm equation is converted into system of equations together with a hyperbolic equation and an elliptic equation. The existence of local weak solution to the Cauchy problem of the second-order Camassa-Holm equations is obtained by the small viscosity method. Then the regularity of solution of the second-order Camassa-Holm equations is studied by a series of a prior estimate. The regularity of solution of two-component Camassa-Holm equations is studied by a series of a prior estimate in chapter 4.