Regularity And Well-posedness Of Solutions Of Two Kinds Of Nonlinear Evolution Equations

In this thesis,we mainly study two kinds of nonlinear evolution equations which have important physical significance.One is the MagnetoHydrodynamic(MHD for short)equations and the other is the generalized Camassa-Holm equation.The MHD model describes the interaction between magnetic field and ideal conducting fluid,the generalized Camassa-Holm equation is a kind of shallow water wave equation which describes the movement of fluid in shallow water environment.For three dimensional incompressible MHD equations,we study the regularity criterion for axisymmetric solutions with nonzero swirl components.For the generalized Camassa-Holm equation,we study its local well-posedness in Sobolev space.This thesis is mainly divided into the following two parts:In the first part,we study the regularity criterion to the MHD equations;In the second part,the local well-posedness of the Cauchy problem of the generalized Camassa-Holm equation in Sobolev space is studied.By using Kato's semigroup approach,we establish the local well-posedness in Hs with s>3/2.