Well-Posedness Of Solutions To The Incompressible Magneto-Hydrodynamic Equations

In this paper,well-posedness of the Cauchy problems for the incompressible Magneto-hydrodynamic(MHD)equations is studied from two different viewpoints.On one hand,we discuss the well-posedness of the Cauchy problems for the MHD equations in different functional spaces,consequently,the well-posedness of the MHD equations in more general spaces is showed,and the well-posed results are improved;On the other hand,we modify the MHD equations, then do research on the well-posedness of the generalized MHD(GMHD) equations.This thesis consists of the following four chapters:In chapter 1,we introduce the backgroud and current advancement of the incompressible MHD equations,and give some important results concerned with well-posedness.In chapter 2,the existence,uniqueness and decay properties for the strong solutions of the MHD system in space PLⁿ∩PL^p,where 1＜p＜n,n≥2, are studied by the semi-group method.Chapter 3 studies the local existence and uniqueness of the mild solutions to the Cauchy problem of the MHD equations in uniformly local L^p spaces L_uloc,ρ^p(Rⁿ)for p＞n andρ＞0,by using the key L_uloc,ρ^p-L_uloc,ρ^q estimates and Banach fixed point theorem.Furthermore,we obtain maximum existence time estimates of the solutions.In chapter 4,using the fractional Laplacian-(â–³)^γ,whereγ＞0,to take the place of the Laplacian-â–³,we can modify the MHD equations.Then the existence and uniqueness of the strong solutions to the Cauchy problem of the GMHD system in the whole space Rⁿ,where n≥2,are proved by applying L^p-L^r estimates based on the semi-group S(t)= e^{-t(-â–³)γ/2}.