The Well-posedness And Nonlinear Stability Of The Solution Of The MHD Equations

The Navier-Stokes equations are derived from the Euler equations if the viscous term is considered.Based on the Navier-Stokes equations,we get the magnetohydrody-namic(MHD)equations which describe the interaction between the viscous or inviscous,compressible or incompressible conductive fluid and the magnetic field if the magnetic field effect is considered.MHD model is applied widely in many fields such as geo-physics,astronomy and engineering,it is one of the simplest models describing the in-teraction between the conductive fluid and magnetic field.In this paper,we consider the ideal incompressible MHD flow with the Coriolis force within the infinite rotating cylinder,the criterion of stability and instability of steady state solution is obtained.Moreover,we consider the well-posedness of weak solutions to the MHD equations in a bounded domain with the Galerkin method and a priori estimate.As the initial data(u0,B0)?((Wm,p(?))2 × Wm,p(?)),the existence and uniqueness of the weak solution(u(·,t),B(·,t))?((Wm,p(?))2×Wm,p(?))are obtained,and we prove the continuous dependence of the solution on the initial data.Furthermore,we study the well-posedness of the strong solution to the N-dimensional ideal MHD equations.As the initial data(u0,B0)?((Hm(?))N ×(Hm(?))N),the existence and uniqueness of the strong solution(u(·,t),B(·,t??((Hm(?))N ×(Hm(?))N)are obtained,and we prove the continuous dependence of the solution on the initial data.