The Study On The Boundary Layer Problem Of The Incompressible MHD Equations

Magnetohydrodynamics(magnetohydrodynamics,MHD)system is a physical model describing the interaction between conductive fluids and electromagnetic fields.In mathematics,it is a strong coupling group of hydrodynamic equations and magnetic field equations.In this paper,by using the asymptotic matched expansion method,the classical energy method,the truncation function method,embedding theorem and the continuum theory,etc,we study the boundary layer problem for the following four cases:the MHD equations with the magnetic field having a non-characteristic perfect conducting wall boundary condition,the MHD equations with the non-characteristic Dirichlet boundary condition for the velocity field and magnetic field,the anisotropic MHD equations with the characteristic Dirichlet boundary condition and the MHD equations in the plane-parallel channel.In Chapter 1,we mainly introduce the physical background,the brief introduction of the model,the research progress and the main content.In Chapter 2,we study the boundary layer problem of the MHD equations with the characteristic Dirichlet boundary condition for the velocity field and the non-characteristic perfect conducting wall boundary condition for the magnetic field,when the space domain is given ?=T~2×[0,h].Since there are two boundaries in the vertical direction,the system exists two boundary layers.Using the multi-scale analysis and the asymptotic matched expansion method,we can obtain the boundary layer equations at order-1 and order 0,and the inner equations at order 0 and order 1,and then utilize the boundary layer functions and inner functions which have been obtained to construct the approximate solutions.At last,with the help of the elaborate energy method,we can get the large time convergence estimate of the approximate solutions as the viscosity coefficient and the magnetic diffusion coefficient tend to zero.In Chapter 3,we consider the boundary layer problem of the viscous incompressible MHD equations with the non-characteristic Dirichlet boundary condition for both the velocity field and the magnetic field in three-dimensional space.Although the viscous and diffuse MHD equations has two boundaries in the vertical direction,but only one boundary layer exists,due to the fact that the viscous equations and the ideal equations differ in only one boundary condition by con-sidering the well-posedness of the ideal MHD equations.By using the asymptotic matched expansion method,the energy method and the continuum theory,we can obtain the short time convergence estimate of the approximate solutions when the viscosity coefficient and the magnetic diffusion coefficient tend to zero.In chapter 4,we study the boundary layer problem of the anisotropic incompressible MHD equations.We mainly consider the asymptotic behavior of the solutions for the three-dimensional MHD equations which have the different viscosity coefficients and magnetic diffusion coefficients in horizontal directions and in the vertical direction and has been endowed with the characteristic Dirichlet boundary condition for both the velocity field and the magnetic field,when the vertical viscosity coefficient and the magnetic diffusion coefficient tend to zero.We will find that the fixed horizontal viscosity coefficient and magnetic diffusion coefficient play an essential role to prevent the instability of the boundary layer.In chapter 5,we study the boundary layer problem for the three-dimensional incompressible MHD equations with the viscosity coefficient and magnetic diffusion coefficient being different.We consider the plane-parallel channel solutions for the viscous and diffuse MHD equations,and then this can formally change the three-dimensional MHD equations to the two-dimensional MHD equations,which can effectively avoid the singularity problem.When the viscosity coefficient?magnetic diffusion coefficient and the thickness of the boundary layer satisfy a certain relationship,we can finally obtain the L~?norm estimate for the error function.