The Infinite Limit Of(Magnetic) Prandtl Number For Incompressible MHD System

The incompressible magnetohydrodynamic(MHD)equation describes the mo-tion of conductive fluids in electromagnetic fields and has a very important physi-cal applications in astrophysics,geophysics,aerodynamics or plasma physics.Be-cause of the strong coupling between the magnetic field and the flow field,as well as the influence of non-linear Lorentz force,it is very challenging to study the math-ematical theory of multi-dimensional MHD equation.Similar to Navier-Stokes e-quation,the regularity and uniqueness of global weak solutions for large initial data is still an open problem in three-dimensional incompressible MHD equation.Some important parameters in MHD equation,such as Rayleigh number,Ekman number,Chandrasekhar number,Prandtl number and magnetic Prandtl number and so on(see[5,46,47,71,72,74]),characterize some important physical phenomena.Thus,it is of great physical significance to further understand the physical mechanism by study-ing its mathematical theory.The magneto-convection model is a coupling equation between the classical MHD equation and the Boussineseq equation.It describes the effects of magnetic field on Rayleigh-Benard thermal convection.Existing research results have shown that the magnetic field has a good stability effect on Rayleigh-Benard convection.In chapter 2,we mainly study the infinite limit of Prandtl number for two-dimensional incompressible magneto-convection.In this chapter,the Prantl number characterizes the ratio of momentum diffusion coefficient(viscosity coefficient)to thermal diffusion coefficient(heat conductivity coefficient).In order to prove the existence and unique-ness of the global strong solutions,the convergence rates and the initial layer width,we first dimensionless the equation(see[72,74]),and then use the fine(weighted)energy estimates(see[43,45,64,69])to obtain the global uniform estimates(inde-pendent of the Prandtl number).In order to better understand the properties of the initial layer,we construct an effective PDE system to describe the motion of the initial layer in the sense of H1-norm by using the method of formal asymptotic expansion.In the dimensionless MHD equation,the magnetic Prandtl number characterizes the ratio of momentum diffusion(viscosity coefficient)to magnetic diffusion(resis?tance coefficient).In Chapter 3,we mainly study the infinite limit of magnetic Prandtl number for three-dimensional incompressible MHD equation.Formally,when the magnetic Prandtl number tends to be infinite,the MHD equation is transformed into an elliptic-parabolic coupled nonlinear equation,which is closely related to magnet?ic relaxation in magnetohydrodynamics(see[50,51,68]).In this chapter,we first prove the existence of global weak solutions for the elliptic-parabolic system with large initial data by using Aubin-Lions weak convergence lemma;then the exponen-tial stability of the system is studied under the condition of small initial H1-norm;finaly,similar to Chapter 2,we use the weighted energy estimates and the asymptotic expansion method to study the motion of the initial layer.