Some Mathematical Problems In MHD

Magnetohydrodynamics(MHD)describes the motion of conductive fluids in magnetic field,which is a coupled system of the Maxwell Equations,based on Ohm's law and Ampere's law,and the Navier-Stokes equations,describing the motion of fluids.It finds wide applications in plasma physics,astrophysics and geophysics.Due to the interaction between the flow field and the magnetic field and the influence of strong nonlinearity such as nonlinear Lorentz magnetic force,the mathematical structure of MHD system is very complicate,and many basic and important problems have not been solved so far.Thus,the mathematical study of MHD is of great theoretical significance.The main purpose of this dissertation is to study some mathematical problems in MHD from the mathematical view of point,including the frozen limit of magnetic lines(i.e.,vanishing magnetic diffusion limit),the dimension reduction of two-dimensional simplified MHD system,and the influence of magnetic Reynolds number and(vertical)background magnetic field on 3-d incompressible flows.First,we study the frozen limit of magnetic lines(i.e.,vanishing limit of magnetic diffusion)of 1-d compressible heat-conducting MHD flows.In MHD theory,when the magnetic field intensity is very high,the charged particle is restrained by the strong magnetic field,and the high-speed moving charged particle will restrain the magnetic field away,magnetic lines of force are frozen in a conducting fluid.Such a phenomenon is known physically as "frozen-in" magnetic field,which often occurs in highly conductive fluids in the universe and geophysics.From the mathematical point of view,the conductivity is inversely proportional to the magnetic diffusion,so that,the magnetic diffusion is sufficiently small in this direction.Due to the lack of dissipation mechanism of magnetic field,there are few theoretical studies on MHD system with vanishing magnetic diffusion.By making a full use of the mathematical structure of one-dimensional equations,the so-called "effective viscous flux" and the material derivative,we successfully overcome the difficulty induced by the lack of of the derivative estimates of magnetic field,and obtain some higher integrability of magnetic field which enable us to exclude presence of vacuum and to study vanishing limit of magnetic diffusion.As byproducts,we also prove the global well-posedness of strong solutions for non-resistive MHD system with large data.The second part of this dissertation is concerned with the dimension reduction of 2-d MHD equations.We consider an initial-boundary value problem of a simplified 2d model for compressible isentropic viscous and non-resistive MHD flows in the thin layer ??:=(0,1)×(0,?)with ?>0.Based on the relative entropy inequality and the regularity of the solutions of 1-d equations,we prove that as the width of thin layer??0,the weak solutions of 2-d MHD equations converge to the strong solutions of 1-d equations,provided that the relative entropy functional,satisfied by the solutions of 2-d and 1-d equations,tends to zero initially.Finally,the third part of this dissertation aims to investigate the influence of magnetic Reynolds number and background magnetic field on 3-d MHD flows.Indeed,a large number of experimental studies and numerical simulations have shown that the two-dimensionalization of 3-d MHD flows becomes more and more significant when the vertical background magnetic field is strengthened(see,for example,[1,5]).It's also worth pointing out that the two-dimensionalization of MHD flows is closely related to the two-dimensional turbulent(see,for example,[23,52]).By virtue of the time-averaging method,we consider the two-dimensionalization of incompressible MHD flows caused by the strong vertical background magnetic field in a three-dimensional periodic domain,and show that the three-dimensional MHD flow exponentially decays in time to the two-dimensional Navier-Stokes flow.