The Low Mach Number Limit For Compressible MHD Equations

This thesis is devoted to the low Mach limit of compressible magnetohydrodynamics (MHD)equations. Namely, the compressible MHD equations converge to the incompressible MHDequations as the low Mach number tends to zero. The MHD equations are concerned with theinteraction between magnetic fields and fluids conductors of electricity. The applications of MHDcovers a very wide range of physical objects, from liquid metals to cosmic plasmas, for example,the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics,high-speed aerodynamics, and plasma physics. The thesis is divided into five chapters.In the first chapter, we first introduce the MHD equations. Then, we give the physicalbackground and the recent development for this model. To justify the asymptotic processrigorously, there are some technical difficulties which need to be overcome. Finally, the mainresults are given.In the second chapter, we briefly introduce the Littlewood-Paley decomposition and Besovspaces as well as some useful inequalities related to Besov spaces.In the third chapter, we generalize a continuation principle for generally hyperbolic singularlimit problems in more general Besov spaces such that it both holds true in the usual Sobolevspaces with high regularity and the Besov spaces with critical regularity.In the fourth chapter, from the error equations, with the aid of the continuation principle, wegive the rigorous proof of main results by using the error energy methods.In the fifth chapter, we present the further prospect on the problem.