WEAK SOLUTIONS AND INCOMPRESSIBLE LIMITS OFMULTI-DIMENSIONAL MAGNETOHYDRODYNAMIC FLOWS

This dissertation addresses mathematical issues regarding the existence of global weak so-lutions of isentropic compressible magnetohydrodynamic flows (MHD), the limit behaviorof isentropic compressible MHD as Mach number vanishes, and the hydrodynamic limit ofVlasov-Maxwell-Boltzemann equations. More precisely, in the first part, global existenceof weak solutions with large initial data to the Cauchy problem of the three-dimensionalcompressible MHD is established through an invading method for the adiabatic exponent&Gamma > 3/2 and constant viscosity coefficients. In the second part, we focus on the connectionbetween the incompressible MHD and the compressible isentropic MHD; it is showed that asMach number vanishes, the compressible isentropic MHD will converge to the incompress-ible MHD. In the third part, using relative entropy estimate about an absolute Maxwellian,we establish an incompressible Electron-Magnetohydrodynamics-Fourier limit for solutionsof the Vlasov-Maxwell-Blotzmann equation considered over any periodic spatial domain inR^3. It is shown that any properly scaled sequence of renormalized solutions of Vlasov-Maxwell-Boltzmann equations has fluctuations that (in the weak L2 topology) converge toan in¯nitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussi-nesq relations. It is shown that every limit point and the magnetic field are governed by aweak solution of an incompressible electron-magnetohydrodynamics system for all time