Some Problems On Compressible Magnetohydrodynamic System

The motion of electrically conducting fluids in the presence of magnetic field is described by the magnetohydrodynamic(MHD)system.The theory of MHD is widely applied in astrophysics,thermonuclear reactions and industry,among others.In this thesis,we consider the solvability to compressible MHD system of non-resistive and inviscid flows respectively.In Chapter 1,we recall some known results on compressible MHD system in multi-dimensional,planar and one-dimensional spaces respectively.In Chapter 2,we consider 1D compressible viscous non-resistive MHD system.For the isentropic case,we prove the existence,uniqueness and stability of weak solutions.Further-more,the exponential decay of solutions to the steady state is obtained in L2-norm and H1-norm respectively.In addition,for the heat-conductive case,we verify global existence and uniqueness of strong solutions and the existence,uniqueness,stability of weak solutions.In Chapter 3,we study planar compressible non-resistive MHD system.Global existence and uniqueness of strong solutions are proved for the isentropic case and heat-conductive case respectively with large initial data.In Chapter 4,we consider a 2D MHD system by assuming that the motion of fluids takes place in the plane and the magnetic field acts on the fluids only in the vertical direction.The existence of global finite energy weak solutions is established without any smallness restriction on the size of the initial data.In particular,we prove the global existence and uniqueness of strong solutions to the first level approximate system.Then we establish suitable uniform estimates and pass to the limits to recover a global weak solution for the original system.In Chapter 5,we consider the inviscid resistive compressible MHD system in 3D.Using the convex integration method,we prove the existence of infinitely many global weak solutions with general initial data.Under a special setting of fluids,we obtain similar results for the heat-conductive case.