Research Of Well-posedness To The Compressible MHD Eqautions

We consider in this paper the compressible magnetohydrodynamics equationswhich describe compressible charged gas ?ows (plasma gas) with the adiabatic expo-nent , and and denote the density, velocity, magnetic field, pressure andthe external force at , respectively. areinitial density, velocity and magnetic field. We denote by ?? and ?? the two viscositycoe?cients of the ?uid, which are assumed to satisfy . When, the equations (0.1) are ideal compressible magnetohydrodynamics (ICMHD)equations; When the velocity satisfies div ,the equations (0.1) are incompress-ible magnetohydrodynamics (MHD) equations; When the equations (0.1) arecompressible Navier-Stokes equations, if in addition, div, the equations (0.1) areclassical incompressible Navier-Stokes equations.In this paper, we talk about the well-posedness to the compressible MHD equationsfrom three sides.Firstly, we focus on local well-posedness in critical Besov spaces for the com-pressible magnetohydrodynamics (CMHD) equations in R≥2. We mainly ap-ply Terence Tao's abstract bootstrap principle to prove existence of the solution tothe compressible MHD equations. By the Littlewood-Paley decomposition, decom-posing every equation of compressible MHD equations (0.1), then utilizing interpo-lation inequality, Young inequality, Bernstein inequality, commutator estimate, prod-uct estimate of two functions, Gronwall lemma and so on, we get for smallenough, the solutions are uniformly bounded in space , and then by using com-pact argument, the convergence of the solutions is obtained. We finally prove that uniqueness of the solution by Osgood lemma.Secondly, local well-posedness in super critical Besov spaces for the ideal com-pressible magnetohydrodynamics (ICMHD) equations in≥2 is studied, and theresult is original, since well-posedness in super critical spaces is only proved for incom-pressible Euler equation[85]. Mainly by means of Littlewood-Paley theory and Bonyparaproduct decomposition, local well-posedness in super critical Besov spaces for theideal compressible magnetohydrodynamics (ICMHD) equations is talked about. At first,applying localization operator to every equation, estimating some a priori estimates,the uniformly boundedness of the solutions in space is obtained. Then using Terence Tao's ab-stract bootstrap principle and the uniqueness of weak-strong convergence, we prove thatFinally, we consider the other side of the well-posedness, that is, the nonexistence ofglobal weak solutions to the compressible magnetohydrodynamics (CMHD) equations.Under suitable assumptions on integrability for the density, velocity and the magneticfield, if the initial datum satisfiesthen the only global weak solutions to the compressible MHD equations correspond tothe zero density and the zero magnetic field and if the initial datum satisfieswith positive nondecreasing function ??(??) on [0,∞), then the global weak solutions tothe compressible MHD equations do not exist.