Low Mach Number Limit For The Compressible MHD Equations And Well-posedness To A Two-phase Fluid Model

This thesis is concerned with the low Mach number on compressible Magnetohydro-dynamics(MHD)equations and well-posedness to a two-phase fluid model.In Chapter 1,we first give a relatively complete introduction to the background and known results of the researches in this thesis.Then we introduce some useful basic concepts,defini-tions and facts of relevant function spaces.Finally,we state our main results and give some comments on the main results.In the first part of the thesis(Chapters 2-4),we consider the low Mach number limit on compressible MHD equations.From the physical point of view,the incompressible MHD equations can be derived by taking the zero Mach limit to the corresponding compressible MHD equations.In this branch,many results have been obtained.Based on the existing literatures,we shall investigate the zero Mach limit to the compressible MHD equations more deeply.With respect to the study of low Mach number limit,one usually need to consider the following main factors:the type of the initial data("good" or "bad" initial value depending whether the oscillation appears or not),the types of defining area(domain with boundary,torus or the whole space),the framework of solution space(classical solution or weak solution),and so on.In our thesis,we mainly concentrate on the case of ill-prepared data in the framework of Besov space.In Chapter 2,we consider the convergence of the compressible isentropic magneto-hydrodynamic equations to incompressible model with ill-prepared initial data in critical Besov spaces.Under the condition that the initial data is small in some norm,we show that the convergence holds globally as the Mach number goes to zero.Moreover,we also obtain the convergence rate.In Chapter 3,we consider the local well-posedness and low Mach number limit for the multidimensional isentropic compressible viscous MHD equations in the whole spaces.First the local well-posedness of solution to the viscous MHD equations with large initial data is established.Then the low Mach number limit is studied for general large data and it is proved that the strong solution of the compressible MHD equations converges to that of the incompressible MHD equations as the Mach number tends to zero.Moreover,the convergence rates are obtained.In Chapter 4,we consider the convergence of the compressible isentropic MHD equations to the corresponding incompressible model with ill-prepared initial data in periodic domain.when the initial data is large,we prove that the compressible flow with small Mach number exists as long as the incompressible one does.Meanwhile,we obtain the convergence result for the slightly compressible solution filtered by applying the group of acoustics.In the second part of the thesis(Chapters 5),we are concerned with a kinetic-fluid model describing the evolutions of disperse two-phase flows.Kinetic-fluid models are widely used in the description of the dynamics of disperse two-phase flows.We shall focus on the model which consists of the Vlasov-Fokker-Planck equations for the particles(disperse phase)coupled with the compressible Navier-Stokes equa-tions for the fluid(fluid phase)through the friction force.We establish the global well-posedness of strong solutions in the whole space R3 when the initial data are a small perturbation of some given equilibrium.Moreover,we obtain the algebraic rates of convergence of solutions toward the equilibrium state.The global well-posedness result still holds for the periodic domain while the convergence rate is improved to be exponential.