The Investigation On The Solutions To The Compressible Two-phase Model With Magnetic Field

This thesis is concerned with a compressible isentropic two-phase model with magnetic field which is called compressible two-phase MHD model.This model is often used to describe the motion of a two-phase mixture under the impact of magnetic field,which has a wide range of applications in science and engineering.In this thesis,we derive the mod-el formally by taking a scaling limit from the Vlasov-Fokker-Planck/compressible MHD equation,and obtain the global well-posedness of strong solution and the time-decay estimates with small initial data in H2(R3).Besides,we obtain the global well-posedness of strong solution and its large-time behavior with small initial data but possibly large oscillations,and obtain the asymptotic behavior of a solution for the outflow problem in a half line R+.The thesis is organized as follows:·In Chapter 1,we present the background for the problem and some relevant results.In later parts of the chapter,we present our main results of the thesis and some associated challenges.·In Chapter 2,we consider the Cauchy problem for the compressible two-phase model with magnetic field in three dimensions.The global existence and uniqueness of strong solution as well as the time decay estimates in H2(R3)are obtained by introducing a new linearized system with respect to(n?-n?,n-n,P-P,u,H)for constants n? 0 and P>0,and doing some new a priori estimates in Sobolev Spaces to get the uniform upper bound of(n-n,n?-n?)in H2(R3)norm.·In Chapter 3,we consider the Cauchy problem for the model in three dimensions.Under some smallness assumptions on the initial data but possibly large oscillations,we obtain the global well-posedness of strong solution as well as its large-time behavior.Compared with Chapter 2 where the perturbation of the initial data around a trivial steady state is small in H2(R3)norm,we only need the smallness of initial energy which allows large oscillations of the initial data here.·In Chapter 4,we consider an outflow problem for the model,and obtain the asymptotic stability and convergence rates for the global solutions towards corresponding stationary solutions if the initial perturbation is small in some weighted Sobolev spaces.The proof is based on the weighted energy method.