Well-posedness Of Compressible Navier-stokes System And Its Related Models

The thesis is focused on the well-posedness of the compressible Navier-Stokes equations,the Magnetohydrdynamics equations coupled with electromagnetic field,and the radiation hydrodynamic equations coupled with radiation field.First,for the two-dimensional non-isentropic Navier-Stokes equations,we establish the existence of global strong solutions and large time behavior to the initial-boundary value problem.Then,we consider the Cauchy problem of three dimensional Magnetohydrdynamics equations and provide a more general sufficient condition for the existence of classical solutions.Finally,we study the Cauchy problem of radiation hydrodynamic equations and establish the well-posedness for the case of non-isentropic constant viscosity coefficients and isentropic density dependent viscosity coefficients,respectively.We mention that the strong and classical solutions are all carried out under the framework of finite energy.This thesis is divide into five chapters,the details of which are as follows:We start with the introductory Chapter 1,where the physical background of different models are introduced and some related mathematical results are given.Then,we present our main results.Some preliminaries will be given at the end of this chapter.Chapter 2 is devoted to the existence theory of global strong solutions for two-dimensional non-isentropic compressible Navier-Stokes equations.When the viscosity coefficientis suitably large,we prove the existence of global strong solutions containing vacuum and establish the corresponding large time behavior.First,we give a particular blow-up criterion in terms of the density and the temperature.Then the global strong solution can be achieved by using the contradiction argument.The main difficulty is to obtain the upper bound of density.Since the effective viscous flux has no boundary condition in this case.The good regularity of effective viscous flux can not be used here.Therefore,we need to make full use of the compatibility condition,the boundary condition and the dissipation property of velocity to overcome this difficulty.In addition,based on the uniform a priori estimates,we can also get the exponential decay rate of strong solutions.In Chapter 3,we are concerned with the Cauchy problem of three dimensional isentropic compressible Magnetohydrdynamics equations.Li-Xu-Zhang[58]and Liu[72]proved the global well-posedness of classical solutions under the assumptions of small initial energy and small combination of initial density and³norm of initial magnetic field,respectively.We establish the global existence of classical solutions provided that thenorm of initial density and thenorm of initial magnetic field are suitably small for some?[1,+?],?[2,+?).This result can be viewed as generalization of Li-Xu-Zhang[58]and Liu[72].The more refined estimates are used to deal with the difficulties caused by the coupling terms and the nonlinear terms.Chapter 4 studies the Cauchy problem of three dimensional non-isentropic radiation hydro-dynamic equations with the constant viscosity coefficients.It is difficult to study the radiation hydrodynamic equations because the coupled photon transport equation is hyperbolic.In order to compensate the lack of positive lower bound of density caused by vacuum,we need to add the compatibility condition of initial data.We prove the existence and uniqueness of local strong solutions in homogeneous Sobolev space for large initial data.At the same time,we establish a blow up criterion which depends only on the fluid field and the gradient of density but not on the radiation field.In Chapter 5,the local existence of classical solutions for the three dimensional isentropic radiation hydrodynamic equations with density dependent viscosity coefficients is investigated.Compared with the constant viscosity coefficients case,the main difficulties come from the degeneracy of time evolution of velocity and the degeneracy of elliptic operator.In addition,the strong coupling of radiation intensity,velocity and density will also bring difficulties.Therefore,we need to deeply explore the intrinsic special structure of system to obtain the uniform a priori estimates.It should be noted that our work is carried out in the inhomogeneous Sobolev space,and the initial compatibility conditions are no longer needed.