Researches On The Well-posedness For Radiative Fluid Model And Two-phase Fluid Model

The nonlinear partial differential equations in the fluid mechanics are some important models describing a motion of fluid flows,such as a combustion model of a viscous reactive and radiative fluid,a radiation hydrodynamics model,a two-phase fluid model and so on.These models basically consist of the Navier-Stokes equations coupling with the other equation.As is well known,the Navier-Stokes equations,as a most basic and typical system,lie in the hot research areas in mathematics and physics at home and abroad,especially,the study of the definite solution problem.This dissertation studies the well-posedness of some classes of dynamical models including a combustion model of a viscous reactive and radiative fluid,infrarelativistic model,Navier-Stokes-Allen-Cahn equations.It is also obtained some meaningful results.In this dissertation,we mainly study the following problems.(1)We study the wellposedness and long-time behavior of a spherically symmetric solution for n dimensional combustion model of a viscous reactive and radiative gas.Under the condition of the initial specific volume v0 satisfying v0 ? L-1 L0v0(x)dx ??0,we establish the global existence and exponential stability of a spherically symmetry solution in Hi(i = 1,2,4).The novelties are as follows.(i)We establish the suitable expression on the specific volume and obtain its uniform upper and lower bounds by the delicate estimates and the embedding theorem.(ii)By means of the embedding theorem and the delicate interpolation technique,we establish the uniform upper and lower bounds of the absolute temperature ?,and then overcome the difficulties caused by the nonlinear terms on ? in pressure P,internal energy e and heat flow Q,and thus obtain the global regularity of solution.(2)We study the wellposedness and asymptotic behavior of a spherically or cylindrically symmetric solution in Hi(i = 1,2,4)for the compressible infrarelativistic model.Under the constitute assumptions of internal energy e,pressure P,absorbing coefficient ?a,scattering coefficient ?s and Planck function B,we establish the global existence and asymptotic behavior of a spherically symmetry solution for n dimensional model,a cylindrically symmetry solution for three dimensional model,respectively.The novelties are as follows.(i)We first establish the uniform upper and lower bounds for the specific volume and the absolute temperature,respectively.(ii)We obtain the expression of the radiative density I and then establish the desired estimates.(iii)In the cylindrically symmetric case,we remove the assumption on the smallness of initial data and improve the previous related results.In addition,in obtaining the asymptotic behavior,we use the important analysis inequality(see Lemma 1.10).(3)We study the existence and uniqueness of the local classical solution for the Cauchy problem of the 3D compressible Navier-Stokes-Allen-Cahn equations and the regularity for the initial boundary value problem of the 3D incompressible densitydependent Navier-Stokes-Allen-Cahn equations,respectively.The novelties are as follows.(i)By virtue of the linearized method and the successive approximate method,we prove the existence of the local strong solution for the Cauchy problem,and then,by the smooth effects or regularity of solution,we prove that the solution is classical.(ii)We give a regularity criterion for local strong solutions for the initial boundary value problem and refine the blow-up criterion in [133].