Some Studies On The Correlation Properties Of Some Equations Coupled With Viscous Fluid Mechanics Equations

Viscous hydrodynamics is the branch of hydrodynamics,mainly studying the macro-scopic movement law of viscous fluid.In 1827,Navier,who promoted the development of viscous hydrodynamics theory,added the viscosity term to the Euler equation,which there-after studied by Cauchy,Poisson et al..Finally,Stokes proposed the momentum equation of viscous fluid(Navier-Stokes equation),and established the basic equation of viscous hydro-dynamics.The development of the relevant models coupled with the viscous hydrodynamic equation has attracted more and more scholar's attention.In Chapter 1.the theoretical background and research progress of the non-isothermal model for compressible nematic liquid crystals,the non-isothermal model for compressible nematic liquid crystals with potential forces,the viscous hydrodynamic equation with the Ko-rteweg stress tensor,and the MHD model for the compressible fluid interface are introduced.In addition,the main results of the paper are presented.In Chapter 2,the Cauchy problem for the three-dimensional non-isothermal model for compressible nematic liquid crystals is considered.Existence of global-in-time smooth so-lutions is established provided that the initial datum is close to a steady state(?,0,d,?).By using the Lq-Lp estimates and the Fourier splitting method,if the initial perturbation are small in H3-norm and bounded in Lq(q E[1,6/5))norm,the optimal decay rates for the first and second order spatial derivatives of solutions are obtained.In addition,the optimal decay rates for the third and fourth order spatial derivatives of director field d in L2-norm are achieved.In Chapter 3,the Cauchy problem for the three-dimensional non-isothermal model for compressible nematic liquid crystals with external potential force is considered.Under the smallness conditions of the initial perturbation of the stationary solution and the potential force in some Sobolev norms,by using the Lq-Lp estimates for the linearized equations and the energy method,the optimal decay rates for the first order spatial derivatives of smooth solutions are obtained if the initial perturbation of the stationary solution is bounded in Lq(q ?[1,6/5))norm.In addition,the optimal decay rate for the second order spatial deriva-tives of director field d in L2-norm is achieved by virtue of the Fourier splitting method.In Chapter 4.the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction ?2?((?(?)xx?'(?))x with ?(?)= ?? are concerned.The model consists of viscous stabilizations due to quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions.The diffusion term ?uxx in the momentum equation may be interpreted as a classical conservative friction term due to particle interactions.The existence result in[78](?=1/2)is extended to 0<? ?1.In addition,the limit ??0 with respect to 0<a ? 1/2 is performed.In Chapter 5,a diffuse interface between compressible fluids is concerned.By entropy estimates and the standard compactness argument,the compactness of weak solutions to the 1D magnetohydrodynamic model for compressible fluids interface with vacuum is investigat-ed.