The Well-posedness Of Some Kinds Of Hydrodynamic Equations Coupled With N-S Or Euler Equations

This paper mainly focuses on the equations coupled with Navier-Stokes or Euler sys-tem,such as magnetohydrodynamic equations,Navier-Stokes-Maxwell equations,Euler-Maxwell equations and micropolar fluids system.Under the influence of the strong mag-netic field,the motion of the liquid metals,strong electrolyte can be described by mag-netohydrodynamic equations.The dynamical behavior of a plasma is mainly governed by the interaction between the charged particles(electrons and ions)and the internal electromagnetic fields produced by the particles themselves,Euler-Maxwell and Navier-Stokes-Maxwell equations could simulate and model such the dynamical behavior of plas-mas.The micropolar fluids describe a class of microstructure related fluids,for example,animal blood,polymeric suspensions,and liquid crystals.This thesis mainly studied the well-posedness of these equations.In Chapter 3,we are concerned with the compressible magnetohydrodynamic equa-tions with Coulomb force in three-dimensional space.We show the asymptotic stability of solutions to the Cauchy problem near the non-constant equilibrium state provided that the initial perturbation is sufficiently small.We prove the global existence of classical solutions near the steady state for the large doping profile.This is the first result about stability of solutions with large doping profile.For the small doping profile,we prove the time decay rates of the solution provided that the initial perturbation belongs to Lp with 1 ?p<3/2.In Chapter 4,the compressible Navier-Stokes-Maxwell system with damping is in-vestigated in R3 and the global existence and large time behavior of solutions to the Cauchy problem are established in the present paper.We first construct the global u-nique solution under the assumptions that the H3 norm of the initial data is small,but the higher order derivatives can be arbitrarily large.If further the initial data belongs to H-s(0 ?<3/2)o B2,?-s(0<s ? 3/2),by a regularity interpolation trick,we obtain the various decay rates of the solution and its higher order derivatives.As an immediate byproduct,the Lp-L2(1 ? p ? 2)type of the decay rates follow without requiring that the Lp norm of initial data is small.In Chapter 5,we consider the global existence and large time behavior of solutions to the Cauchy problem near a constant equilibrium state to the bipolar non-isentropic compressible Euler-Maxwell system in R3,where the background magnetic field could be non-zero.By the combination of the local existence and the a priori estimates via a standard continuity argument,the global existence is established under the assumption that the H3 norm of the initial data is small,but its higher order derivatives could be large.Combining the negative Sobolev(or Besov)estimates with the interpolation estimates,we prove the optimal time decay rates of the solution and its higher order spatial derivatives.In Chapter 6,we are concerned with the compressible micropolar fluids system in three-dimensional space.We consider the asymptotic behavior of the solution to the Cauchy problem near the constant equilibrium state provided that the initial perturbation is sufficiently small.Under some assumptions of the initial data,we show that the solution of the Cauchy problem converges to its constant equilibrium state at the exact same L2-decay rates as the linearized equations,which shows the convergence rates are optimal.The proof is based on the spectral analysis of the semigroup generated by the linearized equations and the nonlinear energy estimates.Both upper and lower bound decay rates are obtained.