On The Global Well-posedness And Large-time Behaviors Of Solutions Of Several Systems Coupled With Navier-Stokes Equations

This thesis mainly studies the well-posedness and large-time behaviors of the solutions of these fluid mechanics equations,that is,the existence,uniqueness,continuous dependence on initial value conditions and large-time behaviors of solutions.In Chapter 3,we consider the incompressible magnetohydrodynamic equations in the whole space.We first show that there exists global mild solutions with small initial data in N-dimensional(N≥2)scaling invariant space.The main technique is the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data.Moreover,we gain the asymptotic stability of solutions.Finally,the existence of self-similar solutions is established provided the initial data are small homogeneous functions inR~N(N≥ 3).In Chapter 4,we are concerned with the viscoelastic Navier-Stokes equations(VNS)with damping in the whole space.At first,we use the implicit function theorem to obtain the existence of global mild solutions with small initial data in N-dimensional(N≥2)scaling invariant space.which yields necessarily continuous dependence of solutions for the initial data.In addition,we derive the asymptotic stability of solutions.Finally,we deduce the regularity criteria of weak solutions to VNS with damping in R~3.In Chapter 5,we use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids,which can be called the compressible viscoelastic Navier-Stokes-Poisson equations.The global unique smooth solution to the Cauchy problem in R~3 is obtained.In particular,we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.In Chapter 6,we consider magneto-micropolar fluid equations.By a refined pure energy method,we first obtain a global existence theorem with the smallness of H3(R~3)norm of the initial data.If the initial data belong to homogeneous Sobolev or Besov spaces,we get the optimal decay rates of the solution and its higher order derivatives.At last,we derive the existence of a global large weak solution to the Cauchy problem in R~2.