Some Problems On Inhomogeneous Viscous Fluid

In this thesis,we study some problems about inhomogeneous viscous fluid(including inhomogeneous incompressible Navier-Stokes equations and compress-ible Navier-Stokes equations).In chapter 1,we recall some classical results on inhomogeneous viscous fluid and give some preliminary results.In chapter 2 we investigate the large time decay estimates and uniform in time propagation of regularity for the global solutions of the inhomogeneous in-compressible Navier-Stokes equations defined on whole plane R2.We use the time weighted energy estimates and dual method to get decay estimates and uniform in time L1(Lip)-estimate,which ensure us to get uniform in time bounds for the propagation of regularity.Similar method is applied to study the corresponding problems on R6 and the initial boundary value problem in a 2-D boulded domain with Dirichlet boundary condition.In chapter 3 we consider the inhomogeneous incompressible Navier-Stokes equations on thin domains T2×εT.It is shown that the weak solutions on T2×εT converge to the strong/weak solutions of the 2D inhomogeneous incompressible Navier-Stokes equations on T2 as ε→ 0 on arbitrary time interval.It is also shown that when the vertical size e is sufficiently small,there exists a unique global strong solution to the initial value problem with relatively large data.In chapter 4 we consider two kinds of singular limit problems on expanding domains.Firstly,we study the inviscid limit for Navier-Stokes equations on ex-panding domains with Dirichlet boundary conditions.As the viscosity goes to zero and the domain expands to the whole space R3,we show that the corre-sponding solutions to the Navier-Stokes equations with suitable initial data and vanishing Dirichlet boundary condition converge to the solution of Euler equa-tions in R3 supplemented with the limit initial data,which is not necessarily with compact support compared to the early work[42].Secondly,we use the idea introduced above to study the incompressible inviscid limit problem for the com-pressible Navier-Stokes equation on expanding domains with Dirichlet boundary conditions,also without assumption of compact support on the initial data.