Navier-Stokes Equations With Density-Dependent Viscosity

We study the initial boundary value problem for Navier-Stokes equations with density-dependent viscosity. This system comes from the physical consideration. We apply some new techniques to deal with the difficulties coming from such system with variable coefficients. We prove the non-formation of vacuum states, existence and uniqueness of weak solution, non-existence of global classical solutions, and obtain the asymptotic behavior and convergence rate of solutions. .In the following chapters, we study these five problems: the non-formation of vacuum states for 1-D isentropic compressible Navier-Stokes equations with density-dependent viscosity, provided that no vacuum states are present initially; the existence and uniqueness of local strong solution to 3-D compressible Navier-Stokes equations whose viscosity and heat conduction coefficients are in general functions of the density and temperature, provided that the initial data satisfy some compatibility conditions; the non-existence of global smooth solutions to n-D compressible Navier-Stokes equations whose viscosity and heat conduction coefficients are in general functions of the density and temperature, when the initial density has compactly support and the initial total momentum is nonzero; the existence and uniqueness of weak solutions to 1-D isentropic compressible Navier-Stokes equations with density-dependent viscosity and discontinuous initial density; the global behavior of solutions to the free boundary problem for 1-D isentropic compressible Navier-Stokes equations with degenerate viscosity.