Self-similar solutions and large time behavior of solutions to the compressible Navier-Stokes equations.

摘要

In this thesis, I study some characteristic properties of general solutions to the compressible Navier-Stokes equations by considering the self-similar solutions to the isothermal compressible Navier-Stokes equations in one dimension and the self-similar solutions and the large-time behavior of solutions to the stokes approximation equations for two dimensional compressible flows.; First, it is shown that for the Navier-Stokes equations of compressible flow in one dimension, any self-similar solutions satisfying the local energy estimates cannot possess finite total energy for all time. In particular, our results imply that there are no self-similar solutions which satisfy the global energy inequality. Second, we consider the two dimensional Stokes approximation equations for compressible flows to emphasize the nonlinearity due to pressure over the nonlinear convection in the Navier-Stokes equations. We investigate both the existence of forward self-similar solutions and the large time dynamical behavior of weak solutions for such a system. We establish the global existence of small solutions to the Cauchy problems for this nonlinear system in some homogeneous Besov spaces, which, in particular, imply the existence of small forward self-similar solutions for such system. Furthermore, the large time asymptotic behavior toward stationary solutions for such nonlinear system is proved for both Cauchy problems and exterior problems.