Nonlinearly stability of solutions on the outer Pressure Problem of Compressible Navier-Stokes System with Temperature-Dependent Heat Conductivity

In this paper, the one-dimensional compressible Navier-Stokes system with outer pressure boundary conditions is investigated. Under some suitable assumptions, we prove that the specific volume and the temperature are bounded from below and above independently of time, and then give the local and global existence of strong solutions. Furthermore, we also obtain the convergence of the global strong solution to a stationary state and the nonlinearly stability of its convergence. It is worth noticing that all the assumptions imposed on the initial data are the same as Takeyuki Nagasawa [Japan.J.Appl.Math.(1988)]. Therefore, our work can be regarded as an improvement of the results of Takeyuki.

On the asymptotic behavior of the one-dimensional motion of the polytropic ideal gas with degenerate heat conductivity

We consider the one-dimensional compressible Navier-Stokes system with constant viscosity and the nonlinear heat conductivity being proportional to a positive power of the temperature which may be degenerate. This problem is imposed on the stress-free boundary condition, which reveals the motion of a viscous heat-conducting perfect polytropic gas with adiabatic ends putting into a vacuum. We prove that the solution of one dimensional compressible Navier-Stokes system with the stress-free boundary condition shares the same large-time behavior as the case of constant heat conductivity.