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

TL;DR

This paper analyzes the long-term behavior of a one-dimensional polytropic ideal gas with degenerate heat conductivity, showing it behaves similarly to the case with constant heat conductivity under stress-free boundary conditions.

Contribution

It establishes the asymptotic behavior of solutions for the compressible Navier-Stokes system with degenerate heat conductivity, extending understanding to nonlinear heat conduction cases.

Findings

Solutions exhibit the same large-time behavior as with constant heat conductivity.

Degenerate heat conductivity does not alter the asymptotic dynamics.

The analysis applies to vacuum and stress-free boundary conditions.

Abstract

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.