1D compressible flow with temperature dependent transport coefficients

Abstract

We establish existence of global-in-time weak solutions to the one dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas (pressure $p=K\theta /\tau$, internal energy $e=c_v\theta$), when the viscosity $\mu$ is constant and the heat conductivity $\kappa$ depends on the temperature $\theta$ according to $\kappa (\theta )=\bar{\kappa }{\theta }^{\beta }$, with $0\le \beta <3/2$. This choice of degenerate transport coefficients is motivated by the kinetic theory of gasses. Approximate solutions are generated by a semi-discrete finite element scheme. We first formulate sufficient conditions that guarantee convergence to a weak solution. The convergence proof relies on weak compactness and convexity, and it applies to the more general constitutive relations $\mu (\theta )=\bar{\mu }{\theta }^{\alpha }$, $\kappa (\theta )=\bar{\kappa }{\theta }^{\beta }$, with $\alpha \ge 0$, $0\le \beta <2$ ($\bar{\mu },\bar{\kappa }$ constants). We then verify the sufficient conditions in the case $\alpha =0$ and $0\le \beta <3/2$. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.

26 pages, 1 figure