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 $p=Kθ/τ$, internal energy $e=c_v θ$), when the viscosity $μ$ is constant and the heat conductivity $κ$ depends on the temperature $θ$ according to $κ(θ) = \bar κθ^β$, with $0\leqβ<{3/2}$. 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 $μ(θ) = \bar μθ^α$, $κ(θ) = \bar κθ^β$, with $α\geq 0$, $0 \leq β< 2$ ($\bar μ, \bar κ$ constants). We then verify the sufficient conditions in the case $α=0$ and $0\leqβ<{3/2}$. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.