1D compressible flow with temperature dependent transport coefficients

TL;DR

This paper proves the existence of global weak solutions for a 1D compressible Navier-Stokes system with temperature-dependent transport coefficients, using finite element approximations and weak compactness methods.

Contribution

It establishes the existence of weak solutions for a 1D compressible flow with degenerate heat conductivity depending on temperature, extending previous results to more general constitutive relations.

Findings

Proved existence of weak solutions under specified conditions.

Developed a semi-discrete finite element scheme for approximation.

Verified convergence conditions for specific parameter choices.

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 $0\leq\beta<{3/2}$. 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 $\alpha\geq 0$, $0 \leq \beta < 2$ ($\bar \mu, \bar \kappa$ constants). We then verify the sufficient conditions in the case $\alpha=0$ and $0\leq\beta<{3/2}$. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.