One-Dimensional Compressible Flow with Temperature Dependent Transport Coefficients

We establish the 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=cᵥθ) when the viscosity μ is constant and the heat conductivity κ depends on the temperature θ according to κ(θ)=κ̄θ^β, with 0≤β<3/2. This choice of degenerate transport coefficients is motivated by the kinetic theory of gasses. Approximate solutions are generated by a semidiscrete 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)=μ̄θ^α, κ(θ)=κ̄θ^β, with α≥0, 0≤β<2 (μ̄, κ̄ constants). We then verify the sufficient conditions in the case α=0 and 0≤β<3/2. The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.