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/t, internal energy e = c_vθ) 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) = μ̄ θ^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.