1D compressible flow with temperature dependent transport coefficients

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 = cvθ), when the viscosity μ is constant and the heat conductivity κ depends on the temperature θ according to κ(θ) = κ̄θ , 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 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) = μ̄θ, κ(θ) = κ̄θ , 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. 2000 Mathematics Subject Classification: Primary 35Q30; 76N10; Secondary 65M12.