One-Dimensional 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 = cvθ), when the viscosity µ is constant and the heat conductivity κ depends on the temperature θ according to κ(θ) =κθ 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 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) =μθ α , κ(θ) =κθ β , with α ≥ 0, 0 ≤ β < 2 (μ,κ 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.

Global Solutions to the One-Dimensional Compressible Navier--Stokes--Poisson Equations with Large Data

“…• The same arguments for Theorems 1.1 and 1.2 can be applied directly to the compressible Navier-Stokes equations which generalize the previous results [18] and [23], where the viscosity coefficient is assumed to be a positive constant. Downloaded 11/17/14 to 131.155.81.2.…”

“…Unlike the small perturbation solutions, such dependence has a strong influence on the solution behavior and thus leads to difficulties in analysis but not for the case of constant coefficients. In fact, for the one-dimensional compressible NavierStokes equations, there are a lot of recent papers on the construction of nonvacuum solutions to the one-dimensional compressible Navier-Stokes equations with densityand temperature-dependent transportation coefficients in various forms; see [1], [5], [18], [19], [21], [22], [23], [24], [25], and the references therein. However, there is a gap between the physical models and the satisfactory existence theory.…”

“…We now review some related results. First, there are some recent results on the construction of nonvacuum, large solutions to the one-dimensional compressible Navier-Stokes equations with constant viscosity coefficient μ and density-and temperature-dependent heat-conductivity coefficient κ; see [18], [23]. A key ingredient in these works is the pointwise a priori estimates on the specific volume which guarantees that no vacuum or concentration of mass occurs.…”

Vanishing Shear Viscosity and Boundary Layer for the Navier–Stokes Equations with Cylindrical Symmetry

“…Jessen and Karper [7] obtained the existence of global weak solutions for one dimensional flow when 0 < q < 3 2 . Tan et al [14] studied the Cauchy problem of the onedimensional Navier-Stokes-Poisson system with density dependent viscosity when there is a certain relationship between the powers of viscosity and heat conductivity.…”

Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data

“…We hope that such a study can shed some light on the construction of global classical, large solutions to the high-dimensional compressible fluid models of Korteweg type. We point out that there are a lot of results on the construction of non-vacuum, large solutions to the one-dimensional compressible Navier-Stokes equations in various contexts, see [28][29][30][31][32][33][34][35][36][37] and the references therein. The method in the present paper is essentially motivated by the work of Y. Kanel' [29].…”