## Abstract

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_{v}θ) 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.

Original language | English (US) |
---|---|

Pages (from-to) | 904-930 |

Number of pages | 27 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - May 14 2010 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics