Existence analysis of a degenerate diffusion system for heat-conducting gases

Abstract

The existence of global weak solutions to a parabolic energy-transport system in a bounded domain with no-flux boundary conditions is proved. The model can be derived in the diffusion limit from a kinetic equation with a linear collision operator involving a non-isothermal Maxwellian. The evolution of the local temperature is governed by a heat equation with a source term that depends on the energy of the distribution function. The limiting model consists of cross-diffusion equations with an entropy structure. The main difficulty is the nonstandard degeneracy, i.e., ellipticity is lost when the gas density or temperature vanishes. The existence proof is based on a priori estimates coming from the entropy inequality and the H^{-1} method and on techniques from mathematical fluid dynamics (renormalized formulation, div-curl lemma).