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

TL;DR

This paper proves the existence of global weak solutions for a complex degenerate diffusion system modeling heat-conducting fluids, addressing challenges from nonstandard degeneracy and entropy structure.

Contribution

It establishes the existence of solutions for a degenerate, cross-diffusion heat system derived from kinetic models, using entropy and fluid dynamics techniques.

Findings

Existence of global weak solutions proven

Addresses degeneracy when density or temperature vanish

Utilizes entropy inequalities and $H^{-1}$ methods

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 fluid 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).