A bound from below on the temperature for the Navier-Stokes-Fourier system

TL;DR

This paper establishes a uniform lower bound on temperature for a class of weak solutions to a variant of the compressible Navier-Stokes-Fourier system, ensuring the temperature remains strictly positive under certain conditions.

Contribution

It introduces a novel approach using a localized entropy inequality and De Giorgi's method to prove temperature positivity in compressible fluid models.

Findings

Temperature remains bounded away from zero for weak solutions.

The method extends to systems with heat conduction and bounded measurable coefficients.

Provides a mathematical foundation for physical stability of the model.

Abstract

We give a uniform bound from below on the temperature for a variant of the compressible Navier-Stokes-Fourier system, under suitable hypotheses. This system of equations forms a mathematical model of the motion of a compressible fluid subject to heat conduction. Building upon the work of [16], we identify a class of weak solutions satisfying a localized form of the entropy inequality (adapted to measure the set where the temperature becomes small) and use a form of the De Giorgi argument for $L^\infty$ bounds of solutions to elliptic equations with bounded measurable coefficients.