Global existence of weak solutions to the compressible Navier-Stokes equations with temperature-depending viscosity coefficients

TL;DR

This paper proves the global existence of weak solutions for 3D compressible Navier-Stokes equations with temperature-dependent viscosity and heat conduction, handling vanishing viscosity at zero temperature.

Contribution

It introduces a novel approach using De Giorgi's iteration to obtain temperature bounds, enabling the proof of existence with temperature-dependent coefficients.

Findings

Established global weak solutions under temperature-dependent viscosity.

Developed uniform estimates via Galerkin approximation and De Giorgi's iteration.

Handled vanishing viscosity as temperature approaches zero.

Abstract

This paper is devoted to the global existence of weak solutions to the three-dimensional compressible Navier-Stokes equations with heat-conducting effects in a bounded domain. The viscosity and the heat conductivity coefficients are assumed to be functions of the temperature, and the shear viscosity coefficient may vanish as the temperature goes to zero. The proof is to apply Galerkin method to a suitable approximate system with several parameters and obtain uniform estimates for the approximate solutions. The key ingredient in obtaining the required estimates is to apply De Giorgi's iteration to the modified temperature equation, from which we can get a lower bound for the temperature not depending on the artificial viscosity coefficient introduced in the modified momentum equation, which makes the compactness argument available as the artificial viscous term vanishes.