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

TL;DR

This paper proves the global existence of weak solutions for the 3D inhomogeneous heat-conducting Navier-Stokes equations with temperature-dependent viscosity, even when viscosity degenerates at zero temperature, using advanced mathematical techniques.

Contribution

It establishes the first global existence result for weak solutions with degenerate viscosity depending on temperature in a bounded domain.

Findings

Global weak solutions exist for large initial data.

The viscosity may vanish at zero temperature, modeling realistic physical scenarios.

The proof employs a three-level approximation, De Giorgi's method, and compactness arguments.

Abstract

In this paper, the initial-boundary value problem to the three-dimensional inhomogeneous, incompressible and heat-conducting Navier-Stokes equations with temperature-depending viscosity coefficient is considered in a bounded domain. The viscosity coefficient is degenerate and may vanish in the region of absolutely zero temperature. Global existence of weak solutions to such a system is established for the large initial data. The proof is based on a three-level approximate scheme, the De Giorgi's method and compactness arguments.