Existence of proper weak solutions to the Navier-Stokes-Fourier system

TL;DR

This paper proves the existence of proper weak solutions for the Navier-Stokes-Fourier system, incorporating thermal effects, and explores their regularity properties, advancing mathematical understanding of fluid dynamics with heat transfer.

Contribution

It establishes the existence of proper weak solutions for the Navier-Stokes-Fourier system with thermal effects, extending the concept of suitable weak solutions to include temperature considerations.

Findings

Existence of proper weak solutions verified.

Local energy inequalities are satisfied.

Temperature regularity results obtained.

Abstract

The existence of proper weak solutions of the Dirichlet-Cauchy problem constituted by the Navier-Stokes-Fourier system which characterizes the incompressible homogeneous Newtonian fluids under thermal effects is studied. We call proper weak solutions such weak solutions that verify some local energy inequalities in analogy with the suitable weak solutions for the Navier-Stokes equations. Finally, we deal with some regularity for the temperature.