Conditional regularity for the Navier-Stokes-Fourier system with Dirichlet boundary conditions

TL;DR

This paper investigates the conditions under which solutions to the Navier-Stokes-Fourier system with inhomogeneous boundary conditions remain regular, focusing on the boundedness of density, temperature, and velocity.

Contribution

It establishes a conditional regularity criterion for the Navier-Stokes-Fourier system with Dirichlet boundary conditions based on boundedness of key physical quantities.

Findings

Solutions stay regular if density, temperature, and velocity modulus remain bounded.

Provides a regularity criterion for inhomogeneous boundary conditions.

Extends previous results to more general boundary conditions.

Abstract

We consider the Navier-Stokes-Fourier system with the inhomogeneous boundary conditions for the velocity and the temperature. We show that solutions emanating from sufficiently regular data remain regular as long as the density $\varrho$, the absolute temperature $\vartheta$, and the modulus of the fluid velocity $|\textbf{u}|$ remain bounded.