A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system

TL;DR

This paper establishes a regularity criterion for weak solutions to the Navier-Stokes-Fourier system, showing that bounded velocity gradient ensures continued regularity, based on weak-strong uniqueness and a priori estimates.

Contribution

It introduces a new regularity criterion linking velocity gradient bounds to solution regularity for the full Navier-Stokes-Fourier system.

Findings

Weak solutions remain regular if velocity gradient is bounded.

The proof utilizes weak-strong uniqueness and parabolic a priori estimates.

Solutions with initial data in Sobolev space W^{3,2} are considered.

Abstract

We show that any weak solution to the full Navier-Stokes-Fourier system emanating from the data belonging to the Sobolev space W^{3,2} remains regular as long as the velocity gradient is bounded. The proof is based on the weak-strong uniqueness property and parabolic a priori estimates for the local strong solutions.