On the continuity in time of solutions to a generalized Navier--Stokes--Fourier system

TL;DR

This paper proves the time continuity of temperature solutions in a generalized Navier--Stokes--Fourier system with non-Newtonian fluids, establishing new regularity results and linking dissipation to stability.

Contribution

It provides the first rigorous proof of temperature time continuity in this setting and connects dissipation with weak stability and regularity.

Findings

Proved global-in-time weak solutions with entropy equality for p≥2.

Established temperature time continuity in L^1(Ω).

Linked dissipation on high level sets to temporal regularity.

Abstract

We consider the flow of a generalized non-Newtonian incompressible heat-conducting fluid in a~bounded two-dimensional domain, subject to Dirichlet boundary conditions for velocity and temperature. The fluid obeys a power-law constitutive relation for the Cauchy stress with exponent~$p$. For $p\geq 2$ and finite-energy initial data, we establish the existence of a global-in-time weak solution that satisfies the entropy equality. The novelty of this work is the rigorous proof of time continuity of the temperature in $L^1(\Omega)$, a property not previously established in this setting. Furthermore, we prove regularity and time continuity for a weak solution of the entropy equation with a convective term and an $L^1$ right-hand side under minimal assumptions on the velocity regularity, in arbitrary spatial dimensions. We show that this continuity is equivalently described by vanishing dissipation on high level sets, a truncated variational inequality for admissible test functions, or the associated equality. This reveals the connection between energy dissipation, weak stability, and temporal regularity.