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

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.