On solutions for a generalized Navier-Stokes-Fourier system fulfilling entropy equality

TL;DR

This paper establishes the existence of solutions satisfying the entropy equality for a generalized Navier-Stokes-Fourier system, advancing understanding of the stability and long-term behavior of non-Newtonian heat-conducting fluids.

Contribution

It proves the existence of entropy-equality solutions for a complex fluid system, addressing challenges in weak solution admissibility and stability analysis.

Findings

Existence of solutions fulfilling the entropy equality.

Overcoming difficulties with special inequalities and test functions.

Foundation for studying stability and convergence to equilibrium.

Abstract

We consider a flow of non-Newtonian heat conducting incompressible fluid in a bounded domain subjected to the homogeneous Dirichlet boundary condition for the velocity field and the spatially inhomogeneous Dirichlet boundary condition for the temperature. The ultimate goal is to show that the fluid converges to equilibrium as time tends to infinity. However, to justify such result, one needs to deal with very special inequalities and very special test functions, which are typically not admissible on the level of weak solutions. In this paper, we show how one can overcome such difficulties. In particular, we show the existence of a solution fulfilling the entropy equality, which seems to be optimal class of solutions in which one should study the stability result.