On the exponential decay in time of solutions to a~generalized Navier-Stokes-Fourier system

TL;DR

This paper proves exponential decay to equilibrium for a class of non-Newtonian, heat-conducting fluids modeled by a generalized Navier-Stokes-Fourier system, establishing stability via a Lyapunov functional.

Contribution

It introduces a Lyapunov functional for the system and demonstrates exponential convergence to equilibrium for power-law fluids with index greater than 11/5.

Findings

Solutions converge exponentially to equilibrium

Existence of a Lyapunov functional for stability

Steady solutions are nonlinearly stable

Abstract

We consider a non-Newtonian incompressible heat conducting fluid with prescribed nonuniform temperature on the boundary and with the no-slip boundary conditions for the velocity. We assume no external body forces. For the power-law like models with the power law index bigger than $11/5$ in three dimensions, we identify a class of solutions fulfilling the entropy equality and converging to the equilibria exponentially in a proper metric. In fact, we show the existence of a Lyapunov functional for the problem. Consequently, the steady solution is nonlinearly stable and attracts all suitable weak solutions.