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

Abstract

We consider a non-Newtonian incompressible heat conducting fluid with a prescribed nonuniform temperature on the boundary and with the no-slip boundary condition for the velocity. We assume no external body forces. For the power-law like models with the power law index bigger than or equal to 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 proper weak solutions.