Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order

TL;DR

This paper investigates the long-term stability and behavior of solutions to higher-order quasilinear doubly degenerate parabolic equations modeling non-Newtonian thin-film flows, revealing different convergence rates depending on fluid rheology.

Contribution

It provides a detailed analysis of the stability and convergence rates of solutions for various non-Newtonian fluid models within a unified framework.

Findings

Solutions near steady states converge in finite time for shear-thickening fluids.

Solutions exhibit polynomial decay to equilibrium for shear-thinning fluids.

Solutions decay exponentially to steady states for Ellis fluids.

Abstract

We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the $H^1(\Omega)$-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate $1/t^{1/\beta}$ for some $\beta > 0$. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in $H^1(\Omega)$.