Asymptotic Stability and the Forcing Term: An Analysis of Non-Newtonian Thin-Film Flows

TL;DR

This paper analyzes the stability and convergence of solutions to non-Newtonian thin-film flow equations with external forces, providing theoretical estimates and numerical validation for the asymptotic behavior of these complex flows.

Contribution

It introduces new analytical techniques for deriving convergence rates of non-autonomous fourth-order degenerate parabolic equations modeling non-Newtonian thin-film flows.

Findings

Derived two-sided convergence rate estimates.

Validated theoretical results with numerical simulations.

Enhanced understanding of stability in non-Newtonian thin-film flows.

Abstract

We study a class of fourth-order quasilinear degenerate parabolic equations under both time-and space-dependent and time-and space-independent forces, modeling non-Newtonian thin-film flow over a solid surface in the "complete wetting" regime. By analyzing the quantitative properties of solutions to non-autonomous differential inequalities and employing refined integral estimates, we derive two-sided convergence rate estimates for the solution. Numerical simulations are further provided to illustrate the consistency of our main results with the observed physical phenomena.