Long-time asymptotics and regularity estimates for weak solutions to a doubly degenerate thin-film equation in the Taylor-Couette setting

TL;DR

This paper analyzes the long-term behavior and regularity of weak solutions to a complex degenerate thin-film equation modeling fluid interfaces between rotating cylinders, demonstrating global existence and convergence to circular shapes.

Contribution

It introduces new regularity estimates and proves global existence and stability results for a doubly degenerate fourth-order parabolic equation in a fluid dynamics context.

Findings

Global existence of positive weak solutions for low initial energy

Polynomial stability with convergence rate to a circle

Regularity estimates for nonlinear degenerate fourth-order parabolic equations

Abstract

We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate parabolic equation for large times arising in fluid dynamical applications. The degeneracy occurs both with respect to the unknown and with respect to the sum of its first and third spatial derivative. The modelling equation appears as a thin-film limit for the interface separating two immiscible viscous fluid films confined between two cylinders rotating at small relative angular velocity. More precisely, the fluid occupying the layer next to the outer cylinder is considered to be Newtonian, i.e. it has constant viscosity, while we assume that the layer next to the inner cylinder is filled by a shear-thinning power-law fluid. Using energy methods, Fourier analysis and suitable regularity estimates for higher-order parabolic equations, we prove global existence of positive weak solutions in the case of low initial energy. Moreover, these global solutions are polynomially stable, in the sense that interfaces which are initially close to a circle, converge at rate $1/t^{1/\beta}$ for some $\beta > 0$ to a circle, as time tends to infinity. In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.