Finite speed of propagation for the thin film equation in spherical geometry

TL;DR

This paper proves that solutions to a thin film equation on a spherical surface propagate at finite speed, establishing the existence of a moving boundary and providing bounds on its rate of movement.

Contribution

It demonstrates finite speed of propagation for a degenerate thin film equation on a sphere and derives bounds on interface movement using entropy estimates.

Findings

Finite speed of propagation for the thin film equation on a sphere.

Existence of a free boundary separating positive and zero solution regions.

Upper bounds on the interface propagation rate.

Abstract

We show that a double degenerate thin film equation, which originated from modeling of viscous coating flow on a spherical surface, has finite speed of propagation for nonnegative strong solutions and hence there exists an interface or free boundary separating the regions where solution $u>0$ and $u=0$. Using local entropy estimates we also obtain an upper bound for the rate of the interface propagation.