Convergence to equilibrium for a thin film equation on a cylindrical surface

TL;DR

This paper analyzes a thin film equation on a cylindrical surface, proving the existence and uniqueness of steady states, their energy-minimizing properties, and the convergence behavior of solutions over time.

Contribution

It establishes the existence, uniqueness, and stability of steady states for the thin film equation on a cylinder, including decay rates of solutions towards equilibrium.

Findings

Unique steady state for each mass with zero contact angle

Solutions converge to steady state with power-law decay

Steady states minimize the energy functional

Abstract

The degenerate parabolic equation u_t + [u^3(u_xxx + u_x - sin x)]_x=0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each given mass there is a unique steady state, given by a droplet hanging from the bottom of the cylinder that meets the dry region at the top with zero contact angle. The droplet minimizes the energy and attracts all strong solutions that satisfy certain energy and entropy inequalities. The distance of any solution from the steady state decays no faster than a power law.