Dynamic Stability of Equilibrium Capillary Drops

TL;DR

This paper studies the stability and long-term behavior of quasi-static capillary drops on a plane, proving existence, convergence to equilibrium, and ruling out topological changes for certain initial shapes.

Contribution

It introduces a mathematical model for contact angle motion of capillary drops and proves global existence and convergence results for star-shaped initial data.

Findings

Drops converge to equilibrium over time.

Topological changes are prevented for certain initial shapes.

The model applies to large-volume drops that are initially star-shaped.

Abstract

We investigate a model for contact angle motion of quasi-static capillary drops resting on a horizontal plane. We prove global in time existence and long time behavior (convergence to equilibrium) in a class of star-shaped initial data for which we show that topological changes of drops can be ruled out for all times. Our result applies to any drop which is initially star-shaped with respect to a a small ball inside the drop, given that the volume of the drop is sufficiently large. For the analysis, we combine geometric arguments based on the moving-plane type method with energy dissipation methods based on the formal gradient flow structure of the problem.