Head and tail speeds of mean curvature flow with forcing

TL;DR

This paper studies the large-time behavior of interfaces under mean curvature flow with a periodic forcing term, focusing on head and tail speeds and their relation to the interface's large-scale motion.

Contribution

It introduces the concept of head and tail speeds for interfaces with forcing, characterizes their dependence on direction, and analyzes the development of long fingers when speeds differ.

Findings

Head and tail speeds depend continuously on propagation direction.

Unique large-scale speed occurs when head and tail speeds are equal.

Interfaces develop long fingers when head and tail speeds differ.

Abstract

In this paper, we investigate the large time behavior of interfaces moving with motion law $V = -\kappa + g(x)$, where $g$ is positive, Lipschitz and $\mathbb{Z}^n$-periodic. It turns out that the behavior of the interface can be characterized by its head and tail speed, which depends continuously on its overall direction of propagation $\nu$. If head speed equals tail speed at a given direction $\nu$, the interface has a unique large-scale speed in that direction. In general the interface develops linearly growing "long fingers" in the direction where the equality breaks down. We discuss these results in both general setting and in laminar setting, where further results are obtained due to regularity properties of the flow.