Long-time behavior of the mean curvature flow with periodic forcing

TL;DR

This paper studies the long-term evolution of mean curvature flow in media with periodic forcing, establishing the existence of traveling waves and convergence of solutions to these waves.

Contribution

It introduces a framework for analyzing forced mean curvature flow in heterogeneous media and proves the existence and convergence to generalized traveling waves.

Findings

Existence of generalized traveling waves with maximal speed

Convergence of solutions to these traveling waves

Conditions under which the long-time behavior is characterized

Abstract

We consider the long-time behavior of the mean curvature flow in heterogeneous media with periodic fibrations, modeled as an additive driving force. Under appropriate assumptions on the forcing term, we show existence of generalized traveling waves with maximal speed of propagation, and we prove the convergence of solutions to the forced mean curvature flow to these generalized waves.