Parabolic free boundary phase transition and mean curvature flow

TL;DR

This paper establishes a connection between a class of parabolic free boundary problems, including the Allen--Cahn equation, and mean curvature flow, showing convergence of solutions to geometric flows as a parameter tends to zero.

Contribution

It introduces a unified framework for parabolic free boundary problems via gradient flows and proves convergence of the free boundary Allen--Cahn equation to mean curvature flow.

Findings

The forced mean curvature flow equation for level surfaces is derived.

The notion of inner gradient flow is introduced to unify free boundary problems.

The free boundary Allen--Cahn equation converges to mean curvature flow as the parameter tends to zero.

Abstract

It is known that there is a strong relation between the parabolic Allen--Cahn equation and the mean curvature flow, in the sense that the parabolic Allen--Cahn equation can be considered as a ``diffused" mean curvature flow. In this work, we derive a forced mean curvature flow \[ v=-H-\partial_\nu\log |\nabla u|+f(u)/|\nabla u|, \] satisfied by level surfaces of any solution to the nonlinear parabolic equation \[ \partial_tu=\Delta u-f(u). \] Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen--Cahn equation \[ \left\{ \begin{alignedat}{2} \partial_tu&=\Delta u\quad&&\text{in}\quad\{|u|<1\} |\nabla u|&=1/\epsilon\quad&&\text{on}\quad\partial\{|u|<1\}, \end{alignedat} \right. \] and confirm that under reasonable assumptions, the $C^{\alpha}$ norm of the forcing term $\partial_\nu\log|\nabla u|$ converges to zero at an algebraic rate as $\epsilon\rightarrow 0$, uniformly in time. This implies that the parabolic free boundary Allen--Cahn equation converges to the mean curvature flow, uniformly (in $\epsilon$ and in time) in the $C^{2,\alpha}$ sense.