Paper
Parabolic free boundary phase transition and mean curvature flow
Authors
Jingeon An, Kiichi Tashiro
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
satisfied by level surfaces of any solution to the nonlinear parabolic equation
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
and confirm that under reasonable assumptions, the norm of the forcing term converges to zero at an algebraic rate as , uniformly in time. This implies that the parabolic free boundary Allen--Cahn equation converges to the mean curvature flow, uniformly (in and in time) in the sense.