Parabolic free boundary phase transition and mean curvature flow

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.