Parabolic free boundary phase transition and mean curvature flow
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914204156428288 |
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| author | An, Jingeon Tashiro, Kiichi |
| author_facet | An, Jingeon Tashiro, Kiichi |
| contents | 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
\[
v=-H-\partial_ν\log |\nabla u|+f(u)/|\nabla u|,
\]
satisfied by level surfaces of any solution to the nonlinear parabolic equation
\[
\partial_tu=Δu-f(u).
\]
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
\[
\left\{
\begin{alignedat}{2}
\partial_tu&=Δu\quad&&\text{in}\quad\{|u|<1\}
|\nabla u|&=1/ε\quad&&\text{on}\quad\partial\{|u|<1\},
\end{alignedat}
\right.
\]
and confirm that under reasonable assumptions, the $C^α$ norm of the forcing term $\partial_ν\log|\nabla u|$ converges to zero at an algebraic rate as $ε\rightarrow 0$, uniformly in time. This implies that the parabolic free boundary Allen--Cahn equation converges to the mean curvature flow, uniformly (in $ε$ and in time) in the $C^{2,α}$ sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parabolic free boundary phase transition and mean curvature flow An, Jingeon Tashiro, Kiichi Analysis of PDEs 35R35, 35N25, 53E10 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 \[ v=-H-\partial_ν\log |\nabla u|+f(u)/|\nabla u|, \] satisfied by level surfaces of any solution to the nonlinear parabolic equation \[ \partial_tu=Δu-f(u). \] 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 \[ \left\{ \begin{alignedat}{2} \partial_tu&=Δu\quad&&\text{in}\quad\{|u|<1\} |\nabla u|&=1/ε\quad&&\text{on}\quad\partial\{|u|<1\}, \end{alignedat} \right. \] and confirm that under reasonable assumptions, the $C^α$ norm of the forcing term $\partial_ν\log|\nabla u|$ converges to zero at an algebraic rate as $ε\rightarrow 0$, uniformly in time. This implies that the parabolic free boundary Allen--Cahn equation converges to the mean curvature flow, uniformly (in $ε$ and in time) in the $C^{2,α}$ sense. |
| title | Parabolic free boundary phase transition and mean curvature flow |
| topic | Analysis of PDEs 35R35, 35N25, 53E10 |
| url | https://arxiv.org/abs/2512.14437 |