Paper · Dec 16, 2025

Parabolic free boundary phase transition and mean curvature flow

Jingeon An-Lacroix, Kiichi Tashiro

math.APWider flowsarXiv:2512.14437
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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 \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 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.

Comments: 20 pages

Mean curvature flowVolume-preserving & constrained flowsFree boundary & capillaryPhase fields & multiphase flows

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere (Jingwen Chen and Pedro Gaspar, 2021)3Chen 2021 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)7Takasao 2022 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)5Fischer 2022 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)6Steinke 2023 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)2Kroemer 2023 Diffusion-aggregation equations and volume-preserving mean curvature flows (Jiwoong Jang and Antoine Mellet, 2025)1Jang 2025 A game approach to free boundary problems of anisotropic forced mean curvature flow equations (Takuya Sato, 2025)4Sato 2025 Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary (Ezequiel Barbosa, Rosivaldo Gonçalves and Luan de Figueiredo, 2026)8Barbosa 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Diffusion-aggregation equations and volume-preserving mean curvature flows

    Jiwoong Jang and Antoine MelletWider flows PDF

    Topics Phase fields & multiphase flows, Free boundary & capillary, Volume-preserving & constrained flows · Shared terms “Cahn equation”, “Allen”, “forcing term”

  2. Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

    Milan Kroemer and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows · Shared terms “Cahn equation”, “Allen”, “gradient flow”

  3. Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

    Jingwen Chen and Pedro GasparWider flows PDF

    Topics Phase fields & multiphase flows · Shared terms “parabolic Allen”, “Cahn equation”, “gradient flow”

  4. A game approach to free boundary problems of anisotropic forced mean curvature flow equations

    Takuya SatoWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Shared terms “forced mean”

  5. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

    Julian Fischer and Alice MarveggioWider flows PDF

    Topics Phase fields & multiphase flows · Shared terms “Cahn equation”, “Allen”, “gradient flow”

  6. Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow

    Pascal SteinkeWider flows PDF

    Topics Phase fields & multiphase flows · Shared terms “Cahn equation”, “Allen”

  7. Existence of weak solution to volume preserving mean curvature flow in higher dimensions

    Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows · Shared terms “Cahn equation”, “Allen”

  8. Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary

    Ezequiel Barbosa, Rosivaldo Gonçalves and Luan de FigueiredoWider flows PDF

    Topics Free boundary & capillary · Shared terms “phase transition”, “Cahn”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.