Paper · Mar 26, 2020

Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn

Huy The Nguyen, Shengwen Wang

math.DGWider flowsarXiv:2003.11886v3
View PDF View HTML arXiv page

As on arXiv

Abstract

The parabolic Allen-Cahn equation is a semilinear partial differential equation linked to the mean curvature flow by a singular perturbation. We show an improved convergence property of the parabolic Allen-Cahn equation to the mean curvature flow, which is the parabolic analogue of the improved convergence property of the elliptic Allen-Cahn to minimal surfaces by Wang-Wei and Chodosh-Mantoulidis. More precisely, we show if the phase-transition level sets are converging in , then they converge in . As an application, we obtain a curvature estimate for parabolic Allen-Cahn equation, which can be viewed as a diffused version of Brakke's and White's regularity theorem for mean curvature flow

View PDF View HTML arXiv page
Submitted
Mar 26, 2020
Last revised
Jun 27, 2023 (v3)
Subjects
math.DG, math.AP

Comments: Modifications made according to referee's comments. Details added for some estimates. Statement and proof of Corollary 1.5 reformulated

Mean curvature flowWeak & singular flowsMinimal & CMC surfacesPhase fields & multiphase flows

Its neighbourhood

Connected papers

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 2016, 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 Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound (Lorenzo Bertini, Paolo Buttà and Adriano Pisante, 2016)6Bertini 2016 On the Entropy of Parabolic Allen-Cahn Equation (Ao Sun, 2018)1Sun 2018 Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere (Jingwen Chen and Pedro Gaspar, 2021)2Chen 2021 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)3Steinke 2023 Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow (Likhit Ganedi, Alice Marveggio and Kerrek Stinson, 2024)8Ganedi 2024 Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn (J. M. Daniels-Holgate, 2025)5Daniels-Holg 2025 Parabolic free boundary phase transition and mean curvature flow (Jingeon An-Lacroix and Kiichi Tashiro, 2025)4An-Lacroix 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)7Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Entropy of Parabolic Allen-Cahn Equation

    Ao SunWider flows

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

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

    Jingwen Chen and Pedro GasparWider flows

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

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

    Pascal SteinkeWider flows

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

  4. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows

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

  5. Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn

    J. M. Daniels-HolgateWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “parabolic Allen”, “Cahn”, “Brakke”

  6. Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound

    Lorenzo Bertini, Paolo Buttà and Adriano PisanteWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Allen Cahn”, “Cahn equation”, “level set”

  7. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows

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

  8. Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

    Likhit Ganedi, Alice Marveggio and Kerrek StinsonWider flows

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

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.