Paper · Dec 20, 2018

On the Entropy of Parabolic Allen-Cahn Equation

Ao Sun

math.DGWider flowsarXiv:1812.08894v3
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As on arXiv

Abstract

We define a (mean curvature flow) entropy for Radon measures in or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen-Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit measure of the parabolic Allen-Cahn equation has unit density for all future time.

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Submitted
Dec 20, 2018
Last revised
Feb 8, 2021 (v3)
Subjects
math.DG, math.AP

Comments: 13 pages. Accepted by Interfaces and Free Boundaries

Mean curvature flowPhase 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)8Bertini 2016 Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow (Tim Laux and Thilo Simon, 2016)4Laux 2016 Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies (Julian Fischer, Tim Laux and Theresa M. Simon, 2020)3Fischer 2020 Second order estimates for transition layers and a curvature estimate for the parabolic Allen-Cahn (Huy The Nguyen and Shengwen Wang, 2020)2Nguyen 2020 Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere (Jingwen Chen and Pedro Gaspar, 2021)1Chen 2021 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)6Fischer 2022 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)5Steinke 2023 Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow (Helmut Abels, Christoph Hurm and Maximilian Moser, 2024)7Abels 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Shared terms “parabolic Allen”, “Allen Cahn”, “Cahn equation”

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

    Huy The Nguyen and Shengwen WangWider flows

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

  3. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies

    Julian Fischer, Tim Laux and Theresa M. SimonWider flows

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

  4. Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow

    Tim Laux and Thilo SimonWider flows

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

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

    Pascal SteinkeWider flows

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

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

    Julian Fischer and Alice MarveggioWider flows

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

  7. Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow

    Helmut Abels, Christoph Hurm and Maximilian MoserWider flows

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

  8. 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 · 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) 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.