Paper · Feb 27, 2020

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

Julian Fischer, Tim Laux, Theresa M. Simon

math.APWider flowsarXiv:2002.11994v2
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Abstract

We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen-Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.

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Submitted
Feb 27, 2020
Last revised
Aug 18, 2020 (v2)
Subjects
math.AP

Comments: 12 pages

Mean curvature flowStability & uniquenessPhase 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 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 Convergence of the Allen-Cahn Equation to multi-phase mean-curvature flow (Tim Laux and Thilo Simon, 2016)2Laux 2016 On the Entropy of Parabolic Allen-Cahn Equation (Ao Sun, 2018)8Sun 2018 The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions (Julian Fischer, Sebastian Hensel, Tim Laux et al., 2020)5Fischer 2020 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)4Hensel 2021 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)1Fischer 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)7Laux 2022 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)3Kroemer 2023 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)6Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

    Julian Fischer and Alice MarveggioWider flows

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same author Julian Fischer · Shared terms “Allen Cahn”, “Cahn equation”, “equation towards”

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

    Tim Laux and Thilo SimonWider flows

    Topics Phase fields & multiphase flows · Same author Tim Laux · Shared terms “Allen Cahn”, “Cahn equation”

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

    Milan Kroemer and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same author Tim Laux · Shared terms “Cahn equation”, “Allen”, “relative entropy”

  4. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same author Tim Laux · Shared terms “Allen Cahn”, “Cahn equation”

  5. The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions

    Julian Fischer, Sebastian Hensel, Tim Laux et al.Wider flows

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same authors Julian Fischer, Tim Laux, Theresa M. Simon

  6. 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 · Same author Tim Laux · Shared terms “Allen Cahn”, “Cahn equation”

  7. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same author Tim Laux · Shared terms “relative entropy”, “Allen Cahn”, “Cahn equation”

  8. On the Entropy of Parabolic Allen-Cahn Equation

    Ao SunWider flows

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

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.