Paper · Jun 23, 2016

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

Tim Laux, Thilo Simon

math.APWider flowsarXiv:1606.07318v2
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Abstract

We present a convergence result for solutions of the vector-valued Allen-Cahn Equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow using sets of finite perimeter. Like their result, ours relies on the assumption that the time-integrated energies of the approximations converge to those of the limit. Furthermore, we apply our proof to two variants of the equation, incorporating external forces and a volume constraint.

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Submitted
Jun 23, 2016
Last revised
Sep 23, 2016 (v2)
Subjects
math.AP
Mean curvature flowPhase fields & multiphase flows

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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 thresholding scheme for multi-phase mean-curvature flow (Tim Laux and Felix Otto, 2016)1Laux 2016 Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow (Keisuke Takasao, 2018)3Takasao 2018 On the Entropy of Parabolic Allen-Cahn Equation (Ao Sun, 2018)6Sun 2018 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)4Fischer 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)5Hensel 2021 Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets (Jakob Fuchs and Tim Laux, 2022)7Fuchs 2022 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)8Fischer 2022 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)2Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of the thresholding scheme for multi-phase mean-curvature flow

    Tim Laux and Felix OttoWider flows

    Topics Phase fields & multiphase flows · Same author Tim Laux · Shared terms “multi phase”, “integrated energy”, “time integrated”

  2. 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”, “valued Allen”, “vector valued”

  3. Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow

    Keisuke TakasaoWider flows

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

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

  5. 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 · Same author Tim Laux · Shared terms “Allen Cahn”, “Cahn equation”, “multi phase”

  6. On the Entropy of Parabolic Allen-Cahn Equation

    Ao SunWider flows

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

  7. Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

    Jakob Fuchs and Tim LauxWider flows

    Topics Phase fields & multiphase flows · Same author Tim Laux · Shared terms “integrated energy”, “time integrated”, “approximation”

  8. 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”

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.