Paper · Jun 6, 2018

Convergence of the Allen-Cahn Equation to the Mean Curvature Flow with -Contact Angle in 2D

Helmut Abels, Maximilian Moser

math.APWider flowsarXiv:1806.02065
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Abstract

We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain , in the situation where an interface has developed and intersects . Here a parameter in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a \textdegree-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on for some and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen-Cahn equation and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the linearized Allen-Cahn operator.

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Submitted
Jun 6, 2018
Subjects
math.AP

Comments: 53 pages

Mean curvature flowSurfacesHomogeneous spacesBoundaries & non-compactFree boundary & capillaryEigenvalues & spectrumPhase 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 Well-Posedness of a Navier-Stokes/Mean Curvature Flow system (Helmut Abels and Maximilian Moser, 2017)4Abels 2017 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)8Fischer 2020 Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with 90°-Contact Angle in Higher Dimensions (Maximilian Moser, 2021)2Moser 2021 Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to 90° (Helmut Abels and Maximilian Moser, 2021)1Abels 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)5Hensel 2021 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)7Fischer 2022 Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow (Helmut Abels, Christoph Hurm and Maximilian Moser, 2024)3Abels 2024 Parabolic free boundary phase transition and mean curvature flow (Jingeon An-Lacroix and Kiichi Tashiro, 2025)6An-Lacroix 2025
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 Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to °

    Helmut Abels and Maximilian MoserWider flows

    Topics Phase fields & multiphase flows, Free boundary & capillary, Eigenvalues & spectrum · Same authors Helmut Abels, Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “contact angle”

  2. Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with °-Contact Angle in Higher Dimensions

    Maximilian MoserWider flows

    Topics Phase fields & multiphase flows, Free boundary & capillary, Eigenvalues & spectrum · Same author Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “contact angle”

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

    Helmut Abels, Christoph Hurm and Maximilian MoserWider flows

    Topics Phase fields & multiphase flows, Eigenvalues & spectrum, Boundaries & non-compact · Same authors Helmut Abels, Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “approximate solutions”

  4. Well-Posedness of a Navier-Stokes/Mean Curvature Flow system

    Helmut Abels and Maximilian MoserWider flows

    Topics Phase fields & multiphase flows · Same authors Helmut Abels, Maximilian Moser · Shared terms “interface”, “Allen Cahn”, “Cahn equation”

  5. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Free boundary & capillary · Shared terms “Allen Cahn”, “contact angle”, “angle condition”

  6. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows

    Topics Phase fields & multiphase flows, Free boundary & capillary, Boundaries & non-compact · Shared terms “Cahn equation”, “Allen”, “interface”

  7. 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”, “interface”

  8. 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”, “linearized Allen”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.