Paper · May 18, 2021

Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to °

Helmut Abels, Maximilian Moser

math.APWider flowsarXiv:2105.08434v2
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Abstract

This paper is concerned with the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain . We assume that a diffuse interface already has developed and that it is in contact with the boundary . The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant -contact angle. For close to ° we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen-Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen-Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.

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Submitted
May 18, 2021
Last revised
May 31, 2021 (v2)
Subjects
math.AP

Comments: 59 pages

Mean curvature flowBoundaries & 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 of the Allen-Cahn Equation to the Mean Curvature Flow with 90^∘-Contact Angle in 2D (Helmut Abels and Maximilian Moser, 2018)1Abels 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)7Fischer 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 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)6Hensel 2021 Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow (Julian Fischer and Alice Marveggio, 2022)8Fischer 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)5An-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 to the Mean Curvature Flow with -Contact Angle in 2D

    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”, “spectral estimate”

  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 “Allen Cahn”, “Cahn equation”, “interface”

  5. 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”, “boundary condition”

  6. 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”

  7. 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”

  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”, “prepared initial”

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) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

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.