Paper · May 14, 2021

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

Maximilian Moser

math.APWider flowsarXiv:2105.07100
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Abstract

We consider the sharp interface limit for the scalar-valued and vector-valued Allen-Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain of arbitrary dimension in the situation when a two-phase diffuse interface has developed and intersects the boundary . The limit problem is mean curvature flow with °-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on for some time . Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen-Cahn equation. Finally, we prove a spectral estimate for the linearized Allen-Cahn operator in both cases in order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument.

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Submitted
May 14, 2021
Subjects
math.AP

Comments: 132 pages

Mean curvature flowHomogeneous spacesBoundaries & non-compactFree boundary & capillaryEigenvalues & spectrumPhase 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)5Laux 2016 Well-Posedness of a Navier-Stokes/Mean Curvature Flow system (Helmut Abels and Maximilian Moser, 2017)6Abels 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 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)2Abels 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)8Hensel 2021 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)7An-Lacroix 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)4Laux 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 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 author Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “contact angle”

  2. 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 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 author Maximilian Moser · Shared terms “Allen Cahn”, “Cahn equation”, “spectral estimate”

  4. 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 · Shared terms “valued Allen”, “vector valued”, “Allen Cahn”

  5. 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”, “valued Allen”

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

    Helmut Abels and Maximilian MoserWider flows

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

  7. 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”

  8. 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”

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) 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) 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)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  2. 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.