Paper · Aug 1, 2021

The occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media

William M Feldman, Peter S Morfe

math.APWider flowsarXiv:2108.00558v3
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Abstract

We construct several examples related to the scaling limits of energy minimizers and gradient flows of surface energy functionals in heterogeneous media. These include both sharp and diffuse interface models. The focus is on two separate but related issues, the regularity of effective surface tensions and the occurrence of zero mobility in the associated gradient flows. On regularity we build on the theory of Goldman, Chambolle and Novaga to show that gradient discontinuities in the surface tension are generic for sharp interface models. In the diffuse interface case we only show that the laminations by plane-like solutions satisfying the strong Birkhoff property generically are not foliations and do have gaps. On mobility we construct examples in both the sharp and diffuse interface case where the homogenization scaling limit of the gradient flow is trivial, i.e. there is pinning at every direction. In the sharp interface case, these are related to examples previously constructed by Novaga and Valdinoci for forced mean curvature flow.

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Submitted
Aug 1, 2021
Last revised
Jan 6, 2023 (v3)
Subjects
math.AP

Comments: 56 pages, multiple revisions made thanks to comments from anonymous reviewer

Mean curvature flowOther curvature flowsVolume-preserving & constrained flowsPhase fields & multiphase flows

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  1. Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

    Likhit Ganedi, Alice Marveggio and Kerrek StinsonWider flows

    Topics Phase fields & multiphase flows · Shared terms “diffuse interface”, “Allen Cahn”, “heterogeneous”

  2. Generalised stochastic curvature flow in , and sharp interface limit for the stochastic Allen-Cahn equation with nonlinear diffusion

    Weijun Xu and Shuhan ZhouWider flows

    Topics Phase fields & multiphase flows, Other curvature flows · Shared terms “Allen Cahn”, “sharp interface”, “direction”

  3. 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 · Shared terms “diffuse interface”, “interface models”, “Allen Cahn”

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

    Helmut Abels, Christoph Hurm and Maximilian MoserWider flows

    Topics Phase fields & multiphase flows · Shared terms “Allen Cahn”, “diffuse interface”, “sharp interface”

  5. 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”, “interface”, “gradient flow”

  6. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows · Shared terms “Allen”, “Cahn”, “gradient flow”

  7. Long time behavior for a curvature flow of networks related to grain bundary motion with the effect of lattice misorientations

    Takashi Kagaya, Masashi Mizuno and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Other curvature flows · Shared terms “surface tension”, “mobility”

  8. 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, Volume-preserving & constrained flows · Shared terms “Allen”, “Cahn”, “gradient flow”

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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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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.