Paper · Jul 15, 2022

The nonlocal mean curvature flow of periodic graphs

Bogdan-Vasile Matioc, Christoph Walker

math.APWider flowsarXiv:2207.07474
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Abstract

We establish the well-posedness of the nonlocal mean curvature flow of order for periodic graphs on in all subcritical little Hölder spaces with . Furthermore, we prove that if the solution is initially sufficiently close to its integral mean in , then it exists globally in time and converges exponentially fast towards a constant. The proofs rely on the reformulation of the equation as a quasilinear evolution problem, which is shown to be of parabolic type by a direct localization approach, and on abstract parabolic theories for such problems.

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Submitted
Jul 15, 2022
Subjects
math.AP

Comments: 32 pages

Mean curvature flowAnisotropic & nonlocal flowsLong-time existence & convergence

Its neighbourhood

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  1. Short time existence and smoothness of the nonlocal mean curvature flow of graphs

    Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias WethWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “quasilinear evolution”, “Hölder”

  2. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Shared terms “periodic graph”

  3. Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

    De Gennaro Daniele, Andrea Kubin and Anna KubinWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Shared terms “exponentially fast”, “converges exponentially”

  4. Convergence of the volume preserving fractional mean curvature flow for convex sets

    Vesa Julin and Domenico Angelo La MannaWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence · Shared terms “exponentially fast”

  5. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows

    Topics Anisotropic & nonlocal flows

  6. Deforming Locally Convex Curves into Curves of Constant -order Width

    Laiyuan Gao, Horst Martini and Deyan ZhangWider flows

    Topics Anisotropic & nonlocal flows, Long-time existence & convergence

  7. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “sufficiently”

  8. A mean curvature flow arising in adversarial training

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    Topics Anisotropic & nonlocal flows

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.