Paper · May 2, 2022

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

Anoumou Attiogbe, Mouahmed Moustapha Fall, Tobias Weth

math.APWider flowsarXiv:2205.01248
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Abstract

We consider the geometric evolution problem of entire graphs moving by fractional mean curvature. For this, we study the associated nonlocal quasilinear evolution equation satisfied by the family of graph functions. We establish, using an analytic semigroup approach, short time existence, uniqueness and optimal Hölder regularity in time and space of classical solutions of the nonlocal equation, depending on the regularity of the initial graph. The method also yields smoothness estimates of the evolving graphs for positive times.

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Submitted
May 2, 2022
Subjects
math.AP, math.DG

Comments: 30 pages

Mean curvature flowAnisotropic & nonlocal flowsStability & uniquenessHarmonic maps & DeTurck trick

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The nonlocal mean curvature flow of periodic graphs

    Bogdan-Vasile Matioc and Christoph WalkerWider flows

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

  2. Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

    Linlin Fan and Peibiao ZhaoWider flows

    Topics Anisotropic & nonlocal flows, Harmonic maps & DeTurck trick · Shared terms “fractional mean”, “classical solution”, “short time”

  3. Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows

    Topics Harmonic maps & DeTurck trick · Shared terms “short time”, “time existence”, “geometric evolution”

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

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Shared terms “fractional mean”, “time existence”

  5. A distributional approach to nonlocal curvature flows

    Filippo Cagnetti, Massimiliano Morini and Dario ReggianiWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”

  6. 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 · Shared terms “fractional mean”

  7. Smoothness up to the free boundary for the -Laplacian evolution equation and the -Gauss curvature flow

    Albert Chau and Ben WeinkoveWider flows

    Topics Harmonic maps & DeTurck trick · Shared terms “smoothness”, “evolution equation”, “short time”

  8. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “short time”, “time existence”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.