Paper · Apr 15, 2022

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

Helmut Abels, Felicitas Bürger, Harald Garcke

math.APWider flowsarXiv:2204.07626
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Abstract

We prove a short time existence result for a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss a mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. The proof is based on a splitting ansatz, solving both equations separately using linearization and a contraction argument. Our result is formulated for the case of immersed hypersurfaces and yields a uniform lower bound on the existence time that allows for small changes in the initial value of the height function.

Comments: 40 pages

Mean curvature flowHarmonic maps & DeTurck trick

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    Helmut Abels, Felicitas Bürger and Harald GarckeWider flows PDF

    Topics Mean curvature flow · Same authors Helmut Abels, Felicitas Bürger, Harald Garcke · Shared terms “scaled mean”, “diffusion”, “coupling”

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

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

  4. Ricci flow on Finsler manifolds

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

  5. Short time existence for harmonic map heat flow with time-dependent metrics

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

  6. An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

    Ari J. Aiolfi, Giovanni da Silva Nunes, Jaime Ripoll et al.Wider flows PDF

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

  7. Prescribed mean curvature flow for noncompact hypersurfaces in Lorentz manifolds

    Luen-Fai TamWider flows PDF

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

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

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    Topics Harmonic maps & DeTurck trick · Shared terms “evolution equation”, “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) 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.