Paper · Nov 1, 2019

Mean curvature flow with positive random forcing in 2-d

William M Feldman

math.APWider flowsarXiv:1911.00488
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Abstract

We consider the forced mean curvature flow in 2-d, finite range of dependence and positive random forcing. We prove flatness and existence of effective speed for initially flat propagating fronts. This is the analogue, in random media, of a result of Caffarelli and Monneau. The main new tools are a large scale Lipschitz estimate for the arrival time function, and a quantitative uniqueness result which does not use uniform local regularity.

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Submitted
Nov 1, 2019
Subjects
math.AP

Comments: 26 pages

Mean curvature flowVolume-preserving & constrained flowsStability & uniqueness

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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 Eternal forced mean curvature flows III - Morse homology (Graham Smith, 2016)4Smith 2016 The occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media (William M Feldman and Peter S Morfe, 2021)1Feldman 2021 Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)6Jang 2021 A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting (Jiwoong Jang, 2023)5Jang 2023 Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus (Daniele De Gennaro, Antonia Diana, Andrea Kubin et al., 2023)8De Gennaro 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)3Hirsch 2023 Quenching for axisymmetric hypersurfaces under forced mean curvature flows (Hiroyoshi Mitake, Yusuke Oka and Hung Vinh Tran, 2024)2Mitake 2024 Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions (Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al., 2026)7Pu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media

    William M Feldman and Peter S MorfeWider flows

    Topics Volume-preserving & constrained flows · Same author William M Feldman · Shared terms “forced mean”, “effective”

  2. Quenching for axisymmetric hypersurfaces under forced mean curvature flows

    Hiroyoshi Mitake, Yusuke Oka and Hung Vinh TranWider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  3. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows

    Topics Volume-preserving & constrained flows, Stability & uniqueness · Shared terms “forced mean”

  4. Eternal forced mean curvature flows III - Morse homology

    Graham SmithWider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  5. A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

    Jiwoong JangWider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  6. Level-set forced mean curvature flow with the Neumann boundary condition

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”, “Lipschitz”

  7. Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

    Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al.Wider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

  8. Stability of the surface diffusion flow and volume-preserving mean curvature flow in the flat torus

    Daniele De Gennaro, Antonia Diana, Andrea Kubin et al.Wider flows

    Topics Volume-preserving & constrained flows, Stability & uniqueness

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) 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) 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)

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.