Paper · Jun 12, 2019

Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran

math.APWider flowsarXiv:1906.04938
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Abstract

We study a level-set mean curvature flow equation with driving and source terms, and establish convergence results on the asymptotic behavior of solutions as time goes to infinity under some additional assumptions. We also study the associated stationary problem in details in a particular case, and establish Alexandrov's theorem in two dimensions in the viscosity sense, which is of independent interest.

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Submitted
Jun 12, 2019
Subjects
math.AP

Comments: 17 pages

Mean curvature flow

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Connected papers

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 A level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)7Giga 2016 On obstacle problem for mean curvature flow with driving force (Yoshikazu Giga, Hung V. Tran and Longjie Zhang, 2018)1Giga 2018 Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (Yoshikazu Giga and Norbert Pozar, 2020)6Giga 2020 Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)2Jang 2021 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)4Bian 2023 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)3Mitake 2023 Quenching for axisymmetric hypersurfaces under forced mean curvature flows (Hiroyoshi Mitake, Yusuke Oka and Hung Vinh Tran, 2024)5Mitake 2024 A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain (Tokuhiro Eto and Yoshikazu Giga, 2024)8Eto 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On obstacle problem for mean curvature flow with driving force

    Yoshikazu Giga, Hung V. Tran and Longjie ZhangWider flows

    Topics Mean curvature flow · Same authors Yoshikazu Giga, Hung V. Tran · Shared terms “large time”, “time behavior”

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

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

    Topics Mean curvature flow · Same authors Hiroyoshi Mitake, Hung V. Tran · Shared terms “large time”, “level set”, “time behavior”

  3. Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals

    Hiroyoshi Mitake and Hung V. TranWider flows

    Topics Mean curvature flow · Same authors Hiroyoshi Mitake, Hung V. Tran · Shared terms “large time”, “level set”, “time behavior”

  4. A level-set method for a mean curvature flow with a prescribed boundary

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows

    Topics Mean curvature flow · Same authors Yoshikazu Giga, Hiroyoshi Mitake · Shared terms “level set”, “set mean”

  5. Quenching for axisymmetric hypersurfaces under forced mean curvature flows

    Hiroyoshi Mitake, Yusuke Oka and Hung Vinh TranWider flows

    Topics Mean curvature flow · Same authors Hiroyoshi Mitake, Hung V. Tran

  6. Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

    Yoshikazu Giga and Norbert PozarWider flows

    Topics Mean curvature flow · Same author Yoshikazu Giga · Shared terms “level set”, “viscosity”

  7. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Mean curvature flow · Same author Yoshikazu Giga · Shared terms “level set”, “viscosity”

  8. A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

    Tokuhiro Eto and Yoshikazu GigaWider flows

    Topics Mean curvature flow · Same author Yoshikazu Giga · Shared terms “set mean”, “level set”, “viscosity”

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)

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.