Paper · Feb 14, 2019

Existence of weak solution for mean curvature flow with transport term and forcing term

Keisuke Takasao

math.APWider flowsarXiv:1902.05269v3
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Abstract

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

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Submitted
Feb 14, 2019
Last revised
Oct 15, 2019 (v3)
Subjects
math.AP, math.DG

Comments: 22 pages

Mean curvature flowWeak & singular flowsLong-time existence & convergencePhase fields & multiphase flows

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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 Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow (Keisuke Takasao, 2018)3Takasao 2018 A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions (Yoshikazu Giga, Fumihiko Onoue and Keisuke Takasao, 2018)6Giga 2018 On obstacle problem for Brakke's mean curvature flow (Keisuke Takasao, 2021)1Takasao 2021 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)8Hensel 2021 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)2Takasao 2022 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)7Steinke 2023 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)5Jang 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)4Laux 2026
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 Brakke's mean curvature flow

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows, Long-time existence & convergence · Same author Keisuke Takasao · Shared terms “forcing term”, “Allen Cahn”, “Cahn equation”

  2. Existence of weak solution to volume preserving mean curvature flow in higher dimensions

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Cahn equation”, “Allen”

  3. Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Allen Cahn”, “Cahn equation”

  4. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Allen Cahn”, “Cahn equation”

  5. Existence of weak solutions to volume-preserving mean curvature flow with obstacles

    Jiwoong JangWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows, Long-time existence & convergence · Shared terms “forcing term”, “Allen Cahn”, “Cahn equation”

  6. A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions

    Yoshikazu Giga, Fumihiko Onoue and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Allen Cahn”, “Cahn equation”

  7. Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow

    Pascal SteinkeWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Cahn equation”, “Allen Cahn”

  8. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Allen Cahn”, “Cahn equation”, “transport”

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.