Paper · Jan 26, 2022

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

Keisuke Takasao

math.APWider flowsarXiv:2201.10850
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Abstract

In this paper, we construct a family of integral varifolds, which is a global weak solution to the volume preserving mean curvature flow in the sense of -flow. This flow is also a distributional BV-solution for a short time, when the perimeter of the initial data is sufficiently close to that of ball with the same volume. To construct the flow, we use the Allen–Cahn equation with non-local term motivated by studies of Mugnai, Seis, and Spadaro, and Kim and Kwon. We prove the convergence of the solution for the Allen–Cahn equation to the family of integral varifolds with only natural assumptions for the initial data.

Comments: 42 pages

Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsPhase fields & multiphase flows

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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 2021, 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 new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)7Hensel 2021 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)8Laux 2022 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)6Steinke 2023 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)3Kroemer 2023 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)2Jang 2025 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)1Chiesa 2025 Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness (Andrea Poiatti, 2025)4Poiatti 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)5Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “integral varifolds”

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

    Jiwoong JangWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Shared terms “Cahn equation”, “Kwon”, “Mugnai”

  3. Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

    Milan Kroemer and Tim LauxWider flows PDF

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

  4. Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

    Andrea PoiattiWider flows PDF

    Topics Phase fields & multiphase flows, Volume-preserving & constrained flows, Weak & singular flows · Shared terms “varifold”, “Cahn equation”, “Allen”

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

    Tim Laux and Keisuke TakasaoWider flows PDF

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

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

    Pascal SteinkeWider flows PDF

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

  7. 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 PDF

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

  8. Weak-strong uniqueness for volume-preserving mean curvature flow

    Tim LauxWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “distributional”

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) 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) 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) 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.