Paper · May 29, 2025

Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

Andrea Chiesa, Keisuke Takasao

math.APWider flowsarXiv:2505.23222v2
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Abstract

In this paper, we propose a new notion of Brakke inequality for volume preserving mean curvature flow. We show the existence of integral varifolds solving the flow globally-in-time in the corresponding Brakke sense using the phase field method. Moreover, such varifolds are solutions to volume preserving mean curvature flow in the -flow sense as well. We thus extend a previous result by one of the authors [25].

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 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)1Takasao 2022 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)6Laux 2022 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)5Kroemer 2023 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)2Jang 2025 Diffusion-aggregation equations and volume-preserving mean curvature flows (Jiwoong Jang and Antoine Mellet, 2025)7Jang 2025 Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness (Andrea Poiatti, 2025)3Poiatti 2025 Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions (Vedansh Arya, Seongmin Jeon and Vesa Julin, 2026)8Arya 2026 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. Existence of weak solution to volume preserving mean curvature flow in higher dimensions

    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

  3. 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”

  4. 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 “Brakke”

  5. 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

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

    Tim LauxWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows

  7. Diffusion-aggregation equations and volume-preserving mean curvature flows

    Jiwoong Jang and Antoine MelletWider flows PDF

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

  8. Eventual regularity of the volume-preserving mean curvature flow in three and two dimensions

    Vedansh Arya, Seongmin Jeon and Vesa JulinWider flows PDF

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

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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.