Paper · Jan 7, 2025

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

Jiwoong Jang

math.APWider flowsarXiv:2501.03455v3
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Abstract

We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn equation with a multiplier to a weak solution to the flow. The choice of the multiplier is motivated from [Mugnai-Seis-Spadaro '16], [Kim-Kwon '20], and [Takasao '23], which enables us to complete the comparison between the multiplier and the forcing that stops the intrusion into the obstacle. We also prove the vanishing of the discrepancy measure by dealing with the forcing term that is now spatially dependent due to the obstacles.

Comments: v2->v3: further details provided, 47 pages, 1 figure

Mean curvature flowVolume-preserving & constrained flowsWeak & 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 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)8Laux 2022 Flat flow solution to the mean curvature flow with volume constraint (Vesa Julin, 2023)7Julin 2023 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)6Kroemer 2023 Diffusion-aggregation equations and volume-preserving mean curvature flows (Jiwoong Jang and Antoine Mellet, 2025)3Jang 2025 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)5Chiesa 2025 Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness (Andrea Poiatti, 2025)4Poiatti 2025 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)2Jang 2025
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 · Shared terms “Cahn equation”, “Kwon”, “Mugnai”

  2. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows, Long-time existence & convergence · Same author Jiwoong Jang · Shared terms “multiplier”

  3. 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 · Same author Jiwoong Jang · Shared terms “Allen Cahn”, “Cahn equation”, “forcing term”

  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 “Takasao”, “Allen Cahn”, “Cahn equation”

  5. 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

  6. 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”, “multiplier”, “Allen”

  7. Flat flow solution to the mean curvature flow with volume constraint

    Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “Mugnai Seis”, “Seis Spadaro”

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

    Tim LauxWider flows PDF

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

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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) 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)

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.