Paper · Sep 9, 2021

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

Sebastian Hensel, Tim Laux

math.APWider flowsarXiv:2109.04233v2
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Abstract

We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.

Comments: 38 pages, added additional references

Mean curvature flowWeak & singular flowsStability & uniquenessPhase 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 Weak-strong uniqueness for the mean curvature flow of double bubbles (Sebastian Hensel and Tim Laux, 2021)2Hensel 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)1Hensel 2021 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)5Laux 2022 Convergence of Allen-Cahn equations to De Giorgi's multiphase mean curvature flow (Pascal Steinke, 2023)7Steinke 2023 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)8Kroemer 2023 Stability of multiphase mean curvature flow beyond circular topology changes (Julian Fischer, Sebastian Hensel, Alice Marveggio et al., 2024)6Fischer 2024 Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness (Andrea Poiatti, 2025)3Poiatti 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. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same authors Sebastian Hensel, Tim Laux · Shared terms “weak strong”, “Allen Cahn”, “strong uniqueness”

  2. Weak-strong uniqueness for the mean curvature flow of double bubbles

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same authors Sebastian Hensel, Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “uniqueness principle”

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

    Andrea PoiattiWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Shared terms “varifold solutions”, “weak strong”, “strong uniqueness”

  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 Tim Laux · Shared terms “Allen Cahn”, “Cahn equation”, “Brakke”

  5. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “solution concept”

  6. Stability of multiphase mean curvature flow beyond circular topology changes

    Julian Fischer, Sebastian Hensel, Alice Marveggio et al.Wider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Sebastian Hensel · Shared terms “weak strong”, “strong uniqueness”, “varifold”

  7. 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 Cahn”, “varifold solutions”

  8. 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, Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “Cahn equation”, “Allen”, “weak strong”

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) 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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.