Paper · Apr 3, 2024

Stability of multiphase mean curvature flow beyond circular topology changes

Julian Fischer, Sebastian Hensel, Alice Marveggio, Maximilian Moser

math.APWider flowsarXiv:2404.02884
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Abstract

We prove a weak-strong uniqueness principle for varifold-BV solutions to planar multiphase mean curvature flow beyond a circular topology change: Assuming that there exists a classical solution with an interface that becomes increasingly circular and shrinks to a point, any varifold-BV solution with the same initial interface must coincide with it, and any varifold-BV solution with similar initial data must undergo the same type of topology change. Our result illustrates the robustness of the relative energy method for establishing weak-strong uniqueness principles for interface evolution equations, showing that it may also be applied beyond certain topological changes.

Comments: 56 pages, 9 figures

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.

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    Topics Phase fields & multiphase flows, Stability & uniqueness · Same authors Julian Fischer, Alice Marveggio · Shared terms “interface”, “relative”, “initial data”

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

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  4. Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

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    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Alice Marveggio · Shared terms “uniqueness principle”, “weak strong”, “interface”

  5. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

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

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

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

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    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “interface”

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

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

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.