Paper · Dec 22, 2022

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

Tim Laux, Kerrek Stinson, Clemens Ullrich

math.APWider flowsarXiv:2212.11939
PDF Abstract page

As on arXiv

Abstract

The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen-Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models, and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak-strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsStability & uniquenessPhase fields & multiphase flows

Its neighbourhood

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 Weak-strong uniqueness for the mean curvature flow of double bubbles (Sebastian Hensel and Tim Laux, 2021)4Hensel 2021 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)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)3Hensel 2021 Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets (Jakob Fuchs and Tim Laux, 2022)8Fuchs 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 A mean curvature flow arising in adversarial training (Leon Bungert, Tim Laux and Kerrek Stinson, 2024)7Bungert 2024 Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow (Likhit Ganedi, Alice Marveggio and Kerrek Stinson, 2024)1Ganedi 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of a heterogeneous Allen-Cahn equation to weighted mean curvature flow

    Likhit Ganedi, Alice Marveggio and Kerrek StinsonWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same author Kerrek Stinson · Shared terms “diffuse interface”, “weak strong”, “Allen Cahn”

  2. 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, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “solution concept”

  3. 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 author Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “Allen Cahn”

  4. 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 author Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “flow calibrations”

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

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

    Tim LauxWider flows PDF

    Topics Weak & singular flows, Stability & uniqueness · Same author Tim Laux · Shared terms “weak strong”, “strong uniqueness”, “relative entropy”

  7. A mean curvature flow arising in adversarial training

    Leon Bungert, Tim Laux and Kerrek StinsonWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same authors Tim Laux, Kerrek Stinson

  8. Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

    Jakob Fuchs and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows, Weak & singular flows · Same author Tim Laux · Shared terms “approximation”

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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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)

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.