Paper · Mar 11, 2020

The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions

Julian Fischer, Sebastian Hensel, Tim Laux, Theresa Simon

math.APWider flowsarXiv:2003.05478v2
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Abstract

We prove that in the absence of topological changes, the notion of BV solutions to planar multiphase mean curvature flow does not allow for a mechanism for (unphysical) non-uniqueness. Our approach is based on the local structure of the energy landscape near a classical evolution by mean curvature. Mean curvature flow being the gradient flow of the surface energy functional, we develop a gradient-flow analogue of the notion of calibrations. Just like the existence of a calibration guarantees that one has reached a global minimum in the energy landscape, the existence of a "gradient flow calibration" ensures that the route of steepest descent in the energy landscape is unique and stable.

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Submitted
Mar 11, 2020
Last revised
Sep 27, 2021 (v2)
Subjects
math.AP

Comments: 104 pages

Mean curvature flowWeak & singular flowsStability & uniquenessPhase fields & multiphase flows

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

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

    Topics Phase fields & multiphase flows, Weak & singular flows, Stability & uniqueness · Same authors Julian Fischer, Sebastian Hensel · Shared terms “weak strong”, “strong uniqueness”, “planar multiphase”

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

    Sebastian Hensel and Tim LauxWider flows

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

  3. 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

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

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

    Sebastian Hensel and Tim LauxWider flows

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

  5. Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow

    Julian Fischer and Alice MarveggioWider flows

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same author Julian Fischer · Shared terms “gradient flow”, “calibrations”

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

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows

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

  7. Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies

    Julian Fischer, Tim Laux and Theresa M. SimonWider flows

    Topics Phase fields & multiphase flows, Stability & uniqueness · Same authors Julian Fischer, Tim Laux, Theresa Simon

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

    Tim Laux and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Tim Laux

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.

Foundational paper in the Phase fields & multiphase flows guide: Fischer–Hensel–Laux–Simon: weak-strong uniqueness for multiphase mean curvature flow.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. 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.