Paper · Oct 18, 2025

Equivalence of weak solution concepts for mean curvature flow

Tim Laux, Anton Ullrich

math.APWider flowsarXiv:2510.16478v2
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Abstract

We provide a connection between weak solution concepts of mean curvature flow. On the one side we have the viscosity solution which is based on the comparison principle. On the other, variational solutions, which are combined Brakke flows and distributional solutions. We prove that if one has a foliation by variational solutions, then the resulting function is the unique viscosity solution. This answers an open question suggested by the work of Evans and Spruck [J. Geom. Anal., 5(1), 1995] and the authors [J. Geom. Anal., 34(12), 2024]. These results show that almost every level set of the viscosity solution is a variational solution. Thus, we establish the equivalence of these solution concepts. Moreover, we show the generic uniqueness of variational solutions.

Comments: 13 pages, 3 figures

Mean curvature flowWeak & singular flowsStability & uniqueness

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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 Distributional solutions to mean curvature flow (Tim Laux, 2021)4Laux 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)3Hensel 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)7Hensel 2021 Weak-strong uniqueness for volume-preserving mean curvature flow (Tim Laux, 2022)6Laux 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)8Laux 2022 Generic level sets in mean curvature flow are BV solutions (Anton Ullrich and Tim Laux, 2023)1Ullrich 2023 Generic mean curvature flow with obstacles (Tim Laux and Keisuke Takasao, 2025)2Laux 2025 Generic Mean Curvature Flow With Rough Weights (Fabius Krämer, Tim Laux and Keisuke Takasao, 2026)5Krämer 2026
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  6. Weak-strong uniqueness for volume-preserving mean curvature flow

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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) 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.