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Cited by 4Geometric Analysis and Curvature FlowsSolidification and crystal growth phenomenaNonlinear Partial Differential Equations

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Varifold solutions to volume-preserving mean curvature flow: existence and weak–strong uniqueness

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Poiatti, Andrea·2026·Mathematische Annalen
DOI 10.1007/s00208-026-03421-9PMC PMC12982210
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Abstract

In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak–strong uniqueness. These solutions extend the ones proposed by Hensel and Laux (J Differ Geom 130:209–268, 2025) for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao (Arch Ration Mech Anal 247:52, 2023), any sharp interface limit of solutions to a slightly modified nonlocal Allen–Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.

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References 35 · Cited by 4

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Metadata

Authors
Poiatti, A.
Affiliations
University of Vienna (AT)
Corresponding author
Poiatti, Andrea
ORCID
Poiatti, Andrea
Venue
Mathematische Annalen
Publisher
Springer Science and Business Media LLC
Published
2026-03-12
Volume
394
Issue
4
Pages
98-98
ISSN
  • 0025-5831 (ISSN-L)
  • 1432-1807
DOI
10.1007/s00208-026-03421-9
PMC
PMC12982210
OpenAlex
W7135029064
Cited by
4
References
35
Citations by year
2026: 3
FWCI
6.59
Type
journal article
Language
en
Open access
Licence
Locations
  • Mathematische Annalen (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion, cc-by)
  • arXiv (Cornell University)
Funders
Austrian Science Fund (10.55776/ESP552)
Topics
  • Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
  • Physical Sciences › Materials Science › Materials Chemistry › Solidification and crystal growth phenomena
  • Physical Sciences › Mathematics › Applied Mathematics › Nonlinear Partial Differential Equations
Keywords
volume preserving mean curvature flow, two-phase flow, weak-strong uniqueness, global existence, sharp interface limit, weak solutions, transport equation, phase volume

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Poiatti, A. (2026). Varifold solutions to volume-preserving mean curvature flow: existence and weak–strong uniqueness. Mathematische Annalen. https://doi.org/10.1007/s00208-026-03421-9