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

TL;DR

This paper introduces a new mathematical solution concept for volume-preserving mean curvature flow that guarantees global existence and uniqueness under certain conditions.

Contribution

The novel weak solution concept combines varifold evolution with phase volume transport and introduces volume-preserving gradient-flow calibrations.

Findings

Sharp interface limits of modified nonlocal Allen–Cahn equations yield the new varifold solutions.

A new calibration concept ensures uniqueness of classical solutions within the varifold framework.

Strong solutions are shown to be calibrated under the new definition.

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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