Varifold solutions to Volume-Preserving Mean Curvature Flow: existence and weak-strong uniqueness

TL;DR

This paper introduces a new weak solution concept for volume-preserving mean curvature flow that guarantees existence and uniqueness, extending previous models and connecting to nonlocal Allen-Cahn equations.

Contribution

It develops a novel varifold-based weak solution framework with volume-preserving calibrations, ensuring global existence and weak-strong uniqueness for the flow.

Findings

Any limit of nonlocal Allen-Cahn solutions is a varifold solution.

Regular strong solutions are calibrated under the new notion.

Classical solutions are unique within the new solution class.

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-Laux [J. Differential 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.