Volume-Preserving Gradient-Flow Calibrations

• Volume-preserving gradient-flow calibrations are auxiliary fields that encode constrained steepest-descent dynamics, exemplified by volume-preserving mean curvature flow (VPMCF).
• They combine extended normals, divergence-compatible velocity fields, and relative-entropy functionals to enforce volume constraints and recover geometric evolution laws near interfaces.
• This framework underpins stability, quantitative convergence, and weak–strong uniqueness across sharp-interface, diffuse-interface, and varifold formulations.

Searching arXiv for the cited works to ground the article in current metadata and identifiers. Volume-preserving gradient-flow calibrations are auxiliary fields used to encode constrained steepest-descent dynamics, most prominently volume-preserving mean curvature flow (VPMCF). In the recent literature, a calibration is typically a tuple $(\xi,B,\vartheta,\lambda)$ or $(\xi,B,\vartheta,\hat\lambda)$ consisting of a short extension of the unit normal, an extended velocity field, a transported signed-distance weight, and a time-dependent Lagrange multiplier; combined with relative-entropy functionals, these objects encode the volume constraint, recover the geometric evolution law near the interface, and yield stability, quantitative convergence, and weak–strong uniqueness for distributional, diffuse-interface, and varifold solutions (Laux, 2022, Kroemer et al., 2023, Poiatti, 11 Jul 2025). Related variational constructions appear in minimizing-movement schemes for VPMCF and in broader mass-conserving gradient flows, although the term “calibration” is used more narrowly there (Julin et al., 2022, Bruna et al., 2020).

1. Constrained geometric setting

The primary geometric model is VPMCF, the evolution of a closed hypersurface that decreases surface area while preserving enclosed volume. In one common sign convention the normal velocity satisfies

$V = -H + \lambda(t),$

where $H$ is the scalar mean curvature and $\lambda(t)$ is the spatially constant Lagrange multiplier enforcing preservation of the enclosed volume. For classical solutions,

$$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$

and the area dissipation law is

$$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$

Accordingly, VPMCF is the $L^2$-gradient flow of perimeter constrained to the manifold of hypersurfaces enclosing fixed volume (Laux, 2022).

The same constrained-gradient perspective also appears in the flat-torus analysis of volume-preserving mean curvature flow and surface diffusion flow. There, VPMCF is the $L^2$-gradient flow of the perimeter under a volume constraint, while surface diffusion flow is the $H^{-1}$-gradient flow of the perimeter and preserves volume automatically. The dissipation identities

$(\xi,B,\vartheta,\hat\lambda)$0

for VPMCF and

$(\xi,B,\vartheta,\hat\lambda)$1

for surface diffusion flow make explicit the Lyapunov structure that calibration arguments later exploit in weak–strong stability theory (Gennaro et al., 2023).

A basic technical point is that sign conventions vary. Some sources write $(\xi,B,\vartheta,\hat\lambda)$2, others $(\xi,B,\vartheta,\hat\lambda)$3, with the same volume-preserving multiplier. The underlying structure is unchanged: the volume constraint projects the mean-curvature descent direction onto the tangent space of the fixed-volume manifold (Julin et al., 2022).

2. Calibration fields and relative-entropy functionals

In the VPMCF calibration framework of Fischer et al.’s constrained extension, a volume-preserving gradient-flow calibration is a tuple $(\xi,B,\vartheta,\hat\lambda)$4 with $(\xi,B,\vartheta,\hat\lambda)$5, $(\xi,B,\vartheta,\hat\lambda)$6, and $(\xi,B,\vartheta,\hat\lambda)$7. The vector field $(\xi,B,\vartheta,\hat\lambda)$8 extends the unit normal of the strong interface and is short away from it,

$(\xi,B,\vartheta,\hat\lambda)$9

while $V = -H + \lambda(t),$0 is solenoidal to first order near the interface,

$V = -H + \lambda(t),$1

The pair $V = -H + \lambda(t),$2 satisfies approximate transport identities, and the geometric evolution law is encoded by

$V = -H + \lambda(t),$3

The signed weight $V = -H + \lambda(t),$4 is negative inside, positive outside, and coercive with respect to distance from the interface. These fields enter a relative entropy

$V = -H + \lambda(t),$5

and a bulk error

$V = -H + \lambda(t),$6

which together quantify interfacial tilt-excess and bulk phase mismatch (Laux, 2022).

The quantitative Allen–Cahn theory modifies this template. There the calibration $V = -H + \lambda(t),$7 is required to satisfy

$V = -H + \lambda(t),$8

in addition to the shortness, transport, and geometric evolution conditions

$V = -H + \lambda(t),$9

The relative energy is defined on the diffuse variable $H$0 by

$H$1

and the bulk error is

$H$2

A central innovation is that the extended velocity field on the interface is decomposed as

$H$3

so that tangential correction enforces the required divergence constraints without any direct closeness estimate between the Allen–Cahn and VPMCF Lagrange multipliers (Kroemer et al., 2023).

The De Giorgi–varifold formulation introduces a further variant. Its calibration $H$4 is almost divergence-free,

$H$5

and, crucially, is normal on the interface to leading order,

$H$6

The scalar $H$7 is defined by

$H$8

and the relative entropy becomes

$H$9

supplemented by the bulk functional

$\lambda(t)$0

This formulation is designed for oriented varifolds coupled to a BV phase indicator (Poiatti, 11 Jul 2025).

3. Construction from strong solutions

A recurring theorem is that sufficiently regular strong solutions are calibrated. In the distributional weak–strong uniqueness framework, if $\lambda(t)$1 is a regular solution with $\lambda(t)$2, $\lambda(t)$3, and

$\lambda(t)$4

then there exists a volume-preserving gradient-flow calibration $\lambda(t)$5. The construction uses the signed distance $\lambda(t)$6 to $\lambda(t)$7, smooth truncations $\lambda(t)$8, and

$\lambda(t)$9

The velocity extension is obtained by solving the Neumann–Laplace problem

$$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$0

and setting $$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$1 after smooth compactly supported extension. On the interface the $$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$2 errors vanish, and the extension identity reduces exactly to the strong VPMCF law (Laux, 2022).

In the quantitative convergence theory for the nonlocal Allen–Cahn equation, the strong reference flow $$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$3 is assumed $$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$4, and a calibration is produced by combining the same signed-distance normal extension with a tangentially corrected interface field. One sets

$$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$5

and chooses

$$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$6

with $$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$7, $$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$8. The tangential component is the decisive change relative to the earlier unconstrained calibration framework of Fischer et al.; it localizes the divergence defect and permits the Gronwall estimate for the relative energy (Kroemer et al., 2023).

The varifold theory replaces the scalar Neumann–Laplace extension by a vectorial Stokes construction. Given a smooth VPMCF interface $$\lambda(t)=\frac{1}{\mathcal H^{d-1}(\Sigma(t))}\int_{\Sigma(t)} H\,d\mathcal H^{d-1},$$9, one solves

$$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$0

extends $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$1 smoothly, and defines $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$2 with a cutoff $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$3. This produces an extended velocity that is divergence-free on the interface and normal there. The paper emphasizes that earlier scalar-potential constructions could yield divergence-free extensions without normality, whereas the Stokes solver enforces both properties simultaneously (Poiatti, 11 Jul 2025).

4. Stability, weak–strong uniqueness, and quantitative convergence

The calibration formalism was introduced to prove weak–strong uniqueness for VPMCF by relative entropy. For a calibrated strong solution $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$4 and any distributional solution $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$5 with measurable normal velocity $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$6 and $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$7-control of the Lagrange multiplier, the stability theorem gives, for almost every $$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$8,

$$\frac{d}{dt}\mathcal H^{d-1}(\Sigma(t)) = -\int_{\Sigma(t)} V^2\,d\mathcal H^{d-1}.$$9

If the initial data agree, then $L^2$0 almost everywhere, establishing weak–strong uniqueness. The proof couples the energy-dissipation inequality with the calibration identities and the transported bulk weight $L^2$1, so that the volume constraint is handled without any direct closeness estimate between the weak and strong multipliers $L^2$2 and $L^2$3 (Laux, 2022).

In the diffuse-interface setting, the same strategy yields quantitative convergence. For the nonlocal Allen–Cahn equation

$L^2$4

with mass constraint on $L^2$5, the calibrated relative entropy satisfies

$L^2$6

With well-prepared initial data $L^2$7, one obtains

$L^2$8

The paper identifies this as the optimal convergence rate and attributes the closure of the Gronwall argument to the new calibration with tangential component in $L^2$9 (Kroemer et al., 2023).

The varifold theory extends weak–strong uniqueness to a De Giorgi-type solution class having unconditional global-in-time existence. A varifold solution $L^2$0 satisfies the volume-preserving energy-dissipation inequality

$L^2$1

together with a transport equation for $L^2$2 and compatibility between $L^2$3 and the oriented varifold. For a calibrated strong solution $L^2$4, the resulting stability estimate is

$L^2$5

If the initial data are classical, then the varifold solution coincides with the strong one. The paper emphasizes that normality of $L^2$6 on the interface is what allows weak–strong uniqueness in the general varifold class, without an a priori integrality assumption (Poiatti, 11 Jul 2025).

5. Minimizing movements and calibration-style certificates

Long before the explicit calibration framework, VPMCF had already been formulated as a constrained minimizing movement. In the Mugnai–Seis–Spadaro construction, the one-step functional is

$L^2$7

and the discrete normal velocity is

$L^2$8

The discrete Euler–Lagrange identity has the form of a weak law $L^2$9, and the limit yields global distributional solutions with volume preserved almost everywhere in time. In this setting the “gradient-flow calibration” is a variational certificate: the discrete minimizer is characterized by stationarity for the constrained steepest descent of perimeter in the ATW/Luckhaus–Sturzenhecker metric approximation (Mugnai et al., 2015).

A later flat-flow construction imposes the volume constraint directly at each discrete step: $H^{-1}$0 The Euler–Lagrange equation on the regular part of the discrete boundary is

$H^{-1}$1

and the second variation inequality provides the key density estimate. The paper describes the resulting energy–dissipation balance as a “gradient-flow calibration”: it quantitatively balances perimeter decrease with the squared constrained metric speed $H^{-1}$2 (Julin, 2023).

The consistency theory for flat flow to VPMCF sharpened this discrete picture. Using the minimizing movement scheme of Mugnai–Seis–Spadaro and Julin–Niinikoski, the paper proves that for $H^{-1}$3-regular initial data the flat flow agrees with the classical solution as long as the latter exists, and that the flow is unique and smooth up to the first singular time. The discrete Euler–Lagrange equation

$H^{-1}$4

and the continuous identity

$H^{-1}$5

are explicitly interpreted as the constrained gradient-flow identity, while the one-step energy comparison is described as playing the role of an energy calibration certifying the minimizing property (Julin et al., 2022).

For star-shaped sets, the same variational theme appears without explicit calibrations. The flow is constructed by approximate minimizing movements for the $H^{-1}$6-regularized energy

$H^{-1}$7

together with the pseudo-distance

$H^{-1}$8

The paper states that it does not construct a calibration in the classical geometric-measure-theory sense; instead it relies on discrete energy-dissipation inequalities, an Euler–Lagrange bound for the discrete multiplier, preservation of strong star-shapedness, and perimeter-stability estimates. These are presented as “calibration-like controls” (Kim et al., 2018).

6. Extensions, analogies, and conceptual boundaries

Not every certification mechanism for a volume-preserving gradient flow is an explicit calibration. In the flat-torus stability theory of strictly stable constant-mean-curvature sets, calibration methods are not explicitly used. Instead, the proof combines strict stability, a spectral gap for the Jacobi operator, the dissipation identities for VPMCF or surface diffusion, modulation by translations, and a quantitative Alexandrov-type inequality interpreted as a Lojasiewicz–Simon inequality with exponent $H^{-1}$9. The paper describes this package as playing a role analogous to calibrations in certifying minimality, but tailored to dynamical stability under constrained gradient flows (Gennaro et al., 2023).

The same broad idea extends beyond mean-curvature-driven motions. A generalized Cahn–Hilliard bending-energy flow,

$(\xi,B,\vartheta,\hat\lambda)$00

is a mass-preserving $(\xi,B,\vartheta,\hat\lambda)$01-gradient flow whose sharp-interface limit is a volume-preserving Willmore flow. The zero-mass projection $(\xi,B,\vartheta,\hat\lambda)$02, equivalently the scalar Lagrange multiplier

$(\xi,B,\vartheta,\hat\lambda)$03

calibrates the diffuse mass constraint to the limiting geometric volume constraint. The paper explicitly describes the methodology as providing a principled calibration from diffuse-interface dynamics to constrained geometric motion (Chen, 2024).

A terminological boundary appears in coarse-grained diffusion models with excluded volume. There the microscopic dynamics are a Wasserstein gradient flow, and coarse graining preserves the gradient-flow structure with macroscopic free energy

$(\xi,B,\vartheta,\hat\lambda)$04

and mobility $(\xi,B,\vartheta,\hat\lambda)$05. The resulting PDE is

$(\xi,B,\vartheta,\hat\lambda)$06

The paper calls this “volume-preserving” only in the sense that the flow is mass-conserving and the excluded-volume term penalizes overcrowding at low volume fraction $(\xi,B,\vartheta,\hat\lambda)$07. It states explicitly that this does not mean incompressible flow: the model remains compressible, and singular free energies enforcing maximum packing do not arise at the first correction order (Bruna et al., 2020).

Taken together, these developments delimit the modern meaning of volume-preserving gradient-flow calibrations. In the narrow sense, they are auxiliary fields that turn constrained geometric evolution into a relative-entropy-stable structure, yielding quantitative convergence and weak–strong uniqueness. In the broader variational sense, they point to a family of certification mechanisms—Euler–Lagrange identities, dissipation inequalities, transported weights, and divergence-compatible velocity extensions—through which fixed-volume gradient flows are represented, approximated, and compared across sharp-interface, diffuse-interface, and coarse-grained regimes.