Two-Phase Volume-Preserving Mean Curvature Flow

• Two-phase volume-preserving mean curvature flow is a geometric evolution law where a nonlocal Lagrange multiplier preserves phase volume while reducing interfacial area.
• It unifies smooth, weak, and phase-field formulations to rigorously model interface dynamics and convergence to constant-mean-curvature equilibria.
• Constructive methods like Allen–Cahn approximations and minimizing-movement schemes effectively enforce the volume constraint in numerical and variational settings.

Two-phase volume-preserving mean curvature flow is the constrained evolution of an interface separating a phase $A(t)$ from its complement, with normal velocity given by mean curvature modified by a spatially constant Lagrange multiplier chosen so that the volume of one phase remains fixed. In scalar notation this is commonly written as $V=H-\lambda(t)$ or $V=-H+\lambda(t)$, depending on sign convention; in vector notation recent works write $\mathbf V=H-\lambda n$ or $v=h+\lambda \nu$. The subject includes smooth geometric evolution, phase-field and minimizing-movement constructions, BV and varifold weak formulations, and stability theory near canonical constant-mean-curvature equilibria. It also borders several distinct models in which curvature appears in the interface law but no global volume constraint is imposed (Poiatti, 11 Jul 2025, Chiesa et al., 29 May 2025).

1. Geometric law and constrained gradient-flow structure

In the classical smooth setting, the interface is a hypersurface $M_t=\partial U_t$ or $\Sigma_t=\partial A(t)$ moving by mean curvature with a nonlocal correction. Representative formulations are

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$

and

$$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$

with the sign determined by the chosen orientation and curvature convention (Sinestrari et al., 22 Jan 2025, Poiatti, 11 Jul 2025). In the set-theoretic formulation used for finite-perimeter evolutions, the same law appears as

$$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$

This is the standard volume-preserving mean curvature flow for a single interface separating two complementary phases (Julin, 2023).

The flow is the constrained $V=H-\lambda(t)$0-gradient flow of area. Its defining identities are exact preservation of enclosed volume and monotone decrease of interfacial area: $V=H-\lambda(t)$1 or, for hypersurfaces spanning two planes,

$V=H-\lambda(t)$2

Accordingly, equilibria are constant-mean-curvature hypersurfaces satisfying $V=H-\lambda(t)$3 (Hartley, 2012, Sinestrari et al., 22 Jan 2025).

The phrase “two-phase” is used in two closely related ways. In the strict PDE sense it refers to a partition into a phase and its complement, encoded by a characteristic function $V=H-\lambda(t)$4 or $V=H-\lambda(t)$5. In a narrower geometric sense it refers to a single closed interface whose enclosed region is one phase and whose exterior is the other. Much of the modern weak theory adopts the first viewpoint, coupling a varifold or measure-valued interface to a BV phase indicator (Chiesa et al., 29 May 2025).

2. Model classes and the scope of the term “two-phase”

In the most direct contemporary formulations, two-phase VPMCF is a one-interface/two-region problem: the evolving phase is $V=H-\lambda(t)$6, the second phase is its complement, and the interface motion is constrained by exact preservation of $V=H-\lambda(t)$7. The recent De Giorgi-type theory makes this explicit by taking the unknowns to be an oriented varifold $V=H-\lambda(t)$8 and a phase indicator $V=H-\lambda(t)$9, with exact volume conservation

$V=-H+\lambda(t)$0

and a transport identity for the phase (Poiatti, 11 Jul 2025).

A standard source of confusion is that not every two-phase mean-curvature-driven problem is volume-preserving. The sharp-interface fluid model

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

couples interface motion to incompressible non-Newtonian flow and includes curvature both in the velocity law and in the surface-tension force

$V=-H+\lambda(t)$2

but it has no global Lagrange multiplier $V=-H+\lambda(t)$3. The advective part is volume-preserving by incompressibility, whereas the $V=-H+\lambda(t)$4 term generally changes phase volume. It is therefore not classical VPMCF (Liu et al., 2011). Likewise, the convective Navier–Stokes/phase-transition model

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

is a curvature-driven phase-transition law, not a volume-preserving correction of material transport; the paper explicitly notes that the added curvature term “allows for a change of total masses” (Abels et al., 2017).

Several papers are methodologically adjacent but not strictly two-phase in this sense. Near-cylinder stability with orthogonal boundary contact is analyzed for a single interface by

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

with preserved enclosed volume and decreasing area, but without coupled bulk phases (Hartley, 2012). Capillary-boundary flows in a half-space use weighted nonlocal terms to preserve enclosed volume or capillary area for one hypersurface with contact angle condition

$V=-H+\lambda(t)$7

again without a genuine two-phase bulk PDE system (Sinestrari et al., 2024).

3. Approximation schemes and constructive mechanisms

A dominant constructive route is nonlocal Allen–Cahn approximation. In Takasao’s phase-field construction, the order parameter $V=-H+\lambda(t)$8 solves

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

with

$\mathbf V=H-\lambda n$0

The multiplier is chosen so that

$\mathbf V=H-\lambda n$1

which is the diffuse analogue of phase-volume preservation. The same model satisfies the exact energy dissipation identity

$\mathbf V=H-\lambda n$2

(Takasao, 2015).

A higher-dimensional variant replaces exact preservation of $\mathbf V=H-\lambda n$3 by a penalty-driven multiplier. The approximating equation is still of Allen–Cahn type with forcing $\mathbf V=H-\lambda n$4, but now

$\mathbf V=H-\lambda n$5

and the energy contains a penalty term

$\mathbf V=H-\lambda n$6

This yields approximate volume preservation at the diffuse level and exact preservation in the sharp-interface limit (Takasao, 2022).

The recent Brakke-flow existence theory also uses a nonlocal Allen–Cahn model on the torus,

$\mathbf V=H-\lambda n$7

with $\mathbf V=H-\lambda n$8. The associated energy splits into diffuse surface energy and a penalty term, and the discrepancy measure

$\mathbf V=H-\lambda n$9

is a central compactness quantity (Chiesa et al., 29 May 2025).

A distinct constructive mechanism is the minimizing-movement scheme with a hard volume constraint. For sets of finite perimeter,

$v=h+\lambda \nu$0

and the time-discrete update is

$v=h+\lambda \nu$1

Here the volume constraint is imposed exactly at each step, rather than via a penalization term. The discrete Euler–Lagrange relation is

$v=h+\lambda \nu$2

which is the implicit-time analogue of $v=h+\lambda \nu$3 (Julin, 2023).

4. Weak formulations, existence theorems, and uniqueness mechanisms

The modern weak theory is organized around several complementary notions.

Framework	Unknowns	Signature property
De Giorgi-type varifold solution	oriented varifold $v=h+\lambda \nu$4, phase indicator $v=h+\lambda \nu$5, scalar $v=h+\lambda \nu$6	unconditional global-in-time existence and weak-strong uniqueness (Poiatti, 11 Jul 2025)
Volume-preserving Brakke-flow	integral varifolds $v=h+\lambda \nu$7, BV phase $v=h+\lambda \nu$8	modified Brakke inequality and global existence by phase field (Chiesa et al., 29 May 2025)
$v=h+\lambda \nu$9-flow	integral varifolds, BV phase, generalized velocity	global weak existence in arbitrary dimension $M_t=\partial U_t$0 (Takasao, 2022)
Flat flow	finite-perimeter sets $M_t=\partial U_t$1	global minimizing-movement solution with hard volume constraint (Julin, 2023)

In the De Giorgi-type theory, a weak solution is a pair $M_t=\partial U_t$2 with a normal speed $M_t=\partial U_t$3, a generalized mean curvature $M_t=\partial U_t$4, a scalar $M_t=\partial U_t$5, exact volume conservation, and the compatibility condition

$M_t=\partial U_t$6

The defining energy-dissipation inequality is

$M_t=\partial U_t$7

This formulation extends Hensel–Laux’s mean-curvature-flow theory to the volume-preserving setting and yields weak-strong uniqueness (Poiatti, 11 Jul 2025).

The Brakke-type theory modifies the classical Brakke inequality to absorb the nonlocal volume term. The limiting velocity is

$M_t=\partial U_t$8

and the Brakke inequality contains a localized error

$M_t=\partial U_t$9

which replaces the formally problematic positive $\Sigma_t=\partial A(t)$0-term. The same construction produces an $\Sigma_t=\partial A(t)$1-flow and a BV phase with preserved volume (Chiesa et al., 29 May 2025).

The higher-dimensional $\Sigma_t=\partial A(t)$2-flow existence theory proves that the Allen–Cahn approximation converges, under natural finite-perimeter-type assumptions on the initial data, to a family of $\Sigma_t=\partial A(t)$3-integral Radon measures $\Sigma_t=\partial A(t)$4 and a phase indicator $\Sigma_t=\partial A(t)$5, with

$\Sigma_t=\partial A(t)$6

Under an additional near-sphericity assumption,

$\Sigma_t=\partial A(t)$7

the flow is for short time a distributional BV-solution of

$\Sigma_t=\partial A(t)$8

(Takasao, 2022).

The flat-flow construction starts from bounded finite-perimeter initial sets $\Sigma_t=\partial A(t)$9 and yields global-in-time weak solutions with

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$0

With an additional perimeter-convergence assumption, the limiting flow is a distributional solution of

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$1

in dimensions $$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$2 (Julin, 2023).

5. Stability, equilibria, and constrained geometries

The constrained flow selects constant-mean-curvature equilibria, and a substantial part of the literature studies their stability. Near cylinders spanning two parallel planes, the graph formulation

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$3

leads to the linearized operator

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$4

Its center eigenspace has dimension $$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$5, and all other eigenvalues are negative provided

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$6

Under this sharp threshold, sufficiently small perturbations exist globally and converge exponentially to a cylinder, even without axial symmetry (Hartley, 2012).

For rotationally symmetric hypersurfaces between two equidistant barriers, the flow

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$7

preserves the enclosed volume and decreases area, while maintaining orthogonal contact with the barriers. If

$$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$8

the flow exists for all $$\partial_t F = -(H-h)\nu, \qquad h(t)=\frac{1}{|\Sigma_t|}\int_{\Sigma_t} H\,d\mu_t,$$9, and a subsequence converges to a revolution hypersurface of constant mean curvature (Cabezas-Rivas et al., 2010).

Capillary and contact-angle variants replace the global average by a geometry-adapted weighted nonlocal term. For convex capillary hypersurfaces in a half-space, the volume-preserving power-mean-curvature flow is

$$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$0

with

$$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$1

For smooth strictly convex initial data with $$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$2, the solution exists for all time and converges smoothly to a spherical cap with the same preserved quantity (Sinestrari et al., 2024).

In asymptotically flat and asymptotically Schwarzschild spaces, VPMCF remains an interface-only constrained law rather than a coupled two-phase bulk PDE, but the stability theory is structurally analogous. If the initial surface is sufficiently round in the asymptotic region of a $$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$3-dimensional asymptotically flat manifold of positive ADM mass, the flow

$$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$4

exists globally and converges smoothly to a stable CMC surface (Sinestrari et al., 22 Jan 2025). In Schwarzschild and asymptotically Schwarzschild settings, if the initial hypersurface is sufficiently close to the relevant coordinate sphere or large isoperimetric hypersurface, VPMCF exists for all time and converges to that CMC equilibrium; the asymptotic argument uses graph representations, linearization, and center-manifold analysis (Gui et al., 1 Nov 2025).

6. Boundary effects, fluid coupling, and computational formulations

Boundary contact introduces additional nonlocal and dynamic structure without changing the basic role of the volume constraint. For a hypersurface in a solid container with moving contact line, the interior law is

$$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$5

while the boundary evolves by

$$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$6

The energy

$$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$7

shows that the volume-preserving mean-curvature term, wetting term, and line tension term belong to a single constrained gradient-flow framework. The resulting PDE is second-order and nonlinear with a dynamic boundary condition, and local well-posedness is proved via a Hanzawa transform and maximal $$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$8-regularity (Abels et al., 2014).

By contrast, fluid-coupled “two-phase flow + mean curvature” models should not be identified with VPMCF unless a global Lagrange multiplier is actually present. In the non-Newtonian sharp-interface system on $$\mathbf V = H-\lambda n, \qquad \lambda(t)=\frac{1}{\mathcal H^{d-1}(\partial A(t))}\int_{\partial A(t)} H\cdot n\,d\mathcal H^{d-1},$$9,

$$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$0

the incompressibility constraint preserves only the advective part of the motion, while the curvature term generally changes phase volume. The interface is described by a varifold measure $$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$1, a BV phase indicator $$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$2, and a Brakke-type inequality with transport, but there is no $$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$3-type nonlocal correction (Liu et al., 2011). The same distinction applies to the Navier–Stokes/phase-transition system

$$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$4

where curvature represents phase transition rather than a volume-preserving modification of material transport (Abels et al., 2017).

Numerically, the constrained law can be discretized directly from the Onsager principle. For a closed polygonal curve $$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$5, the discrete energy and dissipation are

$$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$6

and the volume constraint is imposed through

$$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$7

The augmented Rayleighian yields a DAE system

$$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$8

for the nodal velocities and the multiplier $$V_t = -(H_{E_t}-\overline H_{E_t}), \qquad \overline H_{E_t} = \frac{1}{\mathcal H^n(\partial E_t)}\int_{\partial E_t} H_{E_t}\,d\mathcal H^n.$$9, and the semi-discrete scheme satisfies

$V=H-\lambda(t)$00

This formulation is directly faithful to the single-interface conserved-volume problem and is methodologically relevant to two-phase VPMCF in the narrow one-interface sense, although it does not include bulk two-phase PDE coupling or multiple volume constraints (Liu et al., 2024).

The central conceptual distinction running through the subject is therefore precise. Standard two-phase VPMCF consists of a two-region partition whose interface moves by mean curvature corrected by a global scalar chosen to preserve one phase volume exactly. Modern weak theories realize this through BV phase indicators, varifolds, $V=H-\lambda(t)$01-flows, De Giorgi-type dissipation inequalities, or minimizing movements. Nearby two-phase models may include curvature, surface tension, transport, and even incompressibility, but unless they impose a nonlocal constraint of the form $V=H-\lambda(t)$02, they belong to a different class of interfacial dynamics (Poiatti, 11 Jul 2025, Liu et al., 2011).