---
title: Two-Phase Volume-Preserving Mean Curvature Flow
url: https://www.emergentmind.com/topics/two-phase-volume-preserving-mean-curvature-flow
type: topic
---

# Two-Phase Volume-Preserving Mean Curvature Flow

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 [2507.08783][2505.23222].

## 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 [2501.13091][2507.08783]. 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 [2301.10089].

The flow is the constrained \(L^2\)-gradient flow of area. Its defining identities are exact preservation of enclosed volume and monotone decrease of interfacial area:
\[
\frac{d}{dt}\operatorname{Vol}(\Omega_t)=0,
\qquad
\frac{d}{dt}|\Sigma_t|=-\int_{\Sigma_t}|H-h|^2\,d\mu_t,
\]
or, for hypersurfaces spanning two planes,
\[
\frac{dV}{dt}=0,
\qquad
\frac{d}{dt}|\Omega_t|=-\int_{M^n}(H-h)^2\,d\mu_t.
\]
Accordingly, equilibria are constant-mean-curvature hypersurfaces satisfying \(H\equiv h\) [1205.0339][2501.13091].

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 \(\chi_{A(t)}\) or \(\psi(\cdot,t)\in\{0,1\}\). 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 [2505.23222].

## 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 \(A(t)\), the second phase is its complement, and the interface motion is constrained by exact preservation of \(\mathcal L^d(A(t))\). The recent De Giorgi-type theory makes this explicit by taking the unknowns to be an oriented varifold \(\mu_t\) and a phase indicator \(\chi(\cdot,t)=\chi_{A(t)}\), with exact volume conservation
\[
\int_\Omega \chi(x,t)\,dx=\int_\Omega \chi_0(x)\,dx
\]
and a transport identity for the phase [2507.08783].

A standard source of confusion is that not every two-phase mean-curvature-driven problem is volume-preserving. The sharp-interface fluid model
\[
V_\Gamma=(u\cdot n)n+K_2H
\]
couples interface motion to incompressible non-Newtonian flow and includes curvature both in the velocity law and in the surface-tension force
\[
n\cdot(T^+-T^-)=K_1H,
\]
but it has no global Lagrange multiplier \(\lambda(t)\). The advective part is volume-preserving by incompressibility, whereas the \(K_2H\) term generally changes phase volume. It is therefore not classical VPMCF [1102.4054]. Likewise, the convective Navier–Stokes/phase-transition model
\[
V_{\Gamma(t)}-v|_{\Gamma(t)}\cdot \nu_{\Gamma(t)}=H_{\Gamma(t)}
\]
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” [1705.10832].

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
\[
\frac{\partial \mathbf X}{\partial t}=(h-H)\nu_{\Omega_t},
\]
with preserved enclosed volume and decreasing area, but without coupled bulk phases [1205.0339]. 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
\[
\langle \nu,E_{n+1}\rangle=\cos\theta,
\]
again without a genuine two-phase bulk PDE system [2405.02722].

## 3. Approximation schemes and constructive mechanisms

A dominant constructive route is nonlocal Allen–Cahn approximation. In Takasao’s phase-field construction, the order parameter \(\varphi^\varepsilon\) solves
\[
\varepsilon \varphi_t^\varepsilon
=
\varepsilon \Delta \varphi^\varepsilon
-\frac{W'(\varphi^\varepsilon)}{\varepsilon}
+\lambda^\varepsilon \sqrt{2W(\varphi^\varepsilon)},
\]
with
\[
W(s)=\frac{(1-s^2)^2}{2}, \qquad
k(s)=\int_0^s \sqrt{2W(\tau)}\,d\tau=s-\frac{s^3}{3}.
\]
The multiplier is chosen so that
\[
\frac{d}{dt}\int_\Omega k(\varphi^\varepsilon)\,dx=0,
\]
which is the diffuse analogue of phase-volume preservation. The same model satisfies the exact energy dissipation identity
\[
\frac{d}{dt}\int_\Omega
\left(
\frac{\varepsilon |\nabla \varphi^\varepsilon|^2}{2}
+\frac{W(\varphi^\varepsilon)}{\varepsilon}
\right)\,dx
=
-\int_\Omega \varepsilon (\varphi_t^\varepsilon)^2\,dx
\]
[1511.01687].

A higher-dimensional variant replaces exact preservation of \(\int k(\varphi^\varepsilon)\) by a penalty-driven multiplier. The approximating equation is still of Allen–Cahn type with forcing \(\lambda^\varepsilon(t)\sqrt{2W(\varphi^\varepsilon)}\), but now
\[
\lambda^\varepsilon(t)
=
-\frac{1}{\varepsilon}\alpha
\left(
\int_\Omega k(\varphi_0^\varepsilon)\,dx
-
\int_\Omega k(\varphi^\varepsilon(x,t))\,dx
\right),
\]
and the energy contains a penalty term
\[
E_V^\varepsilon(t)
=
\frac{1}{2\varepsilon}\alpha
\left(
\int_\Omega k(\varphi_0^\varepsilon)\,dx
-
\int_\Omega k(\varphi^\varepsilon(x,t))\,dx
\right)^2.
\]
This yields approximate volume preservation at the diffuse level and exact preservation in the sharp-interface limit [2201.10850].

The recent Brakke-flow existence theory also uses a nonlocal Allen–Cahn model on the torus,
\[
\partial_t\varphi^\varepsilon
=
\Delta \varphi^\varepsilon
-\frac{W'(\varphi^\varepsilon)}{\varepsilon^2}
+\lambda^\varepsilon \sqrt{2W(\varphi^\varepsilon)},
\qquad
\lambda^\varepsilon(t)
=
\frac{1}{\varepsilon^\alpha}
\left(
\int_\Omega k(\varphi_0^\varepsilon)\,dx
-
\int_\Omega k(\varphi^\varepsilon(\cdot,t))\,dx
\right),
\]
with \(\alpha\in(0,1)\). The associated energy splits into diffuse surface energy and a penalty term, and the discrepancy measure
\[
\xi^\varepsilon(\zeta)
=
\frac{1}{\sigma}
\int
\zeta\left(
\frac{\varepsilon |\nabla \varphi^\varepsilon|^2}{2}
-
\frac{W(\varphi^\varepsilon)}{\varepsilon}
\right)
\]
is a central compactness quantity [2505.23222].

A distinct constructive mechanism is the minimizing-movement scheme with a hard volume constraint. For sets of finite perimeter,
\[
\mathcal F_h(E,F)=P(E)+\frac1h\int_E d_F(x)\,dx,
\]
and the time-discrete update is
\[
E_h^{k+1}\in \operatorname*{arg\,min}
\left\{
P(E)+\frac1h\int_E d_{E_h^k}\,dx:\ |E|=|E_h^k|
\right\}.
\]
Here the volume constraint is imposed exactly at each step, rather than via a penalization term. The discrete Euler–Lagrange relation is
\[
\frac{d_F}{h}=-H_E+\lambda
\quad\text{on }\partial^*E,
\]
which is the implicit-time analogue of \(V=-H+\lambda(t)\) [2301.10089].

## 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 \(\mu_t\), phase indicator \(\chi\), scalar \(\lambda\) | unconditional global-in-time existence and weak-strong uniqueness [2507.08783] |
| Volume-preserving Brakke-flow | integral varifolds \(\mu_t\), BV phase \(\psi\) | modified Brakke inequality and global existence by phase field [2505.23222] |
| \(L^2\)-flow | integral varifolds, BV phase, generalized velocity | global weak existence in arbitrary dimension \(d\ge 2\) [2201.10850] |
| Flat flow | finite-perimeter sets \(E_t\) | global minimizing-movement solution with hard volume constraint [2301.10089] |

In the De Giorgi-type theory, a weak solution is a pair \((\mu,\chi)\) with a normal speed \(V\in L^2((0,\infty);L^2(\Omega,d\omega_t))\), a generalized mean curvature \(\mathbf H\), a scalar \(\lambda\in L^2_{\mathrm{loc}}(0,\infty)\), exact volume conservation, and the compatibility condition
\[
\int_\Omega \zeta\cdot d\nabla\chi(\cdot,t)
=
\int_{\Omega\times \mathbb S^{d-1}} \zeta\cdot p\,d\mu_t.
\]
The defining energy-dissipation inequality is
\[
\int_\Omega 1\,d\omega_T
+\frac12\int_0^T\int_\Omega |V|^2\,d\omega_t
+\frac12\int_0^T\int_\Omega |\mathbf H-\lambda \langle \pi_{x,t}\rangle|^2\,d\omega_t
\le
\int_\Omega 1\,d\omega_0.
\]
This formulation extends Hensel–Laux’s mean-curvature-flow theory to the volume-preserving setting and yields weak-strong uniqueness [2507.08783].

The Brakke-type theory modifies the classical Brakke inequality to absorb the nonlocal volume term. The limiting velocity is
\[
v
=
h
-
\lambda
\frac{d\|\nabla\psi(\cdot,t)\|}{d\mu_t}\,\nu,
\]
and the Brakke inequality contains a localized error
\[
r^{d-1}C(1+t_2-t_1)\|\phi\|_{L^\infty},
\]
which replaces the formally problematic positive \(|\lambda|^2\)-term. The same construction produces an \(L^2\)-flow and a BV phase with preserved volume [2505.23222].

The higher-dimensional \(L^2\)-flow existence theory proves that the Allen–Cahn approximation converges, under natural finite-perimeter-type assumptions on the initial data, to a family of \((d-1)\)-integral Radon measures \(\mu_t\) and a phase indicator \(\psi\in BV_{\mathrm{loc}}\cap C_{\mathrm{loc}}([0,\infty);L^1)\), with
\[
\int_\Omega \psi(x,t)\,dx=\mathcal L^d(U_0)
\qquad\forall t\ge 0.
\]
Under an additional near-sphericity assumption,
\[
\mathcal H^{d-1}(M_0)-d\omega_d\,(\mathcal L^d(U_0))^{1-\frac1d}\le \delta_1,
\]
the flow is for short time a distributional BV-solution of
\[
v-h+\lambda \nu=0
\quad\text{on }\partial^*U_t
\]
[2201.10850].

The flat-flow construction starts from bounded finite-perimeter initial sets \(E_0\subset\mathbb R^{n+1}\) and yields global-in-time weak solutions with
\[
|E_t|=|E_0|,\qquad P(E_t)\le P(E_0),
\qquad |E_t\Delta E_s|\le C\sqrt{s-t}.
\]
With an additional perimeter-convergence assumption, the limiting flow is a distributional solution of
\[
V_t=-(H_{E_t}-\overline H_{E_t})
\]
in dimensions \(n\le 6\) [2301.10089].

## 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
\[
\frac{\partial \rho}{\partial t}
=
\sqrt{1+\tilde g_\rho^{ij}\nabla_i\rho\nabla_j\rho}\,(h_\rho-H_\rho),
\qquad
\left.\frac{\partial \rho}{\partial z}\right|_{z=0,d}=0,
\]
leads to the linearized operator
\[
\partial G(0)u
=
\left(\frac{n-1}{R^2}+\Delta_C\right)u
-\frac{n-1}{R^2}\fint_C u\,d\mu_0.
\]
Its center eigenspace has dimension \(n+1\), and all other eigenvalues are negative provided
\[
R>\frac{d\sqrt{n-1}}{\pi}.
\]
Under this sharp threshold, sufficiently small perturbations exist globally and converge exponentially to a cylinder, even without axial symmetry [1205.0339].

For rotationally symmetric hypersurfaces between two equidistant barriers, the flow
\[
\frac{\partial X_t}{\partial t}=(\bar H_t-H_t)N_t
\]
preserves the enclosed volume and decreases area, while maintaining orthogonal contact with the barriers. If
\[
|M| \le \frac{\min\{V,\operatorname{vol}(G)-V\}}
{\|f^{-n}\|_\infty \int_a^b f(z)^n\,dz},
\]
the flow exists for all \(t\ge 0\), and a subsequence converges to a revolution hypersurface of constant mean curvature [1008.4271].

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
\[
\partial_t X=(\phi(t)-H^\alpha)\bar\nu,
\qquad
\langle \nu,E_{n+1}\rangle=\cos\theta,
\]
with
\[
\phi(t)=
\frac{1}{W_{\theta}(\widehat{\Sigma_t})}
\int_{\Sigma_t}
(1-\cos\theta \langle\nu,E_{n+1}\rangle)\,H^\alpha\,dA.
\]
For smooth strictly convex initial data with \(\theta\in(0,\pi/2]\), the solution exists for all time and converges smoothly to a spherical cap with the same preserved quantity [2405.02722].

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 \(3\)-dimensional asymptotically flat manifold of positive ADM mass, the flow
\[
\partial_t F=-(H-h)\nu
\]
exists globally and converges smoothly to a stable CMC surface [2501.13091]. 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 [2511.00435].

## 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
\[
V_\Gamma=H_\Gamma-\overline H,
\]
while the boundary evolves by
\[
V_{\partial D}=a+b\,\varkappa_{\partial D}+\cos(\alpha).
\]
The energy
\[
E(\Gamma)
=
\int_\Gamma 1\,d\mathcal H^2
-
a\int_D 1\,d\mathcal H^2
+
b\int_{\partial\Gamma}1\,d\mathcal H^1
\]
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 \(L^p\)-regularity [1403.1132].

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 \(\mathbb T^d\),
\[
V_\Gamma=(u\cdot n)n+K_2H,
\]
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 \(\mu_t\), a BV phase indicator \(\varphi\in\{\pm1\}\), and a Brakke-type inequality with transport, but there is no \(\lambda(t)\)-type nonlocal correction [1102.4054]. The same distinction applies to the Navier–Stokes/phase-transition system
\[
V_{\Gamma(t)}-v|_{\Gamma(t)}\cdot \nu_{\Gamma(t)}=H_{\Gamma(t)},
\]
where curvature represents phase transition rather than a volume-preserving modification of material transport [1705.10832].

Numerically, the constrained law can be discretized directly from the Onsager principle. For a closed polygonal curve \(\Gamma_h(t)\subset\mathbb R^2\), the discrete energy and dissipation are
\[
\mathcal E_h=\sum_{i=1}^n |x_{i+1}-x_i|,
\qquad
\Phi_h=\frac12\int_{\Gamma_h(t)} |\vec v_h|^2\,ds,
\]
and the volume constraint is imposed through
\[
\frac{dS_h}{dt}=\int_{\Gamma_h(t)} v_{h,n}\,ds=0.
\]
The augmented Rayleighian yields a DAE system
\[
\hat A\dot X=\hat G
\]
for the nodal velocities and the multiplier \(\lambda\), and the semi-discrete scheme satisfies
\[
\frac{d\mathcal E_h}{dt}=-2\Phi_h\le 0.
\]
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 [2404.11935].

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, \(L^2\)-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 \(V=H-\lambda(t)\), they belong to a different class of interfacial dynamics [2507.08783][1102.4054].

Source: https://www.emergentmind.com/topics/two-phase-volume-preserving-mean-curvature-flow