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Surface Stokes-Cahn-Hilliard System: Analysis and Structure-Preserving Discretization

Maxim Olshanskii, Mansur Shakipov

arXiv:2610.06552v1math.APcs.NAmath.NA

TL;DR

The paper addresses analysis of the inertia-free surface Stokes–Cahn–Hilliard system, where prior continuous-dependence results for fluid systems rely on positive density. It proves weak-solution results and develops a structure-preserving finite element method, establishing global well-posedness and conservation properties while noting an open geometric question about Killing fields.

  • Problem

    For inertia-free coupled surface flow and phase separation, prior results for positive-density fluid systems do not directly provide well-posedness and energy-norm continuous dependence from initial time.

    Source: §1 · p. 2

  • Method

    The paper analyzes the weak formulation and develops a TraceFEM with a penalty on approximate Killing components and projections in transport and capillary terms.

    Source: §1 · pp. 1, 3

  • Results

    Global weak solutions are unique for H1 initial data, satisfy energy equality and mass conservation, and depend Lipschitz-continuously on initial data in energy norms, including for different means.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The coupled problem admits a weak-solution theory and a fully discrete method that preserves total phase and an energy identity.

    Source: §1 · pp. 2–3

    Whether intrinsic and extrinsic Killing spaces coincide on a general smooth compact connected boundaryless surface embedded in R3 remains open.

    Source: §6.1 · p. 19

Abstract

from arXiv · show

We study the surface Stokes-Cahn-Hilliard system governing tangential flow and phase separation in viscous fluid membranes. The equations are posed on a closed, stationary $C^2$ hypersurface $\Gamma\subset\mathbb{R}^{d+1}$, $d=2,3$. For constant viscosity and mobility and a class of regular double-well potentials, we prove global existence and uniqueness of weak solutions for initial phase fields in $H^1(\Gamma)$, with the velocity chosen orthogonal to the Killing fields. These solutions satisfy an energy equality and conserve mass. We establish Lipschitz continuous dependence in energy norms on every finite time interval, including for initial data with different means, and derive higher regularity results for $d=2$ under additional smoothness assumptions on the surface, potential, and initial data. We also develop a fully discrete unfitted finite element method that conserves the total phase and satisfies an exact discrete energy identity. The method combines a penalty on approximate Killing fields with projections in the transport and capillary terms to control these modes while preserving the energy balance. Numerical experiments examine the identification of Killing fields and illustrate phase separation on a torus and a liposome with experimentally informed material parameters.

1. Introduction

The paper analyzes an inertia-free surface fluid–phase system and develops a finite element method designed to preserve its conservation and energy structure. Its analysis addresses weak-solution well-posedness, continuous dependence, and the treatment of Killing fields.

1. Introduction

  • Lateral phase separation motivates the model through links to lipid-raft hypotheses, observed coexisting vesicle domains, and applications to functional liposomes and bilayers.Earlier models and studies consider membrane mechanics, bulk flow, and interactions among surface flow, composition, and geometry.Source: §1 · p. 1
  • The study concerns the surface Stokes–Cahn–Hilliard system on a closed stationary C2 hypersurface in dimensions d = 2, 3, with constant viscosity and mobility and regular double-well potentials.The model neglects inertia at small surface Reynolds numbers.Source: §1 · p. 1
  • For the inertia-free problem, the paper establishes energy-norm continuous dependence from the initial time, unlike estimates for positive-density systems that use stronger solution regularity.The analysis assumes regular potentials and uses the regularity available for weak solutions.Source: §1 · p. 2
  • Killing fields leave viscous stress unchanged but can alter phase transport, so the paper selects velocities orthogonal to these fields and controls their discrete counterparts.The capillary force is orthogonal to Killing fields, ensuring compatibility of the momentum equation.Source: §1 · p. 2
  • Global weak solutions are unique for every H1 initial phase field, conserve mass, satisfy an energy identity, and depend Lipschitz-continuously on data, including data with different means.For d = 2, higher regularity follows under corresponding smoothness assumptions on the surface, potential, and initial data.Source: §1 · p. 2
  • The fully discrete TraceFEM conserves total phase and satisfies an energy identity, using a penalty on approximate Killing components and projections in transport and capillary terms.The projections remove the identified Killing component from phase transport while preserving the cancellation underlying the discrete energy law.Source: §1 · pp. 2–3

2. Governing equations

The governing system couples tangential flow and phase separation on a closed hypersurface, with a velocity constraint that resolves the kernel of the surface viscous operator. For smooth solutions, the system has an energy balance and conserves total phase.

2. Governing equations

  • The system seeks tangential velocity orthogonal to the Killing fields, pressure, phase field, and chemical potential for a given initial phase field on the surface.The surface has dimension d = 2 or 3 and is at least C2.Source: §2 · p. 3, §2.1 · p. 4

2.1. The Stokes–Cahn–Hilliard system.

  • The surface viscous operator has Killing fields in its kernel, so the momentum equation determines velocity only up to a Killing field; imposing L2 orthogonality removes this indeterminacy.An alternative is adding a positive zeroth-order friction term, which models dissipation due to friction with an ambient medium.Source: §2.1 · p. 4
  • The capillary force is L2-orthogonal to all Killing fields, making the momentum equation compatible with the viscous operator kernel without added friction.With friction, the energy law gains an additional velocity-dissipation term.Source: §2.1 · p. 4
  • Smooth solutions satisfy an energy balance derived by testing the momentum, phase, and chemical-potential equations and combining their terms.The phase equation contributes the dissipation term involving mobility and the surface gradient of chemical potential.Source: §2.1 · p. 5
  • The total phase is conserved: its value at time t equals its initial value.This follows by integrating the phase-field equation over the surface.Source: §2.1 · p. 5

3. Preliminaries

The preliminaries set conventions for surface Sobolev spaces and differential operators, then collect geometric identities and estimates used in the analysis of tangential fields and Killing fields.

3. Preliminaries

  • The paper uses Lebesgue and Sobolev spaces on the surface, extends differential operators through density, and notes a regularity loss for tangential vector fields.The stated density property applies on C^m surfaces for derivatives up to order m.Source: §3 · p. 5
  • Killing fields satisfy geometric identities involving the Weingarten and Ricci maps, and their weak Bochner–Laplace formulation yields higher regularity.The projection onto Killing fields extends to Lp spaces and is bounded there.Source: §3 · pp. 5–6
  • A Korn inequality controls the H1 norm of tangential fields orthogonal to Killing fields through their symmetric surface derivative.The argument relates the L2-orthogonal complement to the H1-orthogonal complement using finite dimensionality of the Killing space.Source: §3 · p. 6

4. Weak formulation

This section presents the weak formulation and establishes global well-posedness, mass conservation, and an exact energy equality, using a centered convection form to enable cancellation in the energy balance. It also gives higher-regularity results under additional smoothness assumptions in two dimensions.

4. Weak formulation

  • The centered convection form makes the capillary and phase-convection terms cancel when testing with velocity and chemical potential, without requiring pointwise divΓu = 0.This consistent modification supports a structure-preserving numerical method when the divergence constraint is difficult to enforce pointwise.Source: §4 · p. 7
  • Assumption 1 suffices for existence and continuous dependence, while higher-regularity results require additional smoothness of f0.The section identifies regular potentials such as the Ginzburg–Landau potential as examples.Source: §4 · p. 7

4.1. Well-posedness.

  • For C2 surfaces in d=2,3, every initial phase field in H1(Γ) yields a global weak solution satisfying an energy inequality and conserving mass.The energy inequality bounds the dissipation from surface strain and chemical-potential gradients by the initial energy.Source: §4.1 · pp. 7, 10
  • For Γ of class C2 and d ∈ {2, 3}, if (c, µ) ∈ L∞(I; H1(Γ)) × L2(I; H1(Γ)) satisfy (14d), elliptic regularity upgrades c to L2(I; W2,6(Γ)).Source: §4.1 · pp. 10–11
  • A technical lemma controls differences in f0′ for fields with L∞(I; H1(Γ)), L4(I; L∞(Γ)), and L2(I; W1,6(Γ)) regularity.The stated constants depend on the surface, potential, and the fields’ L∞(I; H1(Γ)) norms.Source: §4.1 · p. 11
  • Theorem 4.4 establishes continuous dependence in energy norms for two weak solutions, with its estimate constant depending on the potential, time horizon, surface, material parameters, and solutions.The result applies on C2 surfaces in d=2,3.Source: §4.1 · p. 12
  • For a C2 surface in dimensions 2 or 3, every H1 initial phase field yields a unique weak solution satisfying mass conservation and the energy equality.Mass is conserved for all times in [0, T].Source: §4.1 · p. 14
  • Testing the momentum and phase equations and applying a chain rule upgrades the energy inequality to an exact energy equality for every weak solution.The proof tests the momentum equation with the velocity and the phase equation with the chemical potential.Source: §4.1 · p. 14

4.2. Higher regularity.

  • For Γ of class Cm+4, the parabolic operator ∂t + ϱΔ²Γ maps the stated solution space homeomorphically onto the forcing and initial-data spaces.Its stability constants depend only on Γ, ϱ, m, and T.Source: §4.2 · p. 14
  • In d = 2, if Γ is Cm+4, c0 is Hm+2, and f0 is Cm+2,1_loc, the weak solution gains higher regularity.For m = 0, c lies in L2(I; H4(Γ)) ∩ H1(I; L2(Γ)), µ in L2(I; H2(Γ)), and (u, p) in L2(I; H3(Γ)) × L2(I; H2(Γ)).Source: §4.2 · pp. 14–15
  • For m ≥ 1, higher spatial regularity follows by bootstrapping when f0 is sufficiently regular, with bounds also depending on its Cm+2,1 norm over the solution range.The range bound R is controlled by the m = 0 estimate.Source: §4.2 · pp. 14–15
  • The d = 2 regularity proof first upgrades the Stokes variables, then c, and uses maximal parabolic regularity to obtain c in H4.The key transport estimate uses W2,4/3(Γ) embedding into C0(Γ), which holds for d = 2 but not d = 3.Source: §4.2 · p. 15
  • With sufficiently smooth data, time regularity can also be bootstrapped; the surface being closed and stationary removes boundary compatibility conditions, and only c needs independent initial data.The authors do not pursue the time-regularity statements because c, µ, u, and p gain temporal regularity at different rates.Source: §4.2 · p. 15

5. Discretization

The section develops a structure-preserving TraceFEM on an implicitly represented, unfitted surface, then adds stabilization and projections to control discrete Killing modes while retaining conservation and energy balance.

5.1. TraceFEM.

  • TraceFEM represents the surface as the zero level set of an approximate level-set function and uses an unfitted background mesh.The discrete surface is piecewise smooth; higher-order level-set approximations may use isoparametric TraceFEM.Source: §5.1 · p. 16

5.2. Stabilization and bilinear forms.

  • A mesh-scaled Killing penalty controls discrete Killing modes, while a curvature-corrected derivative avoids differentiating the discontinuous projection.The penalty is sKil(uh, vh) = h^-2(Phuh, Phvh)Γh.Source: §5.2 · p. 16

5.3. The semidiscrete method.

  • Testing the semidiscrete method yields an energy balance for its solution.The formulation initializes the phase through a suitable projection or interpolation and uses a centered discrete transport form.Source: §5.4 · p. 17, §5.3 · p. 17

5.4. Structure preservation.

  • Using the same projection in transport and capillary terms preserves the cancellation underlying the stabilized semidiscrete energy balance.The method also penalizes the approximate Killing component of velocity.Source: §5.4 · p. 17

5.5. Fully discrete method.

  • The fully discrete method has an exact energy equality and conserves total phase; time discretization adds no dissipation beyond the semidiscrete law.Rannacher startup can damp stiff modes for rough initial data, but its startup steps do not satisfy the exact energy identity.Source: §5.5 · p. 18

6. Numerical experiments

The experiments examine eigenvalue-based Killing-field identification and illustrate phase separation and flow on a torus and an experimentally informed liposome.

6.2. Phase separation on a torus.

  • On the torus, small-scale fluctuations grow into larger domains, with substantial interface flow accompanying rearrangement and changes in domain connectivity.At later times, evolution slows as the interface approaches a perimeter-minimizing configuration subject to conserved total phase area.Source: §6.2 · p. 20, Figure 1 · p. 22
  • All torus energy histories decrease; higher mobility accelerates initial energy decay, while viscosity curves do not remain ordered as domain configurations evolve.Neither mobility nor viscosity enters the energy functional; both affect the dynamics and dissipation, so coarsening time and routes to lower energy depend on phase diffusion and viscous transport.Source: §6.2 · pp. 20–21, Figure 2 · p. 23

6.1. Identification of Killing vector fields.

  • On the torus, refinement separates the intrinsic Killing mode from positive modes; the six-dimensional rigid-motion procedure identifies extrinsic fields when Eextr = E.No assumption that the discrete surface has a trivial Killing space is needed; the extrinsic procedure detects only the extrinsic subspace when Eextr ≠ E.Source: §6.1 · pp. 19–20, Table 1 · p. 21

6.3. A phase separated liposome.

  • Liquid-disordered domains coarsen within the liquid-ordered majority phase, with pronounced early deformation and later larger, rounder domains and slower evolution.The simulation is qualitatively consistent with experimental data; quantitative microscopy comparison would require domain statistics across multiple initial realizations.Source: §6.3 · p. 23

A.1. Integration by parts identity.

The appendix establishes a surface integration-by-parts identity under C2 regularity and outlines its weak formulation.

A.1. Integration by parts identity.

  • For a C2 surface, the weak identity relates the inner product of tangential derivatives to the divergence terms and curvature contribution.The stated identity is ((∇Γv)^T, ∇Γw) = (divΓv, divΓw) − (Rv, w).Source: §A.1 · p. 24
  • The proof applies surface integration by parts to the projected derivative expression, using tangentiality and the stated geometric identities.The derivation invokes the tangential-field identity from [34] and identifies R = tr(B)B − BB.Source: §A.1 · pp. 24–25
  • Because the identity involves only first derivatives, the argument extends to v ∈ C1(Γ).The passage explicitly notes this extension after establishing the identity.Source: §A.1 · p. 25

A.2. Maximal parabolic L2-regularity.

This appendix develops maximal parabolic L2-regularity through a Sobolev-space embedding and an abstract parabolic existence result.

A.2. Maximal parabolic L2-regularity.

  • On a C2 surface, the fourth-order parabolic operator defines a linear homeomorphism onto L2(I; H2(Γ)′) × L2(Γ), with stability constants depending on ϱ, Γ, and T.The result is stated for each ϱ > 0.Source: §A.2 · p. 25
  • For m ≥ 0, Xm+2 embeds continuously into C0([0, T]; Hm+2(Γ)).Here Xm+2 = L2(I; Hm+4(Γ)) ∩ H1(I; Hm(Γ)).Source: §A.2 · p. 25
  • The proof uses a convex lower-semicontinuous energy functional and an abstract parabolic result, then applies elliptic regularity to obtain the solution estimate.Uniqueness follows from the earlier parabolic proposition, and boundedness uses the embedding into continuous-in-time Sobolev functions.Source: §A.2 · p. 26
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