Bulk-Surface Cahn–Hilliard Model

• The bulk–surface Cahn–Hilliard model is a coupled system of PDEs that dynamically governs phase separation in the bulk and diffusion along the surface.
• It enforces boundary mass constraints using Lagrange multipliers and dynamic boundary conditions, ensuring physical and mathematical consistency.
• The framework supports rigorous well-posedness analysis and guides numerical simulations in materials science, biology, and fluid dynamics applications.

A Bulk-Surface Cahn–Hilliard model is a coupled system of partial differential equations (PDEs) that describes the evolution of phase separation phenomena within a domain (the bulk) and its boundary (the surface), often incorporating dynamic boundary conditions, surface diffusion, and explicit coupling between bulk and boundary quantities. This class of models generalizes the classical Cahn–Hilliard equation by allowing boundary values to follow their own time-dependent evolution equations and by introducing coupling mechanisms such as boundary mass constraints, Lagrange multipliers, and transmission (Robin or Dirichlet) boundary conditions. These models are widely used for physical systems where interfaces or surfaces play an active, nontrivial role in phase dynamics, including materials science, biology, and fluid dynamics.

1. Mathematical Structure and Dynamic Boundary Conditions

Traditional Cahn–Hilliard equations impose static boundary conditions (such as homogeneous Neumann for mass conservation in the bulk). In contrast, the bulk–surface Cahn–Hilliard model features dynamic boundary conditions, where the boundary itself evolves via a PDE that incorporates surface diffusion, surface potentials, and possibly external forcing. A canonical formulation on a domain Ω with smooth boundary Γ is

$$\begin{aligned} & u_t - \Delta \mu = 0 && \text{in } Q = \Omega \times (0, T), \ & \mu = - F^{-1}u + T(u) + \xi && \text{in } Q, \quad \xi \in \beta(u), \ & u_\Gamma = u|_\Gamma && \text{on } \Sigma = \Gamma \times (0, T), \ & \partial_t u_\Gamma + \partial_\nu u - \Delta_\Gamma u_\Gamma + \beta_\Gamma(u_\Gamma) + T_\Gamma(u_\Gamma) = f_\Gamma && \text{on } \Sigma, \end{aligned}$$

where:

• $u$ is the bulk order parameter,
• $u_\Gamma$ is the surface (trace) order parameter,
• $\mu$ is the bulk chemical potential,
• $\xi \in \beta(u)$ enforces non-smooth (e.g. singular) potentials,
• $T(u)$, $T_\Gamma(u_\Gamma)$ are Lipschitz perturbations,
• $\Delta_\Gamma$ is the Laplace–Beltrami operator on the manifold Γ,
• $\partial_\nu u$ is the outer normal derivative at the boundary.

This framework allows for conservation and diffusion not only within the bulk but also along the surface, capturing independent surface evolution directly coupled to the interior phase behavior (Colli et al., 2014).

2. Boundary Mass Constraints and Introduction of Lagrange Multipliers

The model enforces a boundary mass constraint through a time-dependent inequality, for example

$$k_* \leq \int_\Gamma w_\Gamma u_\Gamma(t) \, d\Gamma \leq k^*, \quad \forall t \in [0, T],$$

where $w_\Gamma$ is a nonnegative weight and $k_*, k^*$ are prescribed bounds. This constraint is nontrivial: dynamic boundary conditions with surface mass evolution can otherwise allow the total boundary mass to drift during the evolution. The imposition of the constraint introduces a Lagrange multiplier $\lambda_\Gamma$ into the surface dynamics.

Correspondingly, the coupled variational formulation requires:

• A bulk Lagrange multiplier, associated with mass conservation in Ω, often interpreted as the mean of the chemical potential,
• A boundary Lagrange multiplier, enforcing the surface constraint via variational inequalities or the indicator function of a convex set in the abstract formulation.

The chemical potential in the bulk thus alters to

$\mu = -F^{-1}u + T(u) + w,$

where $w$ is the (bulk) Lagrange multiplier. For the boundary mass constraint, the multiplier λ or $\xi_\Gamma$ enters into the dynamic boundary equation through a projection or subdifferential formalism (Colli et al., 2014). The introduction of two distinct Lagrange multipliers is essential for ensuring the correct evolution and mathematical well-posedness of the model.

3. Well-Posedness, Abstract Evolution Inclusion, and Solution Regularity

Well-posedness of the bulk–surface Cahn–Hilliard model with a dynamic boundary and mass constraints is established using the theory of evolution equations governed by subdifferentials—doubly nonlinear evolution inclusions. The main analytical approach involves:

• Defining Hilbert spaces for the bulk (e.g., $H(\Omega)$) and for the surface ($H(\Gamma)$), so that the evolving unknowns are pairs $(u, u_\Gamma)$,
• Considering a convex functional $y$ incorporating the double-well (or singular) potential energies and the indicator function for the mass constraint,
• Formulating the PDE system as an abstract inclusion

$$A u'(t) + \partial (y + I_K)(u(t)) \ni P(f(t) - \text{additional terms}),$$

where $A$ arises from the viscoelastic or nonlocal operators, and $I_K$ is the indicator function for the convex set enforcing the boundary mass constraint,

• Employing a priori energy estimates and compactness arguments to obtain existence, uniqueness, and continuous dependence results.

A complete characterization of the solution includes regularity properties, typically

$$u \in H^1(0, T; H(\Omega)) \cap L^2(0, T; V(\Omega)),\quad u_\Gamma \text{ similar in suitable surface spaces}.$$

The regularity is sufficient to interpret the boundary conditions in the classical sense almost everywhere, even under active constraints (Colli et al., 2014).

4. Role of the Laplace–Beltrami Operator in Surface Diffusion

A distinguishing mathematical feature is the appearance of the Laplace–Beltrami operator $\Delta_\Gamma$ in the surface dynamics. In the dynamic boundary equation,

$$\partial_t u_\Gamma + \partial_\nu u - \Delta_\Gamma u_\Gamma + \beta_\Gamma(u_\Gamma) + T_\Gamma(u_\Gamma) = f_\Gamma$$

the operator $\Delta_\Gamma$ governs surface diffusion and smooths the surface profile, precisely reflecting effects such as surface curvature and inhomogeneous lateral diffusion along $\Gamma$. The inclusion of $\Delta_\Gamma$ is crucial for proper energy dissipation and for modeling the redistribution of phases on curved or rough surfaces. In the abstract operator-theoretic setting, the action of $\Delta_\Gamma$ integrates the geometry of the surface into the bulk–surface energy coupling (Colli et al., 2014).

5. Physical Significance and Applications

The mathematical framework for bulk–surface Cahn–Hilliard models with dynamic boundary conditions and mass constraints is highly relevant for describing phase separation processes where interfaces manifest their own independent dynamics. Applications include:

• Microstructure development in alloys with surface segregation,
• Dynamics of thin films or coatings where interfacial mass exchange is non-negligible,
• Biological membranes or other interfaces where surface diffusion, reactions, and boundary mass conservation play a dominant role.

The presence of mass constraints, Lagrange multipliers, and the Laplace–Beltrami operator allows these models to capture quantitative as well as qualitative features observed in experimental systems. From the viewpoint of mathematical analysis, the rigorous treatment via subdifferential calculus and variational inequalities supplies a solid foundation for further developments in both the theory and numerical simulation of coupled bulk–surface processes.

6. Broader Impact and Methodological Developments

The abstract variational-analytic methodologies introduced—particularly the use of doubly nonlinear evolution inclusions and the operator-theoretic perspective—have influenced subsequent analysis of bulk–surface models involving singular potentials, dynamic transmission conditions, and complex boundary constraints. The systematic use of Lagrange multipliers for distributing global constraints across subsystems plays a central role in both the analysis and modeling of phase-field systems with bulk–surface interplay. This framework has set the stage for extensions to nonlocal models, models with chemical reactions, as well as for numerical methods capable of handling dynamic, mass-constrained boundary phenomena.

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This comprehensive setting provides a foundation for both further mathematical analysis and application-driven modeling of multiphase systems with sophisticated interface dynamics, enforcing rigorous mass constraints and boundary coupling in phase-field descriptions (Colli et al., 2014).