Variable-Mobility Cahn–Hilliard Equation

• Variable-mobility Cahn–Hilliard equation is a phase-field model that incorporates concentration-dependent mobility to capture both sharp interface and diffuse phase dynamics.
• It leverages a gradient-flow structure that ensures mass conservation and energy dissipation while handling nonlinear and degenerate dynamics.
• Advanced numerical schemes like IEQ, SAV, and operator splitting enable robust simulations across materials science, mathematical biology, and fluid mixture applications.

The variable-mobility Cahn–Hilliard equation generalizes the classical Cahn–Hilliard model of phase separation by allowing the mobility coefficient to depend nontrivially on the order parameter—typically concentration or density. This creates complex nonlinear, and often degenerate, fourth-order parabolic dynamics reflecting the physical inhomogeneity of diffusion in multi-phase systems. Choices of degenerate mobility enforce vanishing mass flux in pure phases and drive sharp-interface or surface-diffusion limiting dynamics, while concentration-dependent forms admit a broader array of sharp and diffuse interface phenomena. The development, analysis, and simulation of variable-mobility Cahn–Hilliard systems have become central in fields ranging from materials science to mathematical biology, with major advances in existence theory, sharp-interface asymptotics, numerical schemes, and applications to evolving and anisotropic geometries.

1. Mathematical Formulation and Classification

In a bounded domain Ω ⊂ ℝᵈ (often equipped with Neumann or periodic boundary conditions), the variable-mobility Cahn–Hilliard equation is

$$\partial_t u = \nabla \cdot \bigl( M(u) \nabla \mu \bigr), \qquad \mu = -\gamma\Delta u + F'(u),$$

where $u(x,t)$ is an order parameter, $M(u)\ge 0$ is the mobility function (possibly vanishing), $\gamma > 0$ is typically a gradient energy constant, and $F$ is the bulk potential, often a double-well or Flory–Huggins/logarithmic form. The choice of $M(u)$, either non-degenerate (bounded below) or degenerate (vanishing at pure phases), fundamentally alters the PDE dynamics.

Mobility Types:

• Non-degenerate: $M(u)\ge M_*>0$ for all $u$. Ensures uniformly parabolic character; classical energy and regularity theory applies (Conti et al., 2024).
• Degenerate: $M(u)=0$ at some $u$ (e.g., $M(u)=u(1-u)$ or $M(u) = (1-u^2)_+$). Leads to loss of uniform parabolicity, appearance of surface-diffusion regimes, and maximum principles on $u$ (Luong, 2023, Elbar et al., 2024).

Potential Choices:

• Polynomial double-well: $F(u)=\frac14(u^2-1)^2$ or $F(u)=u^2(1-u)^2$.
• Logarithmic (Flory–Huggins): $F(u) = u\ln u + (1-u)\ln(1-u) - \theta (u - 1/2)^2$; enforces $u \in (0,1)$ and captures entropy-dominated mixtures (Yang et al., 2017, Elbar et al., 2024).

2. Analytical Foundations and Gradient-Flow Structure

Variable-mobility Cahn–Hilliard equations are gradient flows of free energy $$E[u] = \int ( \frac12 \gamma |\nabla u|^2 + F(u) ) \, dx$$ in a weighted $H^{-1}$–like metric involving $M(u)$. When $M$ is concave, the weighted Wasserstein geometry admits existence theory via minimizing movement (JKO) schemes (Lisini et al., 2012), ensuring:

• Mass conservation: $\int u(x,t)\, dx$ is time-invariant.
• Energy dissipation: $$dE[u]/dt = -\int_\Omega M(u) |\nabla \mu|^2 dx \leq 0$$.
• Nonnegativity and bounds: For degenerate $M$ (e.g., $M(u)=u(1-u)$), maximum principles naturally enforce $u \in [0,1]$ (Elbar et al., 2024, Guillen-Gonzalez et al., 2023).

The weak solution definitions typically require: $$u \in L^{\infty}(0,T; H^1(\Omega)) \cap L^2(0,T; H^2(\Omega))$$, $\partial_t u \in L^2(0,T; (H^1(\Omega))')$, $\mu \in L^2(0,T; H^1(\Omega))$, together with the weak formulation incorporating $M(u)$ as variable weight (Luong, 2023, Elbar et al., 17 Feb 2025).

For singular or logarithmic potentials, additional entropy functionals (e.g., $\Phi''(u) = 1/M(u)$) provide a priori control of lower-order nonlinearities, compactness, and positivity of $u$ (Luong, 2023, Elbar et al., 2024, Perthame et al., 2019).

3. Sharp-Interface and Asymptotic Limits

Degenerate mobility plays a decisive role in setting the sharp-interface or surface-diffusion limit as the small parameter $\epsilon$ (interface width) vanishes:

• For $M(u)$ fully degenerate at the pure phases (e.g., $u=0,1$), formal asymptotics and matched-layer analysis yield that the mass flux vanishes in the bulk, and interface motion is governed by surface diffusion or Mullins–Sekerka-type problems. For instance, with $M(u)=(1-u^2)^p$, as $p\geq 2$, the interface velocity is dominated by the surface Laplacian of curvature (Lee et al., 2015, O'Connor et al., 2016).
• For lower-order degeneracy ($p=1$, e.g., quadratic), there is an additional porous-medium-type bulk diffusion term in the interface law:

$$v_n = C_1 \Delta_s \kappa + C_2 \mu_1 \partial_n \mu_1,$$

with $\Delta_s$ the surface Laplacian, $\kappa$ curvature, and $\mu_1$ the leading-order chemical potential at the interface (Lee et al., 2015).

• In the anisotropic or spatially inhomogeneous setting, sharp-interface limits yield weighted Mullins–Sekerka or anisotropic Hele–Shaw flows, with interface propagation determined by the anisotropy of the gradient term or potential (Elbar et al., 17 Feb 2025).
• The rigorous nonlocal-to-local limit (as nonlocal kernels concentrate) was established for degenerate mobility $m(u)=u(1-u)$, yielding strong convergence in $L^2(0,T;H^1(\Omega))$ and maximum-principle bounds $0\le u\le1$ in both the nonlocal and classical (local) setting (Elbar et al., 2024).

4. Numerical Methods and Energy-Stable Discretizations

Variable-mobility Cahn–Hilliard systems introduce substantial challenges for time integration and spatial discretization due to degeneracy and nonlinearity. Advances include:

• Energy-stable Time Schemes:
  • Invariant Energy Quadratization (IEQ): Transforms nonlinear terms for unconditionally energy-stable, linear, semi-implicit schemes (Yang et al., 2017).
  • Scalar Auxiliary Variable (SAV): Introduces an explicit auxiliary variable encoding either the energy or (notably) the mobility-dependent dissipation. Novel 'mobility-SAV' schemes (first- and second-order) decouple the degeneracy from linear solves, achieving unconditionally stable, robust discretizations even in strong degeneracy regimes (Bretin et al., 2023).
• IMEX and Operator-Splitting Approaches: Linear constant-coefficient parts are evolved implicitly, with nonlinearities treated explicitly, enabling large time steps and energy decay under suitable splitting (Orizaga et al., 2024).
• Structure-Preserving and Maximum-Principle Numerics: Non-centered discretizations, including upwind discontinuous Galerkin and finite-element schemes, preserve energy dissipation and enforce boundedness of solutions up to the truncation error, crucial in the presence of degeneracy (Acosta-Soba et al., 2023, Guillen-Gonzalez et al., 2023).
• Geometric and Surface/Manifold Discretization: Development of variable-mobility Cahn–Hilliard schemes on evolving and/or curved manifolds (e.g., bulk–surface, closed surfaces), employing TraceFEM and adaptive time stepping, with rigorous proofs of energy stability (Olshanskii et al., 2023, Stange, 22 Jul 2025).

These numerical developments enable simulation of surface-diffusion-driven coarsening, interface nucleation, anisotropic flows, and coupled bulk-surface phenomena with high accuracy and fidelity to physical laws.

5. Existence, Regularity, and Long-Time Behavior

The variable-mobility Cahn–Hilliard framework admits robust existence and regularity theorems under broad conditions. Key results include:

• Global Existence of Weak Solutions: For concave, possibly degenerate $M$ and reasonable initial data, global-in-time mass-conserving nonnegative solutions exist (Lisini et al., 2012, Elbar et al., 2024, Luong, 2023, Perthame et al., 2019, Delgadino, 2015).
• Uniqueness and Regularity:
  • Non-degenerate mobility (e.g., $M(u)\geq M_*>0$) and singular/logarithmic potentials: Weak solution uniqueness, propagation of uniform-in-time regularity, instantaneous $L^\infty$ separation from singularities (e.g., $|u|<1-\delta$ for $t>\tau$), and long-time convergence to equilibrium states (characterized by the steady Cahn–Hilliard equation) (Conti et al., 2024, Stange, 22 Jul 2025).
  • Degenerate mobility: Existence and energy dissipation proven even in the absence of full $H^2$ regularity, with solutions preserving bounds ($u\in[0,1]$) and, in the case of singular potentials, using entropy- or Lyapunov-type functionals to obtain compactness and mass conservation (Elbar et al., 2024, Luong, 2023).
• Bulk-Surface and Evolving Domains: Well-posedness and regularity theory extend to settings with coupled bulk and surface equations and variable, possibly non-degenerate, mobility, underpinned by new elliptic regularity theorems for systems with non-constant coefficients (Stange, 22 Jul 2025, O'Connor et al., 2016).

6. Applications and Extensions

Variable-mobility Cahn–Hilliard models are central to:

• Materials Science: Modelling spinodal decomposition, coarsening, grain boundary motion, and surface-diffusion-limited morphological evolution. Degenerate mobility creates surface-diffusion or Mullins–Sekerka limits, essential for modeling interface-controlled dynamics (Lee et al., 2015, Orizaga et al., 2024).
• Mathematical Biology: Describing living tissue behavior, cell-cell adhesion, or population segregation, often via singular single-well potentials and degenerate mobilities (Perthame et al., 2019, Elbar et al., 2024).
• Fluid Mixtures: Cahn–Hilliard–Navier–Stokes with variable mobility and density yields mass-, bound-, and energy-preserving hydrodynamic simulations of multiphase flows (Acosta-Soba et al., 2023).
• Anisotropic and Nonlocal Systems: Incorporation of anisotropy/inhomogeneity in interface energy, disparate mobility, and long-range interactions, leading to anisotropic Hele–Shaw limits and nonlocal-to-local variational convergence (Elbar et al., 17 Feb 2025, Elbar et al., 2024).
• Evolving and Curved Geometries: Formulations on dynamically evolving surfaces, manifolds, or with coupled surface–bulk dynamics, where mobility scaling fundamentally impacts interface motion and pattern formation (O'Connor et al., 2016, Stange, 22 Jul 2025, Olshanskii et al., 2023).

The flexibility and generality of the variable-mobility Cahn–Hilliard paradigm have led it to underpin phase-field modeling in a broad range of physical, biological, and geometric contexts, with ongoing research focused on refinement of rigorous analysis, identification of limiting interface laws, and the development of robust, physically faithful numerical schemes.