--- title: Time-Fractional Cahn–Hilliard Equation url: https://www.emergentmind.com/topics/time-fractional-cahn-hilliard-equation type: topic --- # Time-Fractional Cahn–Hilliard Equation The time-fractional Cahn–Hilliard equation is a phase-field model for phase separation in which the classical first-order time derivative is replaced by a Caputo derivative of order $\alpha\in(0,1)$. In the literature summarized here, this replacement is used to encode nonlocal memory effects, anomalous diffusion, and subdiffusive coarsening, while preserving the Cahn–Hilliard structure of mass-conserving gradient flow. A representative form is \[ \partial_t^\alpha \phi=\operatorname{div}(m(\phi)\nabla\mu),\qquad \mu=\Psi'(\phi)-\varepsilon^2\Delta\phi, \] with constant-mobility reductions such as \[ \partial_t^\alpha u=\gamma \Delta\!\left(-\varepsilon^2\Delta u+F'(u)\right). \] For this class of models, the main research themes are derivation from constitutive laws with memory, continuous and discrete energy dissipation, weak solution theory under degenerate mobility and singular free energies, sharp-interface asymptotics, and structure-preserving numerical schemes on uniform and nonuniform time grids [2104.03096; 2202.12192; 2108.09908]. ## 1. Governing formulation and constitutive origin In the time-fractional Cahn–Hilliard model derived from continuum mixture theory, conservation of mass is combined with a memory-modified Fick law. Writing $\phi$ for the phase variable and $\mu$ for the chemical potential, the constitutive relation is \[ J(t)=-\int_0^t k(t-s)\,m(\phi(s))\nabla\mu(s)\,ds, \qquad k(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}, \] which leads to \[ \partial_t^\alpha \phi=\operatorname{div}(m(\phi)\nabla\mu),\qquad \mu=\Psi'(\phi)-\varepsilon^2\Delta\phi. \] The Caputo derivative is \[ \partial_t^\alpha \varphi(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\frac{d}{ds}\varphi(s)\,ds. \] This model was derived to describe phase separation with nonlocal memory effects, and the analytical framework was developed for positive and degenerating mobility functions together with Landau, Flory–Huggins, and double-obstacle free energies [2104.03096]. For constant mobility, the equation is frequently written as \[ \partial_t^\alpha u=\gamma \Delta\!\left(-\varepsilon^2\Delta u+F'(u)\right), \] with free energy \[ E(u)=\int_\Omega\left(\frac{\varepsilon^2}{2}|\nabla u|^2+F(u)\right)\,dx. \] The same free-energy density appears in several numerical studies, often with $f(\phi)=F'(\phi)=(\phi^2-1)\phi$ and periodic or homogeneous Neumann boundary conditions [2202.12192; 2506.11817; 2006.02061]. A recurring point in the time-fractional literature is that the adjective “fractional” refers specifically to the time derivative. This distinguishes the Caputo-based model from space-fractional Cahn–Hilliard systems involving fractional Laplacians or fractional powers of elliptic operators, which form a separate branch of the literature [1502.06383; 1904.00931]. ## 2. Energy structure and modified dissipation laws A central departure from the classical Cahn–Hilliard equation is that the standard energy dissipation law \[ \frac{d}{dt}E(u)\le 0 \] is generally not satisfied for time-fractional phase-field models because of the nonlocal-in-time memory effect. The continuous theory therefore replaces direct monotonicity of $E$ by monotonicity of a modified, nonlocal-in-time energy that augments the original Ginzburg–Landau functional by a memory accumulation term [2202.12192]. For the time-fractional Cahn–Hilliard equation, let $-\Delta\Psi(t,x)=u(t,x)-u(0,x)$ with suitable boundary and mean constraints. The modified energy is \[ \widetilde E_{CH}(t)=E(t)+\frac{1}{\gamma\Gamma(1-\alpha)}D_{\Delta,\alpha}(t), \] where \[ D_{\Delta,\alpha}(t) = \frac{\|\nabla\Psi(t,\cdot)\|^2}{2t^\alpha} + \frac{\alpha}{2}\int_0^t \frac{\|\nabla(\Psi(t,\cdot)-\Psi(\tau,\cdot))\|^2}{(t-\tau)^{\alpha+1}} \,d\tau . \] The key monotonicity statement is \[ E(t)\le \widetilde E_{CH}(t)\le \widetilde E_{CH}(s)\le E(0), \qquad 0\le s\le t\le T, \] together with \[ \frac{d}{dt}\widetilde E_{CH}(t)\le 0. \] Moreover, $\widetilde E_{CH}(0)=E(0)$, $\widetilde E_{CH}(t)\to E(t)$ as $t\to\infty$ under convergence to steady state, and $\widetilde E_{CH}(t)\to E(t)$ for fixed $t$ as $\alpha\to1$ [2202.12192]. A related variational formulation introduces \[ \mathcal E_\alpha(t)=E(\Phi)+\frac{\kappa}{2}\mathcal I_t^\alpha\|\nabla\mu\|^2, \] for which the continuous inequality \[ \frac{d\mathcal E_\alpha}{dt}+\frac{\kappa}{2}\omega_\alpha(t)\|\nabla\mu\|^2\le 0 \] serves as the template for variable-step discrete energy laws. This framework is asymptotically compatible with the classical Cahn–Hilliard dissipation law as $\alpha\rightarrow1$ [2201.00920]. These results clarify a common misconception. For time-fractional Cahn–Hilliard dynamics, monotonic decay of the original energy is not the fundamental structure. The fundamental structure is monotonic decay of a modified energy that includes memory. Numerical studies nevertheless report that the original energy can decay, and in the continuous theory it is shown to decay with respect to time in a small neighborhood at $t=0$ [2202.12192; 2506.11817]. ## 3. Weak solutions, degeneracy, and analytical difficulties The analytical theory for the Caputo-time model includes existence, uniqueness, and regularity of weak solutions. In the derivation-and-analysis framework based on continuum mixture theory, weak solutions are obtained by the Faedo–Galerkin method, energy estimates, and compactness theorems, with explicit treatment of degenerating mobility and free energies of Landau, Flory–Huggins, and double-obstacle type [2104.03096]. A major obstacle is the missing chain rule for fractional derivatives: \[ \partial_t^\alpha F(u(t))\neq F'(u(t))\,\partial_t^\alpha u(t) \] in general. To compensate for this, a fractional chain inequality for semiconvex functions is proved and used to derive the energy-type estimates needed for well-posedness. For positive mobility and Landau potential, existence of weak solutions is established, and uniqueness holds for constant mobility. For degenerate mobility and general potentials, the analysis uses regularized problems for $m_\delta$ and $\Psi_\delta$, uniform energy estimates, and entropy-type estimates based on an entropy function $\Phi$ with $\Phi''=1/m$; the resulting weak solutions satisfy $|\phi|\le 1$ almost everywhere [2104.03096]. This analytical picture suggests that the memory term alters the standard gradient-flow toolkit at a structural level rather than merely perturbing coefficients. A plausible implication is that the main technical novelty of the time-fractional theory lies not in the elliptic part of the equation, but in recovering energy control without the usual local-in-time differential identities. ## 4. Sharp-interface asymptotics and coarsening laws Matched asymptotic expansions yield sharp-interface limits that differ from the classical Cahn–Hilliard equation both in kinetics and in timescale separation. For constant mobility, the time-fractional Cahn–Hilliard equation \[ \partial_t^\alpha u=\nabla\big(M(u)\nabla\mu\big),\qquad \mu=-\varepsilon^2\Delta u+F'(u) \] has two distinguished asymptotic regimes [2108.09908]. At the timescale $t=O(1)$, the sharp-interface limit is a fractional Stefan problem. The interface law takes the form \[ \mathrm I^{1-\alpha}V=\frac12[\partial_m\mu_0]_-^+ \qquad \text{on } \Gamma. \] At the longer timescale $t_1=\varepsilon^{-1/\alpha}t=O(1)$, the sharp-interface limit becomes a fractional Mullins–Sekerka problem: \[ \partial_{t_1}^\alpha u_0=\Delta\mu_1 \quad \text{in } \Omega\setminus\Gamma, \] \[ \mu_1=-\kappa \frac{S}{[U]} \quad \text{on }\Gamma, \] \[ \mathrm I^{1-\alpha}V=\frac12[\partial_m\mu_1]_-^+ \quad \text{on }\Gamma. \] For one-sided degenerate mobility $M(u)=1+u$, analogous one-sided fractional Stefan and Mullins–Sekerka limits arise, together with an even slower regime at $t_2=\varepsilon^{-2/\alpha}t$ [2108.09908]. The scaling properties of the sharp-interface models imply coarsening laws that are slower than in the classical case. For constant mobility, the predicted rate is \[ \ell(t)\sim t^{\alpha/3}. \] For one-sided degenerate mobility, a crossover from $\alpha/3$ to $\alpha/4$ is obtained at late times [2108.09908]. This asymptotic picture should be read together with computational observations on transient behavior. One numerical study reports that smaller $\alpha$ can produce faster early coarsening, while another reports that for very small $\alpha$ the initial evolution can be sharper but reaching the steady state takes longer [2006.02061; 2104.03096]. These statements concern early-time and long-time regimes, respectively, and are therefore not mutually inconsistent. ## 5. Structure-preserving discretization Because time-fractional solutions are singular at the initial time and because the energy law is intrinsically nonlocal, the numerical literature is dominated by nonuniform temporal meshes and modified discrete energies. Several families of schemes have been designed to preserve mass, energy decay, and asymptotic compatibility with the classical $\alpha=1$ limit [2006.02061; 2201.00920; 2210.12514; 2506.11817; 2508.17178]. | Scheme | Temporal approximation | Stated properties | |---|---|---| | Convex-splitting scheme [2006.02061] | $L1^+$ on non-uniform meshes | second-order in time, spectrally accurate in space, uniquely solvable, mass preserving, unconditionally energy stable | | Variable-step L1-type schemes [2201.00920] | L1, half-grid L1, averaged L1 | L1 and L1\(_\mathrm{h}\) energy stable; adaptive time stepping; asymptotically compatible as $\alpha\to1$ | | Variable-step FBDF2 [2210.12514] | fractional BDF2 | discrete energy dissipation law; asymptotically compatible energy; adaptive stepping | | Linear relaxation [2506.11817] | L1\(^+\)-CN | linear, second-order accurate in time, unconditionally energy stable | | Refined L2-type scheme [2508.17178] | variable-step L2-type | unique solvability, exact discrete volume conservation, proper energy dissipation laws, optimal convergence rates | In the variable-step L1 analysis, the decisive discrete tools are the discrete orthogonal convolution kernels and discrete complementary convolution kernels. They are used to prove positive definiteness of the discrete fractional derivative and to derive discrete cumulative energies of the form \[ \mathcal E_\alpha[\phi^n] = E[\phi^n]+\frac{\kappa}{2}\sum_{j=1}^n p_{n-j}^{(n)}\|\nabla\mu^j\|^2, \qquad \partial_\tau \mathcal E_\alpha[\phi^n]\le 0. \] The same work reports that L1\(_\mathrm{a}\) can lose energy stability if time-step ratios are not controlled [2201.00920]. For second-order variable-step methods, two developments are especially notable. The FBDF2 scheme introduces a local-nonlocal splitting of the fractional BDF2 formula and proves a discrete energy dissipation law under the weak step-ratio constraint \[ 0.4753\le \tau_k/\tau_{k-1}\overline{\rho}\approx4.7476114, \] eliminating the lower bound used in earlier variable-step L2 theory. The same paper couples this time discretization with a fourth-order compact difference scheme in space and proves exact discrete volume conservation, proper energy dissipation laws, and optimal convergence rates [2508.17178]. A complementary line of work addresses the memory cost directly. By approximating the Laplace spectrum $z^{-\alpha}$ of the fractional kernel with a rational function using the adaptive Antoulas–Anderson algorithm, the history integral is replaced by a small system of ODEs. Applied to the time-fractional Cahn–Hilliard problem, this yields a method with $O(N\log N)$ time and $O(\log N)$ memory, together with error bounds and long-time 2D simulations [2102.05139]. ## 6. Stochastic extensions and neighboring fractional models A stochastic extension of the time-fractional Cahn–Hilliard equation replaces the deterministic right-hand side by a fractionally integrated additive Gaussian noise: \[ {}^{C}\partial_t^{\alpha} u-\Delta(-\Delta u+\Phi(u)) = {}^{I}\partial_t^\gamma \dot W(t), \qquad \alpha\in(0,1),\ \gamma\in[0,1]. \] For this model, a piecewise linear finite element method in space is combined with convolution quadrature in time for both time-fractional operators and an $L^2$-projection for the noise. Strong convergence rates are proved for both the spatially semidiscrete and fully discrete schemes, and the temporal Hölder continuity of the solution is identified as a key ingredient in the error analysis. The paper emphasizes that, unlike the stochastic Allen–Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity adds substantial complexity [2402.03790]. The broader fractional Cahn–Hilliard literature also includes models that are not time-fractional in the Caputo sense. One branch studies space-fractional systems such as \[ \partial_t u+(-\Delta)^s w=0,\qquad w=(-\Delta)^\sigma u+W'(u), \] with homogeneous Dirichlet boundary conditions of solid type [1502.06383], while another treats generalized systems with fractional powers $A^{2r}$ and $B^{2\sigma}$ and characterizes their omega-limit sets in terms of the first eigenvalue of $A$ [1904.00931]. A separate space-fractional Cauchy–Dirichlet theory proves global existence of weak solutions, parabolic smoothing effects, and convergence of each solution to a single equilibrium via a variant of the Łojasiewicz–Simon inequality for the fractional Dirichlet Laplacian [1801.01722]. This suggests that the phrase “fractional Cahn–Hilliard equation” is not uniform across the literature; in current usage, the time-fractional equation is the Caputo-memory model, whereas fractional Laplacian models belong to a distinct class. The time-fractional Cahn–Hilliard equation therefore occupies a specific position within the broader fractional phase-field landscape: it retains the mass-conserving Cahn–Hilliard transport structure, but replaces local time evolution by hereditary dynamics. The resulting theory is organized around nonlocal energy functionals, initial-layer-aware time discretizations, and interface laws in which memory persists even in the sharp-interface limit. Source: https://www.emergentmind.com/topics/time-fractional-cahn-hilliard-equation