Fractional Allen–Cahn Equation

• The fractional Allen–Cahn equation is a nonlocal PDE that replaces classical derivatives with fractional analogs to model phase separation and interface evolution.
• It integrates Caputo time-fractional derivatives and the fractional Laplacian, providing a rigorous framework with proven well-posedness and error estimates.
• Advanced numerical schemes, including finite element discretization and convolution quadrature, enable efficient simulation of anomalous diffusion and nonlocal interactions.

The fractional Allen–Cahn equation is a class of nonlocal partial differential equations used to model phase separation and interface dynamics, where either the time derivative, the spatial Laplacian, or both are replaced by fractional (nonlocal) analogs. Fractional generalizations capture anomalous diffusion, long-range spatial interactions, or subdiffusive temporal memory effects. This subject intersects applied analysis, geometric measure theory, numerical analysis, and scientific computing, and features prominently in recent developments in phase field modeling, interface evolution, and the theory of nonlocal minimal surfaces.

1. Mathematical Formulation and Operator Definitions

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. The fully fractional Allen–Cahn equation couples a Caputo time-fractional derivative of order $0 < \alpha \le 1$ and a fractional Laplacian of order $0 < s < 1$: $${}^C\!\partial_t^\alpha u(x,t) + \epsilon^2(-\Delta)^s u(x,t) + f(u(x,t)) = 0,\quad x\in\Omega,\quad t\in(0,T]$$ with initial and (nonlocal) homogeneous Dirichlet boundary conditions: $$u(x,0) = u_0(x);\quad u(x,t) = 0 \text{ for } x\in\mathbb{R}^n\setminus\Omega,\, t\ge 0.$$ The nonlinearity typically takes the Allen–Cahn form $f(u) = u - u^3$.

Fractional time derivative: The Caputo derivative $\;^C\!\partial_t^\alpha$ is defined by

$${}^C\!\partial_t^\alpha w(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-\tau)^{-\alpha} \partial_\tau w(\tau)\,d\tau;\qquad 0<\alpha<1$$

and reduces to the classical time derivative as $\alpha \to 1$.

Fractional spatial Laplacian: The spectral or integral definition for $(-\Delta)^s$ over $\mathbb{R}^n$ is

$$(-\Delta)^s u(x) = C(n,s) \,\mathrm{P.V.}\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy$$

with normalization $$C(n,s)=2^{2s} s \Gamma(s+n/2)/(\pi^{n/2}\Gamma(1-s))$$.

Energy functional: The associated nonlocal Ginzburg–Landau functional is

$$F_s(u) = \frac{\epsilon^2}{2}|u|_{H^s(\Omega)}^2 + \int_{\Omega} W(u)\,dx,\qquad W(u) = \frac{1}{4}(u^2-1)^2$$

where $$|u|_{H^s}^2 = \iint_{\Omega \times \Omega} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy$$.

2. Well-Posedness and Regularity of Fractional Allen–Cahn Flows

For $f(u)$ globally Lipschitz and $u_0 \in L^2(\Omega)$, one obtains a unique weak solution

$$u \in C([0,T];L^2(\Omega)) \cap C((0,T];\dot H^s(\Omega))$$

satisfying the variational identity

$$\left({}^C\!\partial_t^\alpha u(t),v\right)_{L^2} + \epsilon^2\langle u(t),v\rangle_{H^s} = (f(u(t)),v)_{L^2}$$

for all $v\in\dot H^s(\Omega)$ and almost every $t>0$. For non-smooth data $u_0\in\dot H^q(\Omega)$, $0<q\leq 2$, the solution exhibits the following instantaneous regularization estimates: $$\|u(t)\|_{\dot H^s} \lesssim t^{-\alpha(2-q)/2}\|u_0\|_{\dot H^q},\quad \|\partial_t u(t)\|_{L^2} \lesssim t^{\alpha/2-1}\|u_0\|_{\dot H^q}$$ These bounds are optimal and reflect limited smoothing in the time-fractional, memory-retaining context (Acosta et al., 2019).

3. Numerical Schemes and Discretization Strategies

Spatial discretization: The domain $\Omega$ is triangulated with shape-regular meshes $\mathcal{T}_h$, allowing construction of continuous, piecewise linear finite element spaces $X_h\subset \dot H^s(\Omega)$. The nonlocal (fractional) stiffness matrix is assembled by integrating the bilinear form associated to the Gagliardo semi-norm.

Time discretization: Caputo derivatives are discretized using convolution quadrature, often via the backward Euler formula: $${}^{RL}\partial_t^\alpha w(t_n) \approx \Delta t^{-\alpha}\sum_{j=0}^n \omega_j w(t_{n-j})$$ with convolution weights generated by the expansion $(1-\xi)^\alpha = \sum_{j=0}^\infty \omega_j \xi^j$.

The fully discrete nonlinear system (with $U^n$ representing FE nodal values) reads: $$M(\omega_0 U^n + \cdots + \omega_n U^0) + \epsilon^2 K U^n = M(\omega_0 U^0+\cdots+\omega_n U^0) + M f(U^n)$$ where $M$ and $K$ are the mass and fractional stiffness matrices. For sufficiently small $\Delta t$, this system has a unique solution, obtained via fixed-point iteration in $X_h$.

Error estimates: For $n$-th time-step $t_n = n\Delta t$: $$\|u(t_n) - U_h^n\|_{L^2} \leq C \left[ h^{2\gamma}\left(t_n^{-\alpha(2-q)/2} + |\log h|^2\right) + \Delta t\, t_n^{-1+\alpha q/2} \right]$$ with $\gamma = \min\{s,1/2-\epsilon\}$. Thus, one achieves spatial order $2\gamma$ and temporal order 1 in weak norms, even with only limited temporal regularity. The constants depend on $\alpha$, $s$, $T$, and the Sobolev index $q$.

4. Asymptotic Limits, Energy $\Gamma$-Convergence, and Limit Geometries

As $\alpha\to 1$ and $s\to 1$, the fractional operators recover their classical counterparts, so the limit equation is the standard Allen–Cahn flow: $\partial_t u - \epsilon^2 \Delta u + f(u) = 0.$ As $s\to 0^+$, the rescaled fractional energies (after appropriate renormalization according to

$\tilde F_s(u) = \frac{F_s(u) - \text{const}}{2s}$

) $\Gamma$-converge in $L^1$ to a nonlocal perimeter-type functional: $$F_0(u) = \begin{cases} \int_{\mathbb{R}^n} \ln|\xi|\,|\widehat{u}(\xi)|^2\,d\xi\,, & u = \pm 1\cdot \chi_E\ +\infty, & \text{otherwise} \end{cases}$$ where $\widehat{u}$ denotes the Fourier transform of the zero-extension of $u$ and $E$ is a measurable set. The minimizers tend to characteristic functions, and the limit energy recovers a nonlocal perimeter (Acosta et al., 2019).

Sketch of $\Gamma$-convergence:

• The liminf direction is shown by decomposing the relevant Fourier modes and employing Fatou’s lemma.
• The limsup direction uses indicator functions $u = \pm 1\chi_E$ and the monotone convergence theorem.
• Equi-coerciveness results from uniform Fourier control, ensuring compactness in $L^2(\Omega)$.

This formalism captures the physical tendency for the minimizers of the fractional Allen–Cahn energy to sharpen as $s\to 0$ and vanish outside of two-phase regions, with interface energy described by a log-perimeter functional.

5. Implementation Issues and Computational Considerations

• The nonlocal fractional stiffness matrix $A_h$ is densely populated, leading to $O(N^2)$ storage and arithmetic, but symmetry and Toeplitz (or block Toeplitz) structure can be exploited for efficiency.
• Fast algorithms: FFT-based solvers reduce cost for uniform grids, but for general meshes, hierarchical low-rank factorizations or adaptive quadrature are required.
• The convolution weights in time are precomputed for fast application in the recursion.
• Non-smooth initial data is handled without loss of order due to the limited smoothing property.
• For strongly nonlocal diffusion (small $s$), the interface width measured in mesh units decreases, requiring finer spatial meshes for accurate interface resolution.
• Fixed-point or Newton-type solvers are adopted per nonlinear time step; in each step, the principal arithmetic cost is a sparse (in FE) or dense (in spectral) matrix solve.

6. Connections to Phase-Field Modeling, Minimal Surfaces, and Geometric Flows

The fully fractional Allen–Cahn equation serves as a phase-field approximation of nonlocal minimal interface evolution (see also (Hasani et al., 6 Nov 2025, Millot et al., 2016, Dipierro et al., 2018)). As the interfacial parameter $\epsilon \to 0$, and under suitable scaling, level sets of solutions concentrate along hypersurfaces evolving via (possibly fractional) mean curvature flow, and energy densities equidistribute along nonlocal minimal surfaces.

• For $s < 1/2$, the $\Gamma$-limit of the fractional Ginzburg–Landau energy is the fractional $2s$-perimeter, and limiting interfaces solve (in a weak sense) the prescribed nonlocal mean curvature equation (Millot et al., 2016).
• In the strongly nonlocal regime, the interfacial layer width shrinks with $\epsilon$, and the convergence to sharp interfaces remains valid for arbitrary data with uniformly bounded energy.
• This variational and geometric perspective has direct implications for the study of fractional minimal surfaces, interface pinning, and the nonlocal isoperimetric problem.

7. Summary Table: Model, Discretization, and Key Theoretical Properties

Aspect	Main Formulation / Result	Reference
Model PDE	$${}^C\!\partial_t^\alpha u + \epsilon^2(-\Delta)^s u + f(u) = 0$$	(Acosta et al., 2019)
Operators	Caputo time-fractional; integral fractional Laplacian	(Acosta et al., 2019)
FE Semidiscretization	Piecewise linear FE, nonlocal Gagliardo semi-norm	(Acosta et al., 2019)
Time Discretization	Backward Euler convolution quadrature; weight recursion	(Acosta et al., 2019)
Error Estimates	$$\| u(t_n)-U_h^n \|_{L^2} \lesssim h^{2\gamma}(t_n^{-\alpha(2-q)/2}+|\log h|^2 ) + \Delta t\, t_n^{-1+\alpha q/2}$$	(Acosta et al., 2019)
Asymptotic Regimes	Limits: $\alpha \to 1^-$, $s\to 1^-$ (local AC); $s\to 0^+$ (log-perimeter $\Gamma$-limit)	(Acosta et al., 2019)
Computational Issues	Dense stiffness matrix, precomputable convolution weights, fast matrix solves	(Acosta et al., 2019)

These results establish the fully fractional Allen–Cahn equation as a rigorous and practical model for simulating nonlocal phase field dynamics, with a comprehensive theoretical analysis and robust, efficient numerical frameworks.