An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived from a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in(0,1)$. The Fourier pseudo-spectral method is applied for the spatial approximation. The proposed scheme inherits the energy dissipation law in the form of the modified discrete energy under the sufficient restriction of the time-step ratios. The convergence of the fully discrete scheme is rigorously provided utilizing the newly proved discrete embedding type convolution inequality dealing with the fractional Laplacian. Besides, the mass conservation and the unique solvability are also theoretically guaranteed. Numerical experiments are carried out to show the accuracy and the energy dissipation both for various interface widths. In particular, the multiple-time-scale evolution of the solution is captured by an adaptive time-stepping strategy in the short-to-long time simulation.

翻译：本文针对由负阶Sobolev空间$H^{-\alpha}$, $\alpha\in(0,1)$中的梯度流导出的、含分数阶拉普拉斯算子的空间分数阶Cahn-Hilliard方程，建立了一种隐式变步长BDF2格式。空间逼近采用傅里叶伪谱方法。在时间步长比的充分限制下，所提格式继承了修正离散能量形式的能量耗散规律。利用新证明的、处理分数阶拉普拉斯算子的离散嵌入型卷积不等式，严格给出了全离散格式的收敛性。此外，质量守恒和唯一可解性也得到理论保证。数值实验展示了不同界面宽度下的精度和能量耗散特性。特别地，通过自适应时间步进策略，在短时到长时模拟中捕捉到了解的多时间尺度演化过程。