It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $\lambda \to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrt{\lambda}$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $\lambda$ could be linked to the discretization parameters, yielding appropriate error estimates.

翻译：众所周知，可以通过参数$\lambda\to 0$的Yosida逼近构造含奇异势的非局部Cahn-Hilliard方程的解。通常方法基于紧性论证，但无法提供收敛速度。本文填补了这一空白，得到了显式收敛速度$\sqrt{\lambda}$。证明基于极大单调算子理论以及非局部算子属于希尔伯特-施密特型的观察。该估计可为Galerkin方法提供收敛性结果，其中参数$\lambda$可与离散化参数建立关联，进而得到适当的误差估计。