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，可通过参数$\lambda\to 0$的Yosida逼近构造具有奇异势的非局部Cahn-Hilliard方程的解。常规方法基于紧性论证，无法提供任何收敛速率。本文填补了这一空白，获得了显式收敛速率$\sqrt{\lambda}$。证明基于极大单调算子理论以及非局部算子属于Hilbert-Schmidt型这一观察。该估计可为Galerkin方法提供收敛性结果，其中参数$\lambda$可与离散参数相关联，从而得到恰当的误差估计。