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.

中文翻译：

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非局部 Cahn-Hilliard 方程 Yosida 近似的收敛速度

众所周知，可以通过参数 $\lambda \to 0$ 的 Yosida 近似构造具有奇异势的非局部 Cahn-Hilliard 方程的解。通常的方法基于紧致性论证，并且不提供任何收敛率。在这里，我们填补了这个空白，并获得了显式收敛率 $\sqrt{\lambda }$。证明基于最大单调算子理论和非局部算子是希尔伯特-施密特类型的观察。我们的估计可以为伽辽金方法提供收敛结果，其中参数 $\lambda $ 可以链接到离散化参数，从而产生适当的误差估计。