On the rate of convergence of Yosida approximation for the nonlocal Cahn-Hilliard equation

TL;DR

The paper derives an explicit convergence rate for the Yosida regularization of the nonlocal Cahn–Hilliard equation with a singular potential. By leveraging maximal monotone operator theory and the Hilbert–Schmidt property of the nonlocal kernel, the authors connect $H^{-1}$-gradient-flow-type estimates to $L^2$-convergence, enabling a rate of $\sqrt{\lambda}$ and providing a framework for error analysis in Galerkin discretizations. Under precise structural assumptions on the domain, kernel, and potential, they establish uniform a priori bounds for the regularized problem and then prove the rate via a careful comparison of two regularized solutions and a finite-dimensional decomposition of the nonlocal operator. This result fills a gap by offering a concrete convergence rate rather than mere compactness, with implications for numerical approximations of nonlocal Cahn–Hilliard systems.

Abstract

It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $λ\to 0$. 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 $\sqrtλ$. 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 $λ$ could be linked to the discretization parameters, yielding appropriate error estimates.

Theorems & Definitions (11)

• Remark 2.1
• Lemma 1
• proof
• Theorem 2.1
• Theorem 3.1
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:uniform_estimates']}