Nonlocal-to-Local Convergence of the Cahn-Hilliard Equation and its Operator

We prove convergence of a sequence of weak solutions to the nonlocal Cahn-Hilliard equation to the weak solution to the corresponding local Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth bounded domains with Neumann boundary condition and a W^{1,1}-kernel. The proof is based on an energy method. Additionally, we prove the strong L^p-convergence of the nonlocal operator to a local differential operator together with a rate of convergence. The analysis also includes more singular kernels. > References > [1] H. Abels, C. Hurm. Strong Nonlocal-to-Local Convergence of the Cahn-Hilliard Equation and its Operator. J. Differential Equations, 402: 593-624, 2024. > [2] H. Abels, C. Hurm, P. Knopf. Nonlocal-to-local L^p-convergence of convolution operators with singular, anisotropic kernels. (2026).

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