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arXiv · 2306.06486

On the limit problem arising in the kinetic derivation of the Cahn-Hilliard equation

Charles Elbar·Benoît Perthame·Jakub Skrzeczkowski
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Abstract

The non-local degenerate Cahn-Hilliard equation is derived from the Vlasov equation with long-range attraction. We study the local limit as the delocalization parameter converges to 0. The difficulty arises from the degeneracy which requires compactness estimates, but all necessary a priori estimates can be obtained only on the nonlocal quantities yielding almost no information on the limiting solution itself. We introduce a novel condition on the nonlocal kernel which allows us to exploit the available nonlocal a priori estimates. The condition, satisfied by most of the kernels appearing in the applications, can be of independent interest. Our approach is flexible and systems can be treated as well.

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Charles Elbar, Benoît Perthame, Jakub Skrzeczkowski. 2023-06-10. On the limit problem arising in the kinetic derivation of the Cahn-Hilliard equation. https://arxiv.org/abs/2306.06486

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