Convective nonlocal Cahn-Hilliard equations with reaction terms

Francesco Della Porta; Maurizio Grasselli

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arXiv · 1406.1684

Convective nonlocal Cahn-Hilliard equations with reaction terms

Francesco Della Porta·Maurizio Grasselli

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Abstract

We introduce and analyze the nonlocal variants of two Cahn-Hilliard type equations with reaction terms. The first one is the so-called Cahn-Hilliard-Oono equation which models, for instance, pattern formation in diblock-copolymers as well as in binary alloys with induced reaction and type-I superconductors. The second one is the Cahn-Hilliard type equation introduced by Bertozzi et al. to describe image inpainting. Here we take a free energy functional which accounts for nonlocal interactions. Our choice is motivated by the work of Giacomin and Lebowitz who showed that the rigorous physical derivation of the Cahn-Hilliard equation leads to consider nonlocal functionals. The equations also have a transport term with a given velocity field and are subject to a homogenous Neumann boundary condition for the chemical potential, i.e., the first variation of the free energy functional. We first establish the well-posedness of the corresponding initial and boundary value problems in a weak setting. Then we consider such problems as dynamical systems and we show that they have bounded absorbing sets and global attractors.

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Francesco Della Porta, Maurizio Grasselli. 2014-06-06. Convective nonlocal Cahn-Hilliard equations with reaction terms. https://arxiv.org/abs/1406.1684

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