Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics

Davoli, Elisa; Ranetbauer, Helene; Scarpa, Luca; Trussardi, Lara

doi:10.1016/j.anihpc.2019.10.002

Davoli, Elisa ; Ranetbauer, Helene ; Scarpa, Luca ; Trussardi, Lara

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 627-651.

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Résumé

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.

DOI : 10.1016/j.anihpc.2019.10.002

Classification : 45K05, 35K25, 35K55, 35B40, 76R05
Mots-clés : Nonlocal Cahn-Hilliard equation, Degenerate potential, Singular kernel, Well-posedness, Nonlocal-to-local convergence, Convection

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     title = {Degenerate nonlocal {Cahn-Hilliard} equations: {Well-posedness,} regularity and local asymptotics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {627--651},
     publisher = {Elsevier},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2019.10.002/}
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Davoli, Elisa; Ranetbauer, Helene; Scarpa, Luca; Trussardi, Lara. Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 627-651. doi : 10.1016/j.anihpc.2019.10.002. http://www.numdam.org/articles/10.1016/j.anihpc.2019.10.002/

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