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.
Cite this article
Elisa Davoli, Helene Ranetbauer, Luca Scarpa, Lara Trussardi, Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, pp. 627–651DOI 10.1016/J.ANIHPC.2019.10.002