Calculus of Variations and Geometric Measure Theory

E. Davoli - L. Scarpa - L. Trussardi

Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms

created by davoli on 02 Aug 2019
modified on 04 Sep 2020


Accepted Paper

Inserted: 2 aug 2019
Last Updated: 4 sep 2020

Journal: Archive for Rational Mechanics and Analysis
Year: 2020


We consider a class of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn-Hilliard equation is of viscous type and of pure type.

Keywords: regularity, well-posedness, Neumann boundary conditions, Nonlocal Cahn-Hilliard equation, singular potential, singular kernel, nonlocal-to-local convergence