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

## TL;DR

This paper proves the convergence of nonlocal viscous Cahn-Hilliard equations with Neumann boundary conditions to their local counterparts, even with singular potentials and convolution kernels, using variational and Gamma convergence methods.

## Contribution

It establishes well-posedness and convergence results for nonlocal Cahn-Hilliard equations with singular potentials under Neumann boundary conditions, extending existing theories.

## Key findings

- Solutions to nonlocal equations converge to local solutions as kernels approximate delta functions.
- Well-posedness is proven for equations with singular potentials.
- Convergence analyzed via monotone analysis and Gamma convergence.

## Abstract

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.