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

## TL;DR

This paper establishes existence, uniqueness, and regularity of solutions for a novel class of nonlocal Cahn-Hilliard equations with degenerate potentials, and demonstrates their convergence to local models as the nonlocal kernel approximates a delta function.

## Contribution

It introduces a new existence theory for nonlocal Cahn-Hilliard equations with singular kernels not covered by previous frameworks, including convergence and regularity results.

## Key findings

- Proved existence and uniqueness of solutions.
- Demonstrated convergence to local Cahn-Hilliard equations.
- Established higher regularity and stronger convergence topology.

## Abstract

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.