Nonlocal to local convergence of singular phase field systems of conserved type

TL;DR

This paper proves the existence and nonlocal-to-local convergence of solutions for a singular nonlocal phase field system of conserved type, extending previous work on similar systems and convergence analysis.

Contribution

It establishes the existence of solutions for a nonlocal singular phase field system with a less regular kernel and analyzes the convergence to local systems.

Findings

Proved existence of solutions for a nonlocal singular phase field system.

Demonstrated nonlocal to local convergence of the system.

Extended convergence results to kernels not in W^{1,1}.

Abstract

This paper deals with a singular nonlocal phase field system of conserved type.Colli--K.\ [Nonlinear Anal.\ 190 (2020)] have derived existence of solutions to a singular phase field system of conserved type. On the other hand, Davoli--Scarpa--Trussardi [Arch. Ration. Mech. Anal.\ 239 (2021)] have studied nonlocal to local convergence of Cahn-Hilliard equations. In this paper we prove existence of solutions to a nonlocal singular phase field system of conserved type whose kernel is not $W^{1, 1}$ and focus on nonlocal to local convergence of singular phase field systems of conserved type.