On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation

TL;DR

This paper extends the analysis of phase separation models by applying Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard system, accommodating long-range interactions and singular potentials, and establishing well-posedness.

Contribution

It introduces a novel application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard model with singular potentials, demonstrating well-posedness.

Findings

Model equations are well-posed.

Application of Tikhonov's fixed point theorem in a nonlocal context.

Accommodates singular free energy contributions.

Abstract

This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006) 105-118. The model consists of an initial-boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter and the chemical potential. Singular contributions to the local free energy in the form of logarithmic or double-obstacle potentials are admitted. In contrast to the local model, which was studied by P. Podio-Guidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long-range interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space.