Nonlocal phase-field systems with general potentials

TL;DR

This paper studies nonlocal phase-field models with general potentials, proving well-posedness, boundedness, and the existence of finite-dimensional attractors for both smooth and singular potentials, including physically realistic cases.

Contribution

It extends analysis to include singular potentials like logarithmic functions, establishing well-posedness and attractor existence in these more realistic models.

Findings

Proved well-posedness of the system.

Established uniform boundedness of solutions.

Demonstrated existence of finite-dimensional attractors.

Abstract

We consider a phase-field model where the internal energy depends on the order parameter in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for the order parameter. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, in the case of a potential defined on (-1,1) and singular at the endpoints, the existence of a finite-dimensional global attractor has been proven. Here we examine both the case of smooth potentials as well as the case of physically realistic (e.g., logarithmic) singular potentials. We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional attractors in the present cases as well.