Finite-dimensional global attractor for a nonlocal phase-field system

TL;DR

This paper proves the existence of a finite-dimensional global attractor for a nonlocal phase-field system with a coupled energy balance and a nonlocal ODE, demonstrating long-term boundedness and stability of the system.

Contribution

It introduces a novel analysis of a nonlocal phase-field model with singular potential, establishing the existence of a finite-dimensional global attractor.

Findings

Existence of a bounded absorbing set in the phase space

Proof of a finite-dimensional global attractor

Model captures nonlocal interactions with singular potential

Abstract

We analyze a phase-field system where the energy balance equation is linearly coupled with a nonlinear and nonlocal ODE for the order parameter $\chi$. The latter equation is characterized by a space convolution term which models particle interaction and a singular configuration potential that forces $\chi$ to take values in $(-1,1)$. We prove that the corresponding dynamical system has a bounded absorbing set in a suitable phase space. Then we establish the existence of a finite-dimensional global attractor.