Robust exponential attractors for the modified phase-field crystal equation

TL;DR

This paper studies the modified phase-field crystal equation with inertial effects, establishing the existence of a family of exponential attractors that depend continuously on the relaxation parameter.

Contribution

It introduces and analyzes exponential attractors for the MPFC equation, demonstrating their H"older continuity with respect to the relaxation parameter.

Findings

Existence of exponential attractors for the MPFC equation.

H"older continuity of attractors with respect to the parameter eta.

Framework for understanding long-term dynamics of the MPFC model.

Abstract

We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $\beta\phi_{tt}$. Here $\phi$ is the phase function standing for the number density of atoms and $\beta\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $\beta$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $\mathcal{M}_\beta$ that are H\"older continuous with respect to $\beta$.