Robust exponential attractors for the modified phase-field crystal equation

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 $\beta$.