Robust exponential attractors for the modified phase-field crystal equation

Maurizio Grasselli; Hao Wu

doi:10.3934/dcds.2015.35.2539

Article Contents

2015, Volume 35, Issue 6: 2539-2564. Doi: 10.3934/dcds.2015.35.2539

This issue Previous Article Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint Next Article Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

Robust exponential attractors for the modified phase-field crystal equation

Maurizio Grasselli^1, and
Hao Wu^2,

1.
Dipartimento di Matematica, Politecnico di Milano, 20133 Milano
2.
School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, 200433 Shanghai

Received: November 30, 2013

Revised: March 31, 2014

Published: May 2017

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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 Hölder continuous with respect to $\beta$.

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Mathematics Subject Classification: Primary: 35Q82, 37L25; Secondary: 74N05, 82C26.

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