Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system

Narcisse Batangouna; Morgan Pierre

doi:10.3934/cpaa.2018001

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2018, Volume 17, Issue 1: 1-19. Doi: 10.3934/cpaa.2018001

This issue Previous Article Next Article Unilateral global interval bifurcation for Kirchhoff type problems and its applications

Convergence of exponential attractors for a time splitting approximation of the Caginalp phase-field system

Narcisse Batangouna^1, and
Morgan Pierre^2, ,

1.
Faculté des Sciences et Techniques, Université Marien Ngouabi, BP 69 Brazzaville, République du Congo, France
2.
Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, F-86962 Chasseneuil, France

* Corresponding author: M. Pierre
* Corresponding author: M. Pierre

Received: May 31, 2017

Revised: July 31, 2017

Published: December 2017

This work has been partially supported by the Fédération MIRES.

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We consider a time semi-discretization of the Caginalp phase-field model based on an operator splitting method. For every time-step parameter $ \tau $, we build an exponential attractor $ \mathcal{M}_\tau $ of the discrete-in-time dynamical system. We prove that $ \mathcal{M}_\tau $ converges to an exponential attractor $\mathcal{M}_0$ of the continuous-in-time dynamical system for the symmetric Hausdorff distance as $ \tau $ tends to $0$. We also provide an explicit estimate of this distance and we prove that the fractal dimension of $ \mathcal{M}_\tau $ is bounded by a constant independent of $ \tau $.

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Mathematics Subject Classification: Primary:37L30, 65M12;Secondary:35B41, 80A22.

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