Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework

O. Goubet; N. Maaroufi

doi:10.3934/cpaa.2012.11.1253

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2012, Volume 11, Issue 3: 1253-1267. Doi: 10.3934/cpaa.2012.11.1253

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Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework

O. Goubet^1, and
N. Maaroufi^1,

1.
LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex

Received: October 31, 2010

Revised: May 31, 2011

Published: April 2012

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Abstract

It is well-known that the Ginzburg-Landau equation on $R$ has a global attractor [15] that attracts in $L^\infty_{l o c}(R)$ all the trajectories. This attractor contains bounded trajectories that are analytical functions in space. A famous theorem due to P. Collet and JP. Eckmann asserts that the $\varepsilon$-entropy per unit length in $L^\infty$ of this global attractor is finite and is smaller than the corresponding complexity for the space of functions which are analytical in a strip. This means that the global attractor is flatter than expected. We explain in this article how to establish the Collet-Eckmann Theorem in a Hilbert space framework.

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Mathematics Subject Classification: Primary: 35Q56, 35B41, 37L30.

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