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

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May 2012, 11(3): 1253-1267. doi: 10.3934/cpaa.2012.11.1253

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, France, France

Received November 2010 Revised June 2011 Published December 2011

Abstract
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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.

Keywords: Ginzburg-Landau equation, Entropy by unit length, attractor.

Mathematics Subject Classification: Primary: 35Q56, 35B41, 37L3.

Citation: O. Goubet, N. Maaroufi. Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1253-1267. doi: 10.3934/cpaa.2012.11.1253

References:

[1]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Apllications Vol. 25, North Holland, 1992.
[2]	A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Procceding of Royal Society of Edinburgh, 116A (1990), 221-243.
[3]	H. Brezis, "Analyse Fonctionnelle," Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.
[4]	P. Collet, Thermodynamic limit of the Ginzburg-Landau equation, Nonlinearity, 7 (1994), 1175-1190. doi: 10.1088/0951-7715/7/4/006.
[5]	P. Collet and J. P. Eckmann, Extensive properties of the complex Ginzburg-Landau equation, Commun. Math. Phys., 200 (1999), 699-722. doi: 10.1007/s002200050546.
[6]	P. Collet and J. P. Eckmann, The definition and measurement of the topological entropy per unit volume in parabolic PDEs, Nonlinearity, 12 (1999), 451-473. doi: 10.1088/0951-7715/12/3/002.
[7]	E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 1147-1156.
[8]	J. M. Ghidaglia and B. Heron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Phys. D, 28 (1987), 282-304. doi: 10.1016/0167-2789(87)90020-0.
[9]	J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation I. Compactness methods, Phys. D, 95 (1996), 191-228. doi: 10.1016/0167-2789(96)00055-3.
[10]	J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation II. Contraction methods, Comm. Math. Phys., 187 (1997), 45-79. doi: 10.1007/s002200050129.
[11]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.
[12]	T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.
[13]	A. N. Kolmogorov and V. M. Tikhomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces, Uspehi Mat. Nauk, 14 (1959), 3-86.
[14]	N. Maaroufi, Ph.D thesis, 2010.
[15]	A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains -existence and comparaison, Nonlinearity, 8 (1995), 743-768. doi: 10.1088/0951-7715/8/5/006.
[16]	R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics," Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.
[17]	P. Takac, P. Bollerman, A. Doelman, A. van Harten and E. S. Titi, Analyticity of essentially bounded solutions to semlinear parabolic systems and validity of the Ginzburg-Landau equation, SIAM J. Math. Anal., 27 (1996), 424-448. doi: 10.1137/S0036141094262518.
[18]	M. I. Vishik and V. V. Chepyzov, Kolmogorov $\varepsilon$-entropy of attractors of reaction-diffusion systems, Mat. Sb., 189 (1998), 81-110. doi: 10.1070/SM1998v189n02ABEH000301.
[19]	S. V. Zelik, An attractor of a nonlinear system of reaction-diffusion equations in $\mathbb{R}^{N}$ and estimates for its $\varepsilon$-entropy, Mat. Zametki, 65 (1999), 941-944. doi: 10.1007/BF02675597.
[20]	S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.

show all references

References:

[1]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Apllications Vol. 25, North Holland, 1992.
[2]	A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Procceding of Royal Society of Edinburgh, 116A (1990), 221-243.
[3]	H. Brezis, "Analyse Fonctionnelle," Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.
[4]	P. Collet, Thermodynamic limit of the Ginzburg-Landau equation, Nonlinearity, 7 (1994), 1175-1190. doi: 10.1088/0951-7715/7/4/006.
[5]	P. Collet and J. P. Eckmann, Extensive properties of the complex Ginzburg-Landau equation, Commun. Math. Phys., 200 (1999), 699-722. doi: 10.1007/s002200050546.
[6]	P. Collet and J. P. Eckmann, The definition and measurement of the topological entropy per unit volume in parabolic PDEs, Nonlinearity, 12 (1999), 451-473. doi: 10.1088/0951-7715/12/3/002.
[7]	E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on $\mathbb{R}^{N}$, Differential Integral Equations, 9 (1996), 1147-1156.
[8]	J. M. Ghidaglia and B. Heron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Phys. D, 28 (1987), 282-304. doi: 10.1016/0167-2789(87)90020-0.
[9]	J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation I. Compactness methods, Phys. D, 95 (1996), 191-228. doi: 10.1016/0167-2789(96)00055-3.
[10]	J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation II. Contraction methods, Comm. Math. Phys., 187 (1997), 45-79. doi: 10.1007/s002200050129.
[11]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.
[12]	T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.
[13]	A. N. Kolmogorov and V. M. Tikhomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces, Uspehi Mat. Nauk, 14 (1959), 3-86.
[14]	N. Maaroufi, Ph.D thesis, 2010.
[15]	A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains -existence and comparaison, Nonlinearity, 8 (1995), 743-768. doi: 10.1088/0951-7715/8/5/006.
[16]	R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics," Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.
[17]	P. Takac, P. Bollerman, A. Doelman, A. van Harten and E. S. Titi, Analyticity of essentially bounded solutions to semlinear parabolic systems and validity of the Ginzburg-Landau equation, SIAM J. Math. Anal., 27 (1996), 424-448. doi: 10.1137/S0036141094262518.
[18]	M. I. Vishik and V. V. Chepyzov, Kolmogorov $\varepsilon$-entropy of attractors of reaction-diffusion systems, Mat. Sb., 189 (1998), 81-110. doi: 10.1070/SM1998v189n02ABEH000301.
[19]	S. V. Zelik, An attractor of a nonlinear system of reaction-diffusion equations in $\mathbb{R}^{N}$ and estimates for its $\varepsilon$-entropy, Mat. Zametki, 65 (1999), 941-944. doi: 10.1007/BF02675597.
[20]	S. V. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637. doi: 10.1002/cpa.10068.

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