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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

H. Brezis, "Analyse Fonctionnelle," Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[4]

P. Collet, Thermodynamic limit of the Ginzburg-Landau equation, Nonlinearity, 7 (1994), 1175-1190. doi: 10.1088/0951-7715/7/4/006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on $\mathbbR^n$, Differential Integral Equations, 9 (1996), 1147-1156.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.  Google Scholar

[12]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[14]

N. Maaroufi, Ph.D thesis,, 2010., ().   Google Scholar

[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.  Google Scholar

[16]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics," Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. V. Zelik, An attractor of a nonlinear system of reaction-diffusion equations in $\mathbbR^n$ and estimates for its $\varepsilon$-entropy, Mat. Zametki, 65 (1999), 941-944. doi: 10.1007/BF02675597.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Brezis, "Analyse Fonctionnelle," Théorie et applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar

[4]

P. Collet, Thermodynamic limit of the Ginzburg-Landau equation, Nonlinearity, 7 (1994), 1175-1190. doi: 10.1088/0951-7715/7/4/006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on $\mathbbR^n$, Differential Integral Equations, 9 (1996), 1147-1156.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.  Google Scholar

[12]

T. Kato, "Perturbation Theory for Linear Operators," Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[14]

N. Maaroufi, Ph.D thesis,, 2010., ().   Google Scholar

[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.  Google Scholar

[16]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics," Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. V. Zelik, An attractor of a nonlinear system of reaction-diffusion equations in $\mathbbR^n$ and estimates for its $\varepsilon$-entropy, Mat. Zametki, 65 (1999), 941-944. doi: 10.1007/BF02675597.  Google Scholar

[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.  Google Scholar

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