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

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.
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, (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.

[3]

H. Brezis, "Analyse Fonctionnelle,", Th\'eorie et applications, (1983).

[4]

P. Collet, Thermodynamic limit of the Ginzburg-Landau equation,, Nonlinearity, 7 (1994), 1175. 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. 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. doi: 10.1088/0951-7715/12/3/002.

[7]

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

[8]

J. M. Ghidaglia and B. Heron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation,, Phys. D, 28 (1987), 282. 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. 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. doi: 10.1007/s002200050129.

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, (1988).

[12]

T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980).

[13]

A. N. Kolmogorov and V. M. Tikhomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces,, Uspehi Mat. Nauk, 14 (1959), 3.

[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. doi: 10.1088/0951-7715/8/5/006.

[16]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics,", Applied Mathematical Sciences, (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. 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. doi: 10.1070/SM1998v189n02ABEH000301.

[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. 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. 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, (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.

[3]

H. Brezis, "Analyse Fonctionnelle,", Th\'eorie et applications, (1983).

[4]

P. Collet, Thermodynamic limit of the Ginzburg-Landau equation,, Nonlinearity, 7 (1994), 1175. 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. 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. doi: 10.1088/0951-7715/12/3/002.

[7]

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

[8]

J. M. Ghidaglia and B. Heron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation,, Phys. D, 28 (1987), 282. 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. 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. doi: 10.1007/s002200050129.

[11]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs, (1988).

[12]

T. Kato, "Perturbation Theory for Linear Operators,", Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, (1980).

[13]

A. N. Kolmogorov and V. M. Tikhomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional spaces,, Uspehi Mat. Nauk, 14 (1959), 3.

[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. doi: 10.1088/0951-7715/8/5/006.

[16]

R. Temam, "Infinite-Dimensional Systems in Mechanics and Physics,", Applied Mathematical Sciences, (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. 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. doi: 10.1070/SM1998v189n02ABEH000301.

[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. 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. doi: 10.1002/cpa.10068.

[1]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647

[2]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173

[3]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871

[4]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665

[5]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

[6]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311

[7]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825

[8]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205

[9]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715

[10]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121

[11]

Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507

[12]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579

[13]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329

[14]

Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455

[15]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711

[16]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic & Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57

[17]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575

[18]

Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109

[19]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129

[20]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]