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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 |
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] | |
[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] | |
[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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