no. 2
Partial Differential Equations
On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors
[Sur le comportement en temps grand des solutions d'équations paraboliques d'ordre quatre, et l'entropie de leurs attracteurs]

Présenté par : Haïm Brezis

Efendiev, M.A.1 ; Peletier, L.A.2, 3
1 GSF/Technical University of Münich, Center for Mathematical Sciences, 85747 Garchung-Münich, Germany
2 Mathematical Institute, Leiden University, PO Box 9512, NL-2300 RA Leiden, The Netherlands
3 Centrum voor Wiskunde en Informatica (CWI), NL-1090 GB Amsterdam, The Netherlands
Comptes Rendus. Mathématique, Tome 344 (2007) no. 2, pp. 93-96.

Nous étudions le comportement pour des grandes valeurs du temps des solutions d'une classe d'équations parabolique d'ordre quatre définie sur des domaines non bornés. Les examples spécifiques que nous considérons sont l'équation de Swift–Hohenberg et une généralisation de l'équation de Fisher–Kolmogorov. Nous démontrons l'existence d'un attracteur global dans une topologie locale, et nous obtenons des limites supérieure et inférieure de l'entropie de Kolmogorov.

We study the large time behavior of solutions of a class of fourth order parabolic equations defined on unbounded domains. Specific examples of the equations we study are the Swift–Hohenberg equation and the Extended Fisher–Kolmogorov equation. We establish the existence of a global attractor in a local topology. Since the dynamics is infinite dimensional, we use the Kolmogorov ε-entropy as a measure, deriving a sharp upper and lower bound.

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2006.10.028
Efendiev, M.A. 1 ; Peletier, L.A. 2, 3

1 GSF/Technical University of Münich, Center for Mathematical Sciences, 85747 Garchung-Münich, Germany
2 Mathematical Institute, Leiden University, PO Box 9512, NL-2300 RA Leiden, The Netherlands
3 Centrum voor Wiskunde en Informatica (CWI), NL-1090 GB Amsterdam, The Netherlands 
@article{CRMATH_2007__344_2_93_0,
     author = {Efendiev, M.A. and Peletier, L.A.},
     title = {On the large time behavior of solutions of fourth order parabolic equations and \protect\emph{\ensuremath{\varepsilon}}-entropy of their attractors},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {93--96},
     publisher = {Elsevier},
     volume = {344},
     number = {2},
     year = {2007},
     doi = {10.1016/j.crma.2006.10.028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.10.028/}
}
TY  - JOUR
AU  - Efendiev, M.A.
AU  - Peletier, L.A.
TI  - On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 93
EP  - 96
VL  - 344
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.10.028/
DO  - 10.1016/j.crma.2006.10.028
LA  - en
ID  - CRMATH_2007__344_2_93_0
ER  - 
%0 Journal Article
%A Efendiev, M.A.
%A Peletier, L.A.
%T On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors
%J Comptes Rendus. Mathématique
%D 2007
%P 93-96
%V 344
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.10.028/
%R 10.1016/j.crma.2006.10.028
%G en
%F CRMATH_2007__344_2_93_0
Efendiev, M.A.; Peletier, L.A. On the large time behavior of solutions of fourth order parabolic equations and ε-entropy of their attractors. Comptes Rendus. Mathématique, Tome 344 (2007) no. 2, pp. 93-96. doi : 10.1016/j.crma.2006.10.028. http://www.numdam.org/articles/10.1016/j.crma.2006.10.028/

[1] Dee, G.T.; van Saarloos, W. Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett., Volume 60 (1988), pp. 2641-2644

[2] Efendiev, M.A.; Zelik, S. The attractor for a nonlinear reaction–diffusion system in an unbounded domain, Comm. Pure Appl. Math., Volume LIV (2001), pp. 625-688

[3] Efendiev, M.A.; Zelik, S. Upper and lower bounds for the Kolmogorov entropy of the attractor for an RDE in an unbounded domain, J. Dynam. Differential Equations, Volume 14 (2002), pp. 369-404

[4] Peletier, L.A.; Rottschäfer, V. Pattern selection of solutions of the Swift–Hohenberg equation, Physica D, Volume 194 (2004), pp. 95-126

[5] Peletier, L.A.; Troy, W.C. Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhäuser, Boston, 2001

[6] Swift, J.B.; Hohenberg, P.C. Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, Volume 15 (1977), pp. 319-328

Cité par Sources :