| Title:
|
Attractors for general operators (English) |
| Author:
|
Miranville, Alain |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2003 |
| Pages:
|
31-47 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua. (English) |
| Keyword:
|
global attractor |
| Keyword:
|
minimal attractor |
| Keyword:
|
exponential attractor |
| Keyword:
|
weakly coupled system |
| MSC:
|
35B41 |
| MSC:
|
37L30 |
| idZBL:
|
Zbl 1099.37060 |
| idMR:
|
MR1954502 |
| DOI:
|
10.1023/A:1022902903988 |
| . |
| Date available:
|
2009-09-22T18:12:06Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134515 |
| . |
| Reference:
|
[1] A. Babin, S. Chow: Uniform long-time behavior of solutions of parabolic equations depending on slow time.J. Differential Equations 150 (1998), 264–316. MR 1658609, 10.1006/jdeq.1998.3450 |
| Reference:
|
[2] A. Babin, B. Nicolaenko: Exponential attractors of reaction-diffusion systems in an unbounded domain.J. Dynam. Differential Equations 7 (1995), 567–590. MR 1362671, 10.1007/BF02218725 |
| Reference:
|
[3] A. V. Babin, M. I. Vishik: Attractors of Evolution Equations.North-Holland, Amsterdam, 1992. MR 1156492 |
| Reference:
|
[4] J. W. Cahn: On spinodal decomposition.Acta Metall. 9 (1961), 795–801. 10.1016/0001-6160(61)90182-1 |
| Reference:
|
[5] J. W. Cahn, J. E. Hilliard: Free energy of a nonuniform system I. Interfacial free energy.J. Chem. Phys. 2 (1958), 258–267. 10.1063/1.1744102 |
| Reference:
|
[6] M. Carrive, A. Miranville and A. Piétrus: The Cahn-Hilliard equation for deformable elastic continua.Adv. Math. Sci. Appl. 10 (2000), 530–569. MR 1807441 |
| Reference:
|
[7] M. Carrive, A. Miranville, A. Piétrus and J. M. Rakotoson: The Cahn-Hilliard equation for an isotropic deformable continuum.Appl. Math. Lett. 12 (1999), 23–28. MR 1748727, 10.1016/S0893-9659(98)00143-8 |
| Reference:
|
[8] M. Carrive, A. Miranville, A. Piétrus and J. M. Rakotoson: Weakly coupled dynamical systems and applications.Preprint No 121, Université de Poitiers. MR 1919340 |
| Reference:
|
[9] V. V. Chepyzhov, M. I. Vishik: Attractors of nonautonomous dynamical systems and their dimension.J. Math. Pures Appl. 73 (1994), 279–333. MR 1273705 |
| Reference:
|
[10] A. Eden, C. Foiaş and V. Kalantarov: A remark on two constructions of exponential attractors for $\alpha $-contractions.J. Dynam. Differential Equations 10 (1998), 37–45. MR 1607533, 10.1023/A:1022636328133 |
| Reference:
|
[11] A. Eden, C. Foiaş, B. Nicolaenko and R. Temam: Exponential Attractors for Dissipative Evolution Equations.Masson, 1994. MR 1335230 |
| Reference:
|
[12] M. Efendiev, A. Miranville and S. Zelik: Exponential attractors for a nonlinear reaction-diffusion system in $R^3$.C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), 713–718. MR 1763916, 10.1016/S0764-4442(00)00259-7 |
| Reference:
|
[13] M. Efendiev, A. Miranville and S. Zelik: The infinite dimensional exponential attractor for a nonautonomous reaction-diffusion system.Math. Nachr (to appear). |
| Reference:
|
[14] P. Fabrie, A. Miranville: Exponential attractors for nonautonomous first-order evolution equations.Discrete Contin. Dynam. Systems 4 (1998), 225–240. MR 1617294 |
| Reference:
|
[15] E. Feireisl: Exponentially attracting finite dimensional sets for the processes generated by nonautonomous semilinear wave equations.Funkcial. Ekvac. 36 (1993), 1–10. Zbl 0823.35119, MR 1232076 |
| Reference:
|
[16] C. Foiaş, G. Sell and R. Temam: Inertial manifolds for nonlinear evolution equations.J. Differential Equations 73 (1988), 309–353. MR 0943945, 10.1016/0022-0396(88)90110-6 |
| Reference:
|
[17] C. Galusinski: Thèse.Université Bordeaux-I, 1996. |
| Reference:
|
[18] J. M. Ghidaglia, R. Temam: Attractors for damped nonlinear hyperbolic equations.J. Math. Pures Appl. 66 (1987), 273–319. MR 0913856 |
| Reference:
|
[19] M. Gurtin: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance.Physica D 92 (1996), 178–192. Zbl 0885.35121, MR 1387065, 10.1016/0167-2789(95)00173-5 |
| Reference:
|
[20] J. Hale: Asymptotic Behavior of Dissipative Systems.Math. Surveys and Monographs, Vol. 25, AMS, Providence, 1988. Zbl 0642.58013, MR 0941371 |
| Reference:
|
[21] A. Haraux: Systèmes dynamiques dissipatifs et applications.Masson, 1991. Zbl 0726.58001, MR 1084372 |
| Reference:
|
[22] O. Ladyzhenskaya: Attractors for Semigroups and Evolution Equations.Cambridge University Press, Cambridge, 1991. Zbl 0755.47049, MR 1133627 |
| Reference:
|
[23] A. Miranville: Exponential attractors for nonautonomous evolution equations.Appl. Math. Lett. 11 (1998), 19–22. MR 1609661, 10.1016/S0893-9659(98)00004-4 |
| Reference:
|
[24] A. Miranville: Exponential attractors for a class of evolution equations by a decomposition method.C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 145–150. Zbl 1141.35340, MR 1669003, 10.1016/S0764-4442(99)80153-0 |
| Reference:
|
[25] A. Miranville: Exponential attractors for a class of evolution equations by a decomposition method. II. The nonautonomous case.C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), 907–912. MR 1689877 |
| Reference:
|
[26] A. Miranville: Some generalizations of the Cahn-Hilliard equation.Asymptotic Anal. 22 (2000), 235–259. Zbl 0953.35055, MR 1753766 |
| Reference:
|
[27] A. Miranville: Long time behavior of some models of Cahn-Hilliard equations in deformable continua.Nonlinear Anal. Series B 2 (2001), 273–304. Zbl 0989.35066, MR 1835609, 10.1016/S0362-546X(00)00104-8 |
| Reference:
|
[28] J. Mallet-Paret, G. R. Sell: Inertial manifolds for reaction-diffusion equations in higher space dimensions.J. Amer. Math. Soc. 1 (1988), 805–866. MR 0943276, 10.1090/S0894-0347-1988-0943276-7 |
| Reference:
|
[29] G. R. Sell: Nonautonomous differential equations and topological dynamics, I, II.Trans. Amer. Math. Soc. 127 (1967), 241–262, 263–283. |
| Reference:
|
[30] F. Shuhong: Global attractor for general nonautonomous dynamical systems.Nonlinear World 2 (1995), 191–216. Zbl 0822.34048, MR 1376953 |
| Reference:
|
[31] F. Shuhong: Finite dimensional behavior of periodic and asymptotically periodic processes.Nonlinear Anal. 28 (1997), 1785–1797. Zbl 0873.58044, MR 1432632, 10.1016/S0362-546X(95)00229-O |
| Reference:
|
[32] M. Smiley: Global attractors and approximate inertial manifolds for nonautonomous dissipative equations.Appl. Anal. 50 (1993), 217–241. Zbl 0739.34052, MR 1278326, 10.1080/00036819308840194 |
| Reference:
|
[33] R. Temam: Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed.Springer-Verlag, New-York, 1997. Zbl 0871.35001, MR 1441312 |
| . |
