Abstract
An improvement in the original constructions of exponential attractors is indicated. Namely, when the solution semigroup is α-contractive and satisfies the discrete squeezing property, then even when the invariant set on which the semigroup acts is not compact, the original constructions carries through. We obtain the same conclusion for the construction with Lyapunov dimension for α-constructions.
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REFERENCES
Babin, A. V., and Nicolaenko, B. (1995). Exponential attractors for reaction diffusion equations in unbounded domains. J. Dynam. Diff. Eq. 7, 567–590.
Babin, A. V., and Vishik, M. I. (1989). Attractors for Evolution Equations, Nauka, Moscow (in Russian).
Eden, A. (1990). Local estimates for the Hausdorf dimension of an attractor. J. Math. Anal. Appl. 150(1), 100–119.
Eden, A., and Kalantarov, V. (1996). Finite dimensional attractors for a class of semilinear wave equations. Turk. J. Math. 20(3), 425–450.
Eden, A., and Milani, A. J. (1993). Exponential attractors for extensible beam equations. Nonlinearity 6, 457–479.
Eden, A., and Rakotoson, J. M. (1994). Exponential attractors for a doubly nonlinear equation. J. Math. Anal. Appl. 185(2), 321–339.
Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1990). Ensembles inertiels pour des equations d'evolution dissipatives. C.R. Acad. Si. Paris Ser. I 310, 559–562.
Eden, A., Milani, A. J., and Nicolaenko, B. (1992). Exponential attractors for semilinear wave equations with damping. J. Math. Anal. Appl. 169, 408–419.
Eden, A., Foias, C., and Nicolaenko, B. (1993a). Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations. C.R. Acad. Sci. Paris Ser. I 316, 1211–1215.
Eden, A., Milani, A. J., and Nicolaenko, B. (1993b). Local exponential attractors for models of phase change for compressible gas dynamics. Nonlinearity 6, 93–117.
Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1994a). Exponential Attractors for Dissipative Evolution Equations, Masson, Paris.
Eden, A., Foias, C., and Nicolaenko, B. (1994b). Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations. J. Dynam. Diff. Eq. 6, 301–323.
Hale, J. K. (1988). Asymptotic Behaviour of Dissipative Systems, AMS Math. Surv. Monogr. No. 25, Providence, RI.
Kuratowski, C. (1966). Topology, Academic Press, New York.
Ladyzhenskaya, O. L. (1982). On the finiteness of the dimension of bounded invariant sets for the Navier-Stokes equations and some other dissipative systems. Zapiski Nauchnich Semin. LOMI 115, 137–155.
Temam, R. (1988). Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York.
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Eden, A., Foias, C. & Kalantarov, V. A Remark on Two Constructions of Exponential Attractors for α-Contractions. Journal of Dynamics and Differential Equations 10, 37–45 (1998). https://doi.org/10.1023/A:1022636328133
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DOI: https://doi.org/10.1023/A:1022636328133