Abstract
In this paper we present a new construction of exponential attractors based on the control of Lyapunov exponents over a compact, invariant set. The fractal dimension estimate of the exponential attractor thus obtained is of the same order as the one for global attractors estimated through Lyapunov exponents. We discuss various applications to Navier-Stokes systems.
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Ben-Artzi, A., Eden, A., Foias, C., and Nicolaenko, B. (1992). Hölder continuity for the inverse of Mane's projection.J. Math. Anal. Appl. (in press).
Constantin, P., and Foias, C. (1985). Global Lyapunov exponents, Kaplan-Yorke formulas and the dimension of the attractors for 2D Navier-Stokes equations.Comm. Pure Appl. Math. 38, 1–27.
Constantin, P., Foias, C., and Temam, R. (1988). On the dimension of the attractors in two-dimensional turbulence.Physica D 30, 284–296.
Constantin, P., Foias, G., Nicolaenko, B., and Temam, R. (1989).Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Appl. Math. Sciences Vol. 70, Springer-Verlag, New York.
Doering, C. R., Gibbon, J. D., Holm, D. D., and Nicolaenko, B. (1987). Exact Lyapunov dimension of the universal attractor for the CGL equation.Phys. Rev. Lett. 59, 2911–2914.
Eden, A. (1990). On the Burgers' original model of turbulence.Nonlinearity 3, 557–566.
Eden, A., and Milani, A. J. (1992). Exponential attractors for extensible beam equations. Preprint, ASU.
Eden, A., and Rakotoson, J. M. (1992). Exponential attractors for doubly non-linear parabolic equations. Preprint, Inst. Sci. Comp., Indiana University.
Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1990). Ensembles inertiels pour des équations dévolution dissipatives.C.R. Acad. Sci. Paris Ser. I 310, 559–562.
Eden, A., Foias, C., and Temam, R. (1991a). Local and global Lyapunov exponents.J. Dynam. Diff. Eq. 3, 133–177.
Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1991b). Inertial sets for dissipative evolution equations. I. Construction and applications. IMA preprint No. 812. (Book to appear,Appl. Math. Series, Masson, Paris, 1994.)
Eden, A., Molani, 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., Nicolaenko, B., and She, Z. S. (1993a). Exponential attractors and their relevance to fluid mechanical systems.Physica D 63, 350–360.
Eden, A., Milani, A. J., and Nicolaenko, B. (1993b). Exponential attractors for models of phase change for compressible gas dynamics.Nonlinearity 6, 93–117.
Eden, A., Foias, C., Nicolaenko, B., and She, Z. S. (1993c). Exponential attractors and their relevance to fluid mechanical systems.Physica D 63, 350–360.
Fabrie, P., and Nicolaenko, B. (1993). Exponential attractors for models of natural convection in porous media.Proc. Acad. Sci. Paris Ser. I 316, 1271–1276.
Falconer, K. (1990).Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, Chichester.
Foias, C., and Temam, R. (1979). Some analytic and geometric properties of the solutions of the Navier-Stokes equations.J. Math. Pures Appl. 58, 339–368.
Hale, J. (1988).Asymptotic Behavior of Dissipative Systems, Math. Surv. Monogr. 25, Am. Math. Soc., Providence, RI.
Mallet-Paret, J. (1978). Negatively invariant sets of compact maps and an extension of a theorem of Cartwright.J. Diff. Eq. 22, 331.
Mañé, R., (1981). On the dimension of the compact invariant sets of certain non-linear maps. InLecture Notes Math. 898, Springer-Verlag, New York, pp. 230–242.
Sauer, T., Yorke, J. A., and Casdagli, M. (1991). Embedology.J. Stat. Phys. 65(3/4), 579–616.
Sell, G. R. (1989). Hausdorff and Lyapunov dimensions for gradient systems. InThe Connection Between Infinite Dimensional and Finite Dimensional Dynamical Systems, Contemporary Mathematics, Vol. 99, pp. 85–92.
Temam, R. (1988).Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences Vol. 68, Springer-Verlag, New York.
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Eden, A., Foias, C. & Nicolaenko, B. Exponential attractors of optimal Lyapunov dimension for Navier-Stokes equations. J Dyn Diff Equat 6, 301–323 (1994). https://doi.org/10.1007/BF02218532
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DOI: https://doi.org/10.1007/BF02218532