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Zbl 1040.37069
Dung, L.; Nicolaenko, B.
Exponential attractors in Banach spaces.
(English)
[J] J. Dyn. Differ. Equations 13, No. 4, 791-806 (2001). ISSN 1040-7294; ISSN 1572-9222/e

Let $E$ be a Banach space, $U\subset E$ an open set and $S:U\rightarrow E$ a $C^1$-map. The authors consider the discrete dynamical system (DS) $\{S^n\}_{n=1}^{\infty}$ generated by $S$, extending the theory of exponential attractors from such DS in Hilbert space [{\it A. Eden, C. Foias, B. Nicolaenko} and {\it R. Temam}, Exponential attractors for dissipative evolution equations, Research in Applied Mathematics 37, Chichester: Wiley, Paris: Masson (1994; Zbl 0842.58056)] on Banach spaces. The following requirements are postulated: 1. the semiflow is $C^1$ in some absorbing ball, and 2. the linearized semiflow at every point inside the absorbing ball is splitting into the sum of a compact operator plus a contraction.
MSC 2000:
*37L30 Attractors and their dimensions
35B41 Attractors
35Q30 Stokes and Navier-Stokes equations
47H20 Semigroups of nonlinear operators

Keywords: Banach spaces, discrete dynamical systems, exponential attractors

Citations: Zbl 0842.58056

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