×

Exponential attractors in Banach spaces. (English) Zbl 1040.37069

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 [A. Eden, C. Foias, B. Nicolaenko and 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:

37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
35Q30 Navier-Stokes equations
47H20 Semigroups of nonlinear operators

Citations:

Zbl 0842.58056
PDFBibTeX XMLCite
Full Text: DOI