By A. Eden, C. Foias, B. Nicolaenko, Roger Temam

Exponential attractors is a brand new quarter of dynamical structures, pioneered via the authors of this publication, and has develop into an exceptionally full of life resource of study either from the natural in addition to utilized and numerical issues of view. contains quite a few functions for Navier-Stokes equations and plenty of different similar partial differential equations of mathematical physics.

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**Extra info for Exponential Attractors for Dissipative Evolution Equations**

**Sample text**

So, from now on, we assume that the non-linear operator S{t) exists and for t > 0: S {t) : H —> D {A ) is continuous. 6) and also, we assume the existence of a compact, invariant absorbing set, of the form B = { u e H : |u|ji < po and l|u||o(^i/2) < p i}. (A ‘ /2), makes it clear that V is compactly imbedded in H , For notational simplicity, we denote ll^ll = ll«llv = \A^^'^u \h and |u| = |u |^ . 1), we further assume that the non-linear term R : D {A ) H IS continuous. 10) and that there exists a compact, invariant subset AT of B, and a real number /3 G ( 0 ,1 / 2 ] such that, for every u and v in X , |ii(a) - ii(v)| < Cq\A^{u — u)|.

C ( J S { s ) M , = M . 65) where ce is a constant that depends only on ci,C2 ,C3 and ¿0 = ¿J® < ( 1 / 8 )^®. This finishes the proof that A i is an exponential attractor for ({5 (f)}t > o ,X ). 3. 11), which in turn will determine all the remaining estimates. Of course, the method applied here will be modified slightly to give better estimates for fractal dimension of the exponential attractors in Chapter 5, where we will make the estimates more explicit by determining them in terms of relevant physical parameters.

1 ) admits an exponential attractor M whose fractal dimension can be estimated by cIf ( M ) < d F { M * ) + 1. 58) P ro o f. 1 , we obtain that the map 5 * = 5 (t*) has an exponential attractor Л4*, on X , such that h { s : : x , M ^ ) < e ^ R = c46:^. 60) Moreover, where C5 = In + 1^ / l n ( l / 4 i ^ ) is constant that can be estimated using Ci,C2 and C3 . Now, we set M = \J S {t)M ,. 61) Q