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5) is applicable. 14) nv(2/d+l oT .. 15) cT , . < (u) . ) n+1 ~ n+1 (2/d+Tan+1 / n e)d -—- 1<*<* (* n + 1 ->* n__(e) * , n x ( 2 / d + l a n + 1 e) ^ ~n+d / - , . — d xn+l. ( u , (u . ) )e n + 1 * . x nv(e) X 28 P. CONSTANTIN, C . FOIAS, R. 1 w e can replace L (u) = L ( S P ~ 1 ( u ) ) o ... o L(S(u)) S 0 by S p , L(u) L(u) and this allows us to render certain quantities smaller b y choosing large. 18) io. 20) where (Mae) < | *(e) 4>(e) = e n v ( e ) , 0

50) Sup S u p | u ( x ) | u€X x€fi . 48) can be derived using G and Re . 4. is less If X or equal is an attractor bounded (4. 49) SO). 2. 51) where Re < c' G(l + Re ~ c' is b an appropriate 1/2 ) constant. Proof. According to an inequality of S. 52) Therefore |u| < c ||u|| 1 / 2 |Au| 1/2 , V u € D (A) . L (fi)J Re < c MN , with (t l M = lim sup t+°° Sup ( 2 Sup ( 9 u €X o N = l i m sup t-*~ \ / 2 v X/ 1 t •'o (t i \ /70 j ||S(T)U |AS(x)u || | dx) ? 53) 2 dx i ^ l u j 2 2 M < (-4^t-r9-) v 4 X ^ + 1/4 1/4 ?

2) is satisfied with L(u) replaced by i) Let LfsP" 1 ^)) 0 L(SP~2(u)) o ... o L(Su)) o L(u) . 4) are also satisfied, the numbers k and m above 25 ATTRACTORS REPRESENTING TURBULENT FLOWS being replaced by k p and m p ( ) . We infer from this remark that, by replacing if necessary S by S p , we can assume that the number k is arbitrarily small. 9) &d k < (l |)xd+l and ii) Iterated coverings Let e > 0 be fixed. ) centered at u. € X of radius r. ) n x) . 2) is less or equal to n . ) = u. + B. ,N , and, for each included in the sum Su where E.

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