By P. Constantin

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**Extra resources for Attractors Representing Turbulent Flows**

**Example text**

E . < /2F y f or f „ . _ , _ . ) = y . E. 17) Thus R n < m i n f ^ ^- , < p for 0 < t < tx . 19) |w(t)|2 + v rt ' o 2c' ||w(s)|| 2 ds < |w | 2 e x p ( — | R4 t) . 20) Sup o^t

The best estimate of the dimension of X that can be conjectured £ o 3 is of the form of c( . ) . We are not able at the moment to derive this d improved form of the estimate which was emphasized to us by R. Kraichnan. b. 45) - Tr «V(u) 0 Q v I || 4 K1 II 2 - I (B(4> 3,u),cj> J ) . ) ,u) = f (((*>.. u dx 3 J 3 3 Jo 3 3 £ l*J 2 ' V < | ) il 2 J Z L (fi) J l/(fl) ' U l oo UD ll< UI |u|w D L (fi) L (fi) ( ) Obviously <€> < € and £, < <&d> . Other relations between <&d> and <"£ = > will be given elsewhere.

2. If n is the smallest integer such that y. + ... + y > 0 and y, + ... 34) is larger or equal to y, (u ) + ... + y (u ) „ , 1 o n o w n + , v u -Vn< u o' Observe first that y for p large enough, +1 (u ) < V +1 <0 , v u G X . c v tx . 37) n + Sup u Q €X Vt. (u ) + . 6. 38) X into S(t) H , t > 0 , and such that S ( t ) X = X , V t > 0 . 2, to this situation. Replacing replacing t by pt > 0 S by and apply one of the theorems S^ , p large, amounts to . 2) ; we set, in this case w . (t) = Sup GO . (L(t,u) ) .