By P. Constantin
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Additional info for Attractors Representing Turbulent Flows
Other definitions of the Kolmogorov dissipation length corresponding to other definitions of the average rate of dissipation could be considered, for instance ATTRACTORS REPRESENTING TURBULENT FLOWS = vX^ /2 <€> lim 1 sup Sup (i f ||S(T)U uo ^X l j o corresponding to an average in space and time or t-H» ° || 2 49 dx)
1 and we only need to verify that Sup u>, (L (u)) < 1 for a p sufficiently u€X a p P. CONSTANT IN, C. FOIAS, R. 32) ; this is obvious. 34). 2 replacing again S and L(u) by S p and L (u) , p sufficiently large ; a is then replaced by a -(p) which is less than 1 for p large since lim sup (a n + 1 (p)} 1 / P = e y n + 1 < 1 , as yn+1 p-)»oo y < — + ... 35) j OJ0 (p) nY(2/d+T oT +1 (p) e) ^ 2a Max X n+1 - U«n n does s-nY(e) . 36) (2/5+1 a n+1 (p) e ) d ^(2/d+l « n + 1 (p) e) < 6 nx(e) , with 2 2 d (d+l) d/2 9 = d £ Max (S0(p) a„,,(p) ") .
We also note that Q m depends on ^i"m''^m ' although the dependence is not explicitly indicated. 1 used above. 1. 30) Q T A every (T where = I . . family (*,. A . . m l is A I the m A * and the data. > m l orthogonal 45 in H , and let . . A I + . . ,$ A ... A * projector m ) in , = |*1 i + I A I A . . linear A I A T of H , A ... A $m|2Tr(T0Q) m' , space by onto H be the Tm the spanned . Proof. ,$ ly, remembering that the scalar product in AmH is <•, A ... A V T A ... ^} so that (Tm(^ A ...
Attractors Representing Turbulent Flows by P. Constantin