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By P. Constantin

ISBN-10: 0821823159

ISBN-13: 9780821823156

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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) = vx sup ||u| | 2 , 1 u€X corresponding to a supremum in time and an average in space of the associated dissipation lengths are denoted : 1/4 1/4 3 3 An estimate of the dimension of X I in term of ( .

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 ...

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Attractors Representing Turbulent Flows by P. Constantin

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