By M. Francaviglia
Offering a logically balanced and authoritative account of the several branches and difficulties of mathematical physics that Lagrange studied and built, this quantity offers updated advancements in differential goemetry, dynamical structures, the calculus of diversifications, and celestial and analytical mechanics.
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Sample text
3) (and the same for p), u(xi,x ,0) 2 = uq(xi,x ). 4) Here / represents volume forces applied to the fluid and maintaining turbulence. Lower bound on the dimension of the attractor A special form of / will be chosen below but at this point / is arbitrary. 6) λχ = L being the first eigenvalue of the Stokes problem in Ω (periodic boundary condition, see [15]). 2 The case of interest (turbulent flow) corresponds to the case where the Grashof number is large. e. a = 1, and G is large). Here we are interested in the case where a < 1 and both G and 1/a are large.
With the previous notations we have m L Y,X > c i ( a / ? ) / m / form> 2 2 P p=l where c\ is a numerical Proof. 3 5 3 > — αβ constant. 4)' we see that < C Q X / , for χ > 1, where CQ is a numerical constant. Hence cp~ (m) > 3 (c cvy5m) / 2 3 0 2 l for m > φ(1). 4)' we see that φ{\) < 26/(αβ) and this shows t h a t 26 (p~ (m) > ( c o ^ m ) / for m > ——. 2. 2. A c o l l e c t i v e S o b o l e v i n e q u a l i t y Let us first show t h e following lemma. 9) \p=m+l This kind of inequality was first proved by Brezis and Gallouet [3].
314] is no more valid since the shape of Ω =]0, ^ [ x ] 0 , 2 π Σ [ is varying as α goes to 0. 2. Lower bound A lower bound on the dimension of the global attractor was derived in [1]. 5) on / is satisfied. 3) possess a stationary solution us of the form (U(x2),0), ( ρ = 0, and we have since (us · V)us = 0) : -vU"(x ) 2 = g(x ) U(x ) =—2 2 f ν JQ (x — s)g(s)ds — 2 X \ J 2TLU JQ sg(s)ds + const. 10) so that ι U(x )dx 2 2 = 0. 11) Lower bound on the dimension of the attractor 39 The proof of [1] is based on an estimate of the number of unstable modes for this stationary solution us- This yields a lower bound on the dimension of the unstable manifold A4 of this solution, and thus a lower bound on the U dimension of the attractor A since A D M.