By Paul Bamberg, Shlomo Sternberg

This article breaks new flooring in offering and using refined arithmetic in an uncomplicated atmosphere. aimed toward physics scholars, it covers the idea and actual functions of linear algebra and of the calculus of a number of variables, quite the outside calculus. the outside differential calculus is now being famous through mathematicians and physicists because the most sensible approach to formulating the geometrical legislation of physics, and the frontiers of physics have already started to reopen basic questions on the geometry of area and time. protecting the fundamentals of differential and imperative calculus, the authors then practice the speculation to attention-grabbing difficulties in optics, electronics (networks), electrostatics, wave dynamics, and at last to classical thermodynamics. The authors undertake the "spiral strategy" of training (rather than rectilinear), protecting an analogous subject numerous occasions at expanding degrees of class and variety of software.

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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.