# Analisi funzionale. Teoria e applicazioni by Haim Brezis

By Haim Brezis

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N . I* + Subtracting equation 11 from equation i , we deduce that Ln B*(cri - a , , ) $ * has a I',, constant value -0,on A . Now, using the relationship p:(x) = 1, we deduce that 1,: satisfy the Gibbs state relationships on A . 0 The relaxation equations ( R E w ) have been used for large eddy simulations 1531; they where found very efficient. 3. 1. s~ f ' t r t lc~\~)lirtiotl cclr~cltiorr. Let 11s begin with the simple case of a finite-dimensional dynamical systern: where ( 1 : R" + R" is a srnooth mapping.

So that: The same arguments apply to the other invariants. For example. since r,too converges weakly towards C(x) = J'zdv,(z), we have, for the energy, E ( T , w o ) + E ( Z ) ,which is the energy of the Young measure v, and thus: We shall denote by (**) the set of constraints (associated to the constants of the motion) other than (*), that C has to satisfy: = {energy constraint, angular momenturn constraint (eventually)). (**) Thus we see that the constants of the motion bring the constraints (*), (**) on the possible long-time limits.

S11. I c , ~ , c ~ l . c,cr,vcJ. \. only rr tlistinct v:ilues ( 1 1 . . . t i , , . cudybe clcscribc~l~ n i ~ c r o ~ c o p i c iin~ ltcr-~ns l y ol' ;I set ol' local ~~ruhuhilitics /)I ( I . x ) . . . I ) , , ( / . X I . In other wonls. the system hiis ;ilrc;tdy u ~ i r l c r g o n ~ tine-scale vorticity oscillations. The lociilly ;~vcragcdvol-ticity is 6 ( t . x ) = C ,t i , I), ( t . x ) and u is the corresponding vclocily licld. s arc transported hy u, and we s~ipposeIllill. in i~dclitio~l, they under-go a difl'usion process.