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Additional resources for Areas and logarithms (Topics in mathematics)

Example text

L’id´ee de G. M´etivier et S. 1, pour commencer l’´etude en se restreignant ` a un nombre fini de modes de Fourier. Cela signifie que l’on peut commencer par consid´erer des mod`eles en dimension finie. La premi`ere partie de [51] (et la section 5 de [49]) est ainsi d´evolue `a l’´etude du syst`eme dynamique suivant, `a deux ´echelles en temps : ε∂t uε = A(S)uε + εQ(S)(uε , uε ) (24) ∂t Sε = F (S)uε d´ ef sur le domaine D = [0, L1 ] × · · · × [0, Ld] o` u les Li sont des r´eels strictement positifs.

En outre, supposons que (uε,0 , cε,0 )ε>0 tend vers (u0 , c0 ) dans H s (Ω) avec div u0 = 0 et ∇c0 = 0. On suppose ici que s s0 + 2. Soit alors u la solution du syst`eme incompressible (2), ´el´ement de C 0 ([0, T0 [, H s (Ω)) ∩ C 1 ([0, T0 [, H s−1 (Ω)) pour un certain T0 > 0 (qui est infini si d = 2). Alors pour tout T < T0 , il existe ε0 tel que pour tout ε ε0 , l’´equation (4) admet une unique solution sur le temps [0, T ], telle que (7) uε → u dans C 0 ([0, T ], H s−1 (Ω)), et ∇cε → 0 dans C 0 ([0, T ], H s−2 (Ω)).

2. — Soit Ω un domaine r´egulier de l’espace Rd , et soit s un entier d strictement plus grand que +1. Il existe une constante C telle que si (u0 , c0 ) sont des 2 ´el´ements de H s (Ω) (u0 s’annulant au voisinage du bord par exemple), alors il existe un temps T et une unique solution (u, c) dans l’espace C 0 ([0, T ], H s (Ω)) au syst`eme (3). C En outre on a la minoration suivante sur le temps d’existence : T · (u0 , c0 ) H s (Ω) ´ ASTERISQUE 299 (926) ´ ´ RESULTATS RECENTS SUR LA LIMITE INCOMPRESSIBLE 33 Dans ce th´eor`eme (et partout dans ce texte), l’espace H s (Ω) d´esigne l’espace des fonctions de carr´e int´egrable sur Ω, dont toutes les d´eriv´ees jusqu’`a l’ordre s sont de carr´e int´egrable.

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Areas and logarithms (Topics in mathematics) by a i markushevich

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