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An extension of the Beale Kato Majda criterion for the Euler equations


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Niveau: Supérieur, Doctorat, Bac+8
An extension of the Beale-Kato-Majda criterion for the Euler equations Fabrice Planchon ? Abstract The Beale-Kato-Majda criterion asserts that smooth solutions to the Euler equations remain bounded past T as long as ∫ T 0 ???∞dt is finite, ? being the vorticity. We show how to replace this by a weaker statement, on supj ∫ T 0 ?∆j??∞dt, where ∆j is a frequency localization around |?| ≈ 2j . Introduction The incompressible Euler equations read ? ?? ?? ∂u ∂t + u · ?u = ??p, ? · u = 0, u(x, 0) = u0(x), x ? Rn, t ≥ 0. (1) These equations are known to be locally well-posed for data u0 ? Hs, s > n 2 +1, or more generallyW sp with s? np > 1 (see [6] and references therein). In a celebrated paper, Beale-Kato-Majda gave the following criterion for blow- up: if blow-up occurs at time T , then necessarily, ∫ T 0 ???∞dt = +∞,(2) where ? = ?? u is the vorticity. One may rephrase it as: if ∫ T 0 ???∞dt < +∞,(3) ?Laboratoire Analyse, Geometrie & Applications, UMR 7539, Institut Galilee, Univer- site Paris 13, 99 avenue J.

  • criterion asserts

  • can control ∫

  • solution can

  • beale-kato-majda criterion

  • past time

  • remain bounded past



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R´esume´.We propose a simple method to accelerate significatively the convergence of P n1 Sn= 1/jlog(n) towards Euler’s constantγ: we construct linear combinations of j=1 consecutive values ofSn, to which are assigned certain binomial weights, obtained thanks to a classical integral representation ofγSnand certain special polynomials, of Legendre type. As a special case, we recover an approximation ofγdue to Elsner, obtained by a different method. Our approach also applies to the number log(4) which, as Sondow has noted, is in a sense an alternating analogue ofγ; this enables us to produce an apparently new expression ofπ/4 as an infinite product, which can be viewed as an analogue of Vacca’s series forγ. Finally, although this method cannot prove the irrationality ofγ, it is similar to the one used by Alladi and Robinson to prove the irrationality of log(2) by means of Legendre polynomials.
1.Introduction La constante d’Eulerγmmlelamiti,eolsrestd´eniecoeuqn+, de la suite n1n1¶¶µ µ X X 1 1j+ 1 Sn=log(n) =log j j j j=1j=1 Lavitessedeconvergenceesttre`slente,enO(1/ntrled´eveloonpvpoeimtesnumoemnol)c, asymptotique k X 1B2j1 γ=Sn++ + R(n, k), 2j 2n2j n j=1 ¡¢ 2k ou`lesB2jsont les nombres de Bernoulli etR(n, k) =O(k/πen)k/n(la constante ` dans leOseitdne´epdnnatedeketn: voir [6]). Aksitnam´xno,eagenepgn´easrmnome´e 2 en revanche le choixn=kcelere`ledluclacc´eaγa`nvcogeeritsuuieqniurfoennetuanss 2k la vitesse 1/kmais au prix du calcul des nombres de Bernoulli, qui ne sont pas entiers et dont la croissance est essentiellement factorielle. Sansmˆemeparlerduprobl`emetoujoursouvertdel´eventuelleirrationalite´deγ, on peut plussimplementsedemandercomment,`apartirdelasommeSn, calculer rapidementγ enutilisantdesquantite´scombinatoiresmoinscomplexesquelesnombresdeBernoulli,en particulierdesentiers.Ilexistedenombreusesfac¸onsdere´soudreceprobl`eme(dontcelle deSondow[8]´evoqu´eeplusbas).Uneme´thodeparticulie`rementint´eressanteae´t´edonne´e 1