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By D. Cioranescu, Jacques-Louis L. Lions

This booklet comprises the written types of lectures added on the grounds that 1997 within the recognized weekly seminar on utilized arithmetic on the collage de France in Paris, directed by way of Jacques-Louis Lions. it's the 14th and final of the sequence, as a result of contemporary and premature loss of life of Professor Lions. The texts during this quantity deal normally with quite a few elements of the speculation of nonlinear partial differential equations. They current either theoretical and utilized ends up in many fields of transforming into significance equivalent to Calculus of adaptations and optimum regulate, optimization, procedure concept and keep watch over, operations study, fluids and continuum mechanics, nonlinear dynamics, meteorology and weather, homogenization and fabric technology, numerical research and clinical computations The e-book is of curiosity to all people from postgraduate, who needs to stick with the latest growth in those fields.18.07

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Additional resources for Nonlinear Partial Differential Equations and their Applications: Collège de France Seminar Volume XIV

Example text

Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. in Mathematical Physics, 94 (1984), 61-66. [2] H. Br~zis, Analyse fonctionnelle. Th6orie et applications, Collection Math6matiques Appliqu6es pour la MMtrise, Masson, Paris, 1983. -Y. Chemin, Fluides parfaits incompressibles, Ast6risque, 230 (1995). -Y. Chemin, A remark on the inviscid limit for two-dimensionnal incompressible fluid, Comm. Part. Diff. , 21 (1996), 1771-1779. [5] R. Danchin, Poches de tourbillon visqueuses, Journal de Math6matiques Pures et Appliqu~es, 76 (1997), 609-647.

Soit (i) Pour tout t >__ 0 et ~' < e, Oat,v est dans C l+e,. { f 0 = 0} une dquation non ddgdndrde de On ~ et ft,v = fo o ~t,-1 . AIors { ft,v = 0} est une dquation non ddgdndrde de Of~t,v, fv C Llo ~ (JR +; CX+e') uniformdment en v et f , tend vers fo dans Llo~176 C 1+~') 1orsque v tend vers O. 3. Le gradient de v0 n'est alors pas n6cessairement born6 et peut exploser comme un logarithme au voisinage de 34 Limite non visqueuse pour les fluides incompressibles axisymdtriques certains points. 6 - Soit v ~ E H 1 un champ de vecteur axisymdtrique ~ divergence nuUe tel que w ~ w ~ c L 2 fq L ~ .

I1 suffit d'utiliser l'invariance de (NSv) par la transformation v(t,x) -, Av(A2t, Ax) pour le voir. Dans le cas L ~ , le "bon exposant" serait 5/2. Au logarithme pros, c'est ce que nous avons obtenu. Rappelons que dans [14], l'exposant trouv6 6tait 11/4. R~fdrences bibliographiques [1] J. Beale, T. Kato et A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. in Mathematical Physics, 94 (1984), 61-66. [2] H. Br~zis, Analyse fonctionnelle. Th6orie et applications, Collection Math6matiques Appliqu6es pour la MMtrise, Masson, Paris, 1983.

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