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Comptes Rendus Mathématique
Volume 335, n° 1
pages 17-22 (2002)
Doi : S1631-073X(02)02426-3
Received : 19 Mars 2002 ;  accepted : 28 April 2002
Log-Lipschitz regularity and uniqueness of the flow for a field in (Wn/p+1,ploc(Rn))n
Régularité Log-Lipschitz et unicité du flot pour les champs de vecteurs (Wn/p+1,ploc(Rn))n
 

Enrique Zuazua
Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain 

Note presented by Pierre-Louis Lions

Abstract

We consider the initial value problem ẋ=b(x),t>0;x(0)=x0, with x=x(t)Rn. We prove that local existence and uniqueness of solutions holds when the field b belongs to (Wn/p+1,ploc(Rn))n. This case corresponds to the limit regularity one in Sobolev terms since uniqueness may fail when b(Ws,ploc(Rn)) with s <n /p +1 but holds immediately when s >n /p +1 because of the Sobolev imbedding from (Ws,ploc(Rn))n into the space of locally Lipschitz fields. The proof of uniqueness relies on a Log-Lipschitz continuity property we prove for vector fields in this Sobolev class. When p =2 the proof is carried out by means of Fourier series, decomposing the field into the low and high frequencies. When p ≠2 the proof uses Trudinger's inequality and the strategy of proof of Morrey's theorem. To cite this article: E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 17-22.

The full text of this article is available in PDF format.
Résumé

On considère le problème de Cauchy pour un système d'équations différentielles ordinaires ẋ=b(x),t>0 ;x(0)=x0 où l'état x=x(t)Rn et où b est un champ de vecteurs dans (Wn/p+1,ploc(Rn))n. On démontre que, pour tout x0Rn, il existe une unique solution locale (en temps). Ceci correspond à un cas limite du point de vue de l'appartenance à des espaces de Sobolev. En effet, si s <n /p +1 il existe des champs de vecteurs b(Ws,ploc(Rn))n pour lesquels l'unicité n'est pas satisfaite. Par contre, lorsque s >n /p +1 l'unicité est trivialement vraie car b est localement Lipschitz grâce aux inclusions de Sobolev. La preuve consiste à démontrer que le champ de vitesses vérifie une condition de continuité de type Log-Lipschitz permettant de vérifier que la condition classique d'unicité d'Osgood est satisfaite. Lorsque p =2 la preuve se fait à l'aide des séries de Fourier. Lorsque p ≠2 on utilise l'inégalité de Trudinger et la stratégie de la preuve du théorème de Morrey. Pour citer cet article : E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 17-22.

The full text of this article is available in PDF format.


© 2002  Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All Rights Reserved.
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