Log-Lipschitz regularity and uniqueness of the flow for a field in $ \mathbf{(}\mathrm{W}_{\mathrm{loc}}^{\mathbf{n/p+1,p}}\mathbf{(}\mathbb{R}^{\mathbf{n}}\mathbf{))}^{\mathbf{n}}$

Enrique Zuazua

doi:10.1016/S1631-073X(02)02426-3

Log-Lipschitz regularity and uniqueness of the flow for a field in

{(W_{loc}^{𝐧 / 𝐩 + 1, 𝐩} (ℝ^{𝐧}))}^{𝐧}

[Régularité Log-Lipschitz et unicité du flot pour les champs de vecteurs

{(W_{loc}^{𝐧 / 𝐩 + 1, 𝐩} (ℝ^{𝐧}))}^{𝐧}

]

Enrique Zuazua ¹

¹ Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 17-22.

Résumé
Abstract

On considère le problème de Cauchy pour un système d'équations différentielles ordinaires $\dot{x} = b (x), t > 0; x (0) = x_{0}$ où l'état $x = x (t) \in ℝ^{n}$ et où b est un champ de vecteurs dans ${(W_{loc}^{n / p + 1, p} (ℝ^{n}))}^{n}$ . On démontre que, pour tout $x_{0} \in ℝ^{n}$ , 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 $b \in {(W_{loc}^{s, p} (ℝ^{n}))}^{n}$ 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.

We consider the initial value problem $\dot{x} = b (x), t > 0; x (0) = x_{0},$ with $x = x (t) \in ℝ^{n}$ . We prove that local existence and uniqueness of solutions holds when the field b belongs to ${(W_{loc}^{n / p + 1, p} (ℝ^{n}))}^{n}$ . This case corresponds to the limit regularity one in Sobolev terms since uniqueness may fail when $b \in (W_{loc}^{s, p} (ℝ^{n}))$ with s<n/p+1 but holds immediately when s>n/p+1 because of the Sobolev imbedding from ${(W_{loc}^{s, p} (ℝ^{n}))}^{n}$ 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.

Reçu le : 2002-03-19
Accepté le : 2002-04-28
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02426-3

Affiliations des auteurs :

Enrique Zuazua ¹

¹ Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain

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Enrique Zuazua. Log-Lipschitz regularity and uniqueness of the flow for a field in $ \mathbf{(}\mathrm{W}_{\mathrm{loc}}^{\mathbf{n/p+1,p}}\mathbf{(}\mathbb{R}^{\mathbf{n}}\mathbf{))}^{\mathbf{n}}$. Comptes Rendus. Mathématique, Volume 335 (2002) no. 1, pp. 17-22. doi : 10.1016/S1631-073X(02)02426-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02426-3/

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