Zero-noise solutions of linear transport equations without uniqueness: an example

Stefano Attanasio; Franco Flandoli

doi:10.1016/j.crma.2009.04.027

Partial Differential Equations/Probability Theory

Zero-noise solutions of linear transport equations without uniqueness: an example
[Solutions à bruit nul des équations linéaires de transport : un exemple]

Présenté par : Marc Yor

Stefano Attanasio ¹ ; Franco Flandoli ²

¹ Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
² Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, Via Buonarroti 1, 56127 Pisa, Italy

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 753-756.

Résumé
Abstract

On considère un exemple unidimensionnel classique d'équation de transport linéaire sans unicité des solutions faibles. En présence d'une perturbation donnée par un bruit multiplicatif convenablement choisi, l'équation se révèle bien posée. On identifie les deux solutions de l'équation déterministe obtenues dans la limite ou le bruit s'annule. On prouve aussi que la solution de viscosité nulle existe et qu'elle est différente des deux autres.

We consider a classical one-dimensional example of linear transport equation without uniqueness of weak solutions. Under a suitable multiplicative noise perturbation, the equation is well posed. We identify the two solutions of the deterministic equation obtained in the zero-noise limit. In addition, we prove that the zero-viscosity solution exists and is different from them.

Reçu le : 2008-12-02
Accepté le : 2009-04-29
Publié le : 2009-05-27

DOI : 10.1016/j.crma.2009.04.027

Affiliations des auteurs :

Stefano Attanasio ¹ ; Franco Flandoli ²

¹ Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
² Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, Via Buonarroti 1, 56127 Pisa, Italy

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     author = {Stefano Attanasio and Franco Flandoli},
     title = {Zero-noise solutions of linear transport equations without uniqueness: an example},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {753--756},
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     number = {13-14},
     year = {2009},
     doi = {10.1016/j.crma.2009.04.027},
     language = {en},
}

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Stefano Attanasio; Franco Flandoli. Zero-noise solutions of linear transport equations without uniqueness: an example. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 753-756. doi : 10.1016/j.crma.2009.04.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.027/

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