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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2008-12-02
Accepted: 2009-04-29
Published online: 2009-05-27

DOI: 10.1016/j.crma.2009.04.027

Author's affiliations:

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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     title = {Zero-noise solutions of linear transport equations without uniqueness: an example},
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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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