Optimal regularity for the 2D Euler equations in the Yudovich class

We analyze the optimal regularity that is exactly propagated by a transport equation driven by a velocity field with a BMO gradient. As an application, we study the 2D Euler equations in case the initial vorticity is bounded. The sharpness of our result for the Euler equations follows from a variati...

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Bibliographic Details
Published in:Journal de mathématiques pures et appliquées 2024-11, Vol.191, p.103631, Article 103631
Main Authors: De Nitti, Nicola, Meyer, David, Seis, Christian
Format: Article
Language:English
Subjects:
Bahouri–Chemin's vortex patch
Euler equations
Littlewood–Paley
Non smooth vector fields
Propagation of regularity
Transport equation
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Summary:We analyze the optimal regularity that is exactly propagated by a transport equation driven by a velocity field with a BMO gradient. As an application, we study the 2D Euler equations in case the initial vorticity is bounded. The sharpness of our result for the Euler equations follows from a variation of Bahouri and Chemin's vortex patch example. Nous analysons la régularité optimale propagée par une équation de transport avec un champ de vitesse ayant un gradient BMO. Nous appliquons notre résultat aux équations d'Euler en dimension 2 dans le cas où la vorticité initiale est bornée. L'optimalité de nos résultats découle d'une variante de l'exemple de la poche de tourbillon de Bahouri et Chemin.
ISSN:0021-7824
DOI:10.1016/j.matpur.2024.103631