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Comptes Rendus Mathématique
Volume 354, n° 1
pages 51-55 (janvier 2016)
Doi : 10.1016/j.crma.2015.10.005
Received : 4 September 2015 ;  accepted : 8 October 2015
Applications of Bourgain–Brézis inequalities to fluid mechanics and magnetism
Applications des inégalités de Bourgain–Brézis à la mécanique des fluides et au magnétisme
 

Sagun Chanillo a , Jean Van Schaftingen b , Po-Lam Yung c
a Department of Mathematics, State University of New Jersey, Rutgers, NJ 08854, USA 
b Institut de recherche en mathématique et en physique, Université catholique de Louvain, chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium 
c Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong 

Abstract

As a consequence of inequalities due to Bourgain–Brézis, we obtain local-in-time well-posedness for the two-dimensional Navier–Stokes equation with velocity bounded in spacetime and initial vorticity in bounded variation. We also obtain spacetime estimates for the magnetic field vector through improved Strichartz inequalities.

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

À partir d'inégalités de Bourgain–Brézis, nous démontrons le caractère bien posé localement dans le temps des équations de Navier–Stokes avec vitesse bornée en espace-temps et un tourbillon initial à variation bornée. Nous obtenons également des estimations en espace-temps pour le champ magnétique grâce à des inégalités de Strichartz améliorées.

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

 S.C. was partially supported by NSF grant DMS 1201474. J.V.S. was partially supported by the Fonds de la recherche scientifique, FNRS grant J.044.13. P.-L.Y. was partially supported by a direct grant for research from the Chinese University of Hong Kong (4053120). We thank Haïm Brézis for several comments that improved the paper.



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