Comptes Rendus
no. 1
Partial differential equations
Applications of Bourgain–Brézis inequalities to fluid mechanics and magnetism
[Applications des inégalités de Bourgain–Brézis à la mécanique des fluides et au magnétisme]

Présenté par : Haïm Brézis

Sagun Chanillo1 ; Jean Van Schaftingen2 ; Po-Lam Yung3
1 Department of Mathematics, State University of New Jersey, Rutgers, NJ 08854, USA
2 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
3 Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 51-55.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.005

Sagun Chanillo 1 ; Jean Van Schaftingen 2 ; Po-Lam Yung 3

1 Department of Mathematics, State University of New Jersey, Rutgers, NJ 08854, USA
2 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
3 Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong 
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Sagun Chanillo; Jean Van Schaftingen; Po-Lam Yung. Applications of Bourgain–Brézis inequalities to fluid mechanics and magnetism. Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 51-55. doi : 10.1016/j.crma.2015.10.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.005/

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Cité par Sources :

☆ 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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