Comptes Rendus
no. 9-10
Partial Differential Equations
Existence of global strong solutions for the barotropic Navier–Stokes system with large initial data on the rotational part of the velocity
[Existence de solutions fortes globales pour le système de Navier–Stokes compressible avec des données initiales grandes sur la partie rotationnelle de la vitesse]

Présenté par : the Editorial Board

Boris Haspot1
1 Ceremade, UMR CNRS 7534, université de Paris Dauphine, place du Maréchal DeLattre De Tassigny, 75775 Paris cedex 16, France
Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 487-492.

Nous montrons lʼexistence de solutions fortes globales pour le système de Navier–Stokes compressible en dimension N⩾2 avec des données initiales grandes sur la partie rotationnelle de la vitesse. Suivant Chemin et Gallagher (2009, 2011) [3,4], nous cherchons a exhiber des données initiales u0 telles que la projection sur les champs de vecteurs à divergence nulle Pu0 soient grandes dans B∞,∞−1 (qui est le plus large espace invariant par le scaling des équations) et telle que ces données initiales génèrent des solutions fortes globales. Le fait que lʼhypothèse de petitesse dans Chemin et Gallagher (2009) [3] a lieu sur le terme non linéaire de convection nous permet de décomposer la solution des équations de Navier–Stokes compressible comme la somme dʼune vitesse incompressible et dʼune vitesse purement compressible. Combinant la notion de quasi-solution introduite dans Haspot [8,9,7], nous obtenons lʼexistence de solutions fortes globales avec des données initiales à la fois grande pour la partie irrationnelle et la partie rotationnelle.

We show the existence of global strong solutions for the compressible Navier–Stokes system in dimension N⩾2 with large initial data on the rotational part of the velocity. By following Chemin and Gallagher (2009, 2011) [3,4], we aim at exhibiting large initial data u0 such that the projection on the divergence field Pu0 is large in B∞,∞−1 (which is the largest space invariant by the scaling of the equations) and such that these initial data generate global strong solution. The fact that the smallness hypothesis in Chemin and Gallagher (2009) [3] holds on the nonlinear term of convection enables us to split the solution of the compressible Navier–Stokes equations in the sum of an incompressible solution and of a purely compressible solution. Combining the notion of quasi-solution introduced in Haspot [8,9,7], we obtain the existence of global strong solution for the shallow water system for large initial velocity both on the irrotational and rotational part.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.04.017

Boris Haspot 1

1 Ceremade, UMR CNRS 7534, université de Paris Dauphine, place du Maréchal DeLattre De Tassigny, 75775 Paris cedex 16, France 
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Boris Haspot. Existence of global strong solutions for the barotropic Navier–Stokes system with large initial data on the rotational part of the velocity. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 487-492. doi : 10.1016/j.crma.2012.04.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.04.017/

[1] F. Charve; R. Danchin A global existence result for the compressible Navier–Stokes equations in the critical Lp framework, Archive for Rational Mechanics and Analysis, Volume 198 (2010) no. 1, pp. 233-271

[2] F. Charve, B. Haspot, Existence of strong solutions in larger space for the shallow-water system, Advances in Differential Equations, submitted for publication.

[3] J.-Y. Chemin; I. Gallagher Wellposedness and stability results for the Navier–Stokes equations in R3, Annales de lʼInstitut H. Poincaré, Analyse non Linéaire, Volume 26 (2009) no. 2, pp. 599-624

[4] J.-Y. Chemin; I. Gallagher; M. Paicu Global regularity for some classes of large solutions to the Navier–Stokes equations, Annals of Mathematics, Volume 173 (2011) no. 2, pp. 983-1012

[5] B. Haspot Existence of global strong solutions in critical spaces for barotropic viscous fuids, Archive for Rational Mechanics and Analysis, Volume 202 (2011) no. 2, pp. 427-460

[6] B. Haspot Existence of strong solutions in critical spaces for barotropic viscous fluids in larger spaces, Journal of Differential Equations, Volume 251 (2011) no. 8, pp. 2262-2295

[7] B. Haspot Existence of strong global solutions for the shallow-water equations with large initial data (preprint) | arXiv

[8] B. Haspot Existance of global strong solutions for the Saint-Venant system with large initial data on the irrotational part of the velocity, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) no. 5–6, pp. 249-254

[9] B. Haspot Global existence of strong solution for shallow water system with large initial data on the irrotational part (preprint) | arXiv

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