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Comptes Rendus Mathématique
Volume 350, n° 9-10
pages 487-492 (mai 2012)
Doi : 10.1016/j.crma.2012.04.017
Received : 20 Mars 2012 ;  accepted : 26 April 2012
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
 

Boris Haspot
Ceremade, UMR CNRS 7534, université de Paris Dauphine, place du Maréchal DeLattre De Tassigny, 75775 Paris cedex 16, France 

Abstract

We show the existence of global strong solutions for the compressible Navier–Stokes system in dimension   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   such that the projection on the divergence field   is large in   (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 [[7], [8], [9]], we obtain the existence of global strong solution for the shallow water system for large initial velocity both on the irrotational and rotational part.

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Résumé

Nous montrons lʼexistence de solutions fortes globales pour le système de Navier–Stokes compressible en dimension   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   telles que la projection sur les champs de vecteurs à divergence nulle   soient grandes dans   (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 [[7], [8], [9]], nous obtenons lʼexistence de solutions fortes globales avec des données initiales à la fois grande pour la partie irrationnelle et la partie rotationnelle.

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