We show the existence of global strong solutions for the compressible Navier–Stokes system in dimension with large initial data on the irrotational part of the velocity. We introduce a new notion of quasi-solutions when the initial velocity is assumed to be irrotational, these last one exhibit regularizing effects both on the velocity and in a very surprising way also on the density (indeed the density is a priori governed by a hyperbolic equation). We would like to point out that this smoothing effect is purely non-linear and is absolutely crucial in order to deal with the pressure term as it provides new damping effects in high frequencies. In particular this new damping effect enables us to deal with a Van der Waals pressure.
Nous montrons lʼexistence de solutions fortes globales pour le système de Navier–Stokes compressible en dimension avec des données initiales grandes sur la partie irrotationnelle de la vitesse. Nous introduisons une nouvelle notion de quasi-solution lorsque la vitesse initiale est supposée irrotationnelle, cette dernière exhibe à la fois des effets régularisants sur la vitesse mais aussi de manière très surprenante sur la densité (en effet la densité est à priori gouvernée par une équation hyperbolique). Nous aimerions faire remarquer que cet effet régularisant est purement non linéaire et est absolument crucial afin de traiter la pression puisquʼil fournit un effet dʼamortissement en hautes fréquences. En particulier ce nouvel effet dʼamortissement nous permet de traiter le cas dʼune pression Van der Waals.
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Boris Haspot 1
@article{CRMATH_2012__350_5-6_249_0,
author = {Boris Haspot},
title = {Existence of global strong solutions for the {Saint-Venant} system with large initial data on the irrotational part of the velocity},
journal = {Comptes Rendus. Math\'ematique},
pages = {249--254},
publisher = {Elsevier},
volume = {350},
number = {5-6},
year = {2012},
doi = {10.1016/j.crma.2012.03.007},
language = {en},
}
TY - JOUR AU - Boris Haspot TI - Existence of global strong solutions for the Saint-Venant system with large initial data on the irrotational part of the velocity JO - Comptes Rendus. Mathématique PY - 2012 SP - 249 EP - 254 VL - 350 IS - 5-6 PB - Elsevier DO - 10.1016/j.crma.2012.03.007 LA - en ID - CRMATH_2012__350_5-6_249_0 ER -
%0 Journal Article %A Boris Haspot %T Existence of global strong solutions for the Saint-Venant system with large initial data on the irrotational part of the velocity %J Comptes Rendus. Mathématique %D 2012 %P 249-254 %V 350 %N 5-6 %I Elsevier %R 10.1016/j.crma.2012.03.007 %G en %F CRMATH_2012__350_5-6_249_0
Boris Haspot. Existence of global strong solutions for the Saint-Venant system with large initial data on the irrotational part of the velocity. Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 249-254. doi : 10.1016/j.crma.2012.03.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.03.007/
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