Existence of global strong solutions for the Saint-Venant system with large initial data on the irrotational part of the velocity

Abstract

We show the existence of global strong solutions for the compressible Navier–Stokes system in dimension N⩾2 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.