Global existence of strong solution for viscous shallow water system with large initial data on the irrotational part

We are interested in studying the Cauchy problem for the viscous shallow-water system in dimension N≥2, we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations. More precisely our smallness assumption on the initial data is supercritical for the scaling of the equations. It allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension N=2. To do this, we introduce the notion of quasi-solutions which consists in solving the pressureless viscous shallow water system. We can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density. This smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the “quasi-solutions”. Indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations. To finish we prove the existence of global strong solution with large initial data when N≥2 provided that the Mach number is sufficiently large.