Existence of global strong solutions for the barotropic Navier–Stokes system with large initial data on the rotational 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 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.