Fujita Kato solution for compressible Navier-Stokes equation with axisymmetric initial data and zero Mach number limit

In this paper we investigate the question of the existence of global strong solution for the compressible Navier Stokes equations for small initial data such that the rotational part of the velocity Pu0 belongs to Ḣ N 2 −1. We show then an equivalence of the so called Fujita Kato theorem to the case of the compressible Navier-Stokes equation when we consider axisymmetric initial data in dimension N = 2, 3. The main difficulty is relied to the fact that in this case the velocity is not Lipschitz, in consequence we have to study carefully the coupling between the rotational and irrotational part of the velocity. In a second part, following the arguments developed in [13] we adress the question of convergence to the incompressible model (for illprepared initial data) when the Mach number goes to zero.