Inviscid limit for axisymmetric stratified Navier-Stokes system

This paper is devoted to the study of the Cauchy problem for the stratified Navier-Stokes system in space dimension three. In the first part of the paper, we prove the existence of a unique global solution $(v_{\nu },{\rho }_{\nu })$ for this system with axisymmetric initial data belonging to the Sobolev spaces $H^s\times H^{s-2}$ with $s>5\mathrm{/}2.$ The bounds of the solution are uniform with respect to the viscosity. In the second part, we analyse the inviscid limit problem. We prove the strong convergence in the space $L_{\text{loc}}^{\mathrm{\infty }}({\text{RR}}_{+};H^s\times H^{s-2})$ of the viscous solutions $(v_{\nu },{\rho }_{\nu }{)}_{\nu >0}$ to the solution $(v,\rho )$ of the stratified Euler system.