Inviscid limit for the axisymmetric stratified Navier–Stokes system

Abstract

This paper is devoted to the study of the Cauchy problem for the stratified Navier–Stokes system in three-dimensional space. 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 space $H^s\times H^{s-2}$ with $s>5/2.$ The bounds on the solution are uniform with respect to the viscosity. In the second part, we analyse the inviscid limit problem. We prove that the viscous solutions $(v_{\nu },{\rho }_{\nu }{)}_{\nu >0}$ converge strongly in the space $L_{\text{loc}}^{\infty }({\mathbb{R}}_{+};H^s\times H^{s-2})$ to the solution $(v,\rho )$ of the stratified Euler system.