Inviscid limit for axisymmetric stratified Navier-Stokes system

TL;DR

This paper establishes the existence of unique global solutions for the axisymmetric stratified Navier-Stokes system and proves their strong convergence to the Euler system solutions as viscosity vanishes.

Contribution

It provides the first rigorous analysis of the inviscid limit for axisymmetric stratified Navier-Stokes equations with uniform bounds.

Findings

Global existence of solutions with axisymmetric initial data.

Uniform bounds independent of viscosity.

Strong convergence to stratified Euler solutions as viscosity tends to zero.

Abstract

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 $s>5/2.$ 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 $L^{\infty}_{\text{loc}}(\RR_+; H^{s}\times H^{s-2})$ of the viscous solutions $(v_\nu,\rho_\nu)_{\nu>0}$ to the solution $(v,\rho)$ of the stratified Euler system.