Optimal rate of convergence in Stratified Boussinesq system

TL;DR

This paper investigates the convergence rate of viscous solutions to inviscid solutions in the 2D stratified Navier-Stokes system, establishing optimal rates and extending previous results to a broader setting.

Contribution

It extends convergence results for the stratified Navier-Stokes system, proving optimal rates in $L^p$ spaces and generalizing prior work from $L^2$ to $L^p$.

Findings

Convergence rate for vortices is optimal in $L^p$ space, given by $( u t)^{1/(2p)}$.

Global strong estimates are established uniformly with respect to viscosity.

Results extend previous convergence findings to a more general stratified setting.

Abstract

We study the vortex patch problem for $2d-$stratified Navier-Stokes system. We aim at extending several results obtained in \cite{ad,danchinpoche,hmidipoche} for standard Euler and Navier-Stokes systems. We shall deal with smooth initial patches and establish global strong estimates uniformly with respect to the viscosity in the spirit of \cite{HZ-poche, Z-poche}. This allows to prove the convergence of the viscous solutions towards the inviscid one. In the setting of a Rankine vortex, we show that the rate of convergence for the vortices is optimal in $L^p$ space and is given by $(\mu t)^{\frac{1}{2p}}$. This generalizes the result of \cite{ad} obtained for $L^2$ space.