Title: Optimal rate of convergence in Stratified Boussinesq system,
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.