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.

中文翻译：

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分层 Boussinesq 系统中的最优收敛速度

我们研究了 $2d-$stratified Navier-Stokes 系统的涡旋补丁问题。我们的目标是扩展在 \cite{ad,danchinpoche,hmidipoche} 中获得的几个结果，用于标准 Euler 和 Navier-Stokes 系统。我们将本着 \cite{HZ-poche, Z-poche} 的精神处理平滑的初始补丁并统一建立关于粘度的全局强估计。这允许证明粘性解决方案向无粘性解决方案收敛。在 Rankine 涡流的设置中，我们表明涡流的收敛速度在 $L^p$ 空间中是最优的，并且由 $(\mu t)^{\frac{1}{2p}}$ 给出。这概括了为 $L^2$ 空间获得的 \cite{ad} 的结果。