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Inspired by the recently published paper \cite{Hassainia-Hmidi}, the currentpaper investigates the local well-posedness for the generalized $2d-$Boussinesqsystem in the setting of regular/singular vortex patch. Under the conditionthat the initial vorticity $\omega_{0}={\bf 1}_{D_0}$, with $\partial D_0$ is aJordan curve with a H\"older regularity $C^{1+\EE},\;0<\EE<1$ and the densityis a smooth function, we show that the velocity vector field is locallywell-posed and we also establish local-in-time regularity persistence of theadvected patch. Although, in the case of the singular patch, the analysis israther complicated due to the coupling phenomena of the system and thestructure of the patch. For this purpose, we must assume that the initialnonlinear term is constant around the singular part of the patch boundary.

Local persistence of geometric structures of the inviscid nonlinear Boussinesq system