Local persistence of geometric structures of the inviscid nonlinear Boussinesq system

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