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We establish global-posedness in time for the viscous Boussinesq equations intwo dimensions of space with temperature-dependent diffusivity in the frameworkof a smooth vortex patch. We also provide the inviscid limit for velocity,temperature, and associated flow toward the system studied very recently in\cite{Paicu-Zhu} as soon as the viscosity goes to zero, and quantify the rateof convergence.

Inviscid limit for the viscous 2D Boussinesq system with temperature-dependent diffusivity