Global regularity for 2D Boussinesq temperature patches with no diffusion

This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space ${\mathbb{R}}^2$. We prove that for initial patches with $W^{2,\mathrm{\infty }}$ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound ${\mathrm{\nabla }}^{2}u$, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the $C^{1+\gamma }$ regularity result in [23], $0<\gamma <1$, using the scale of Sobolev spaces for the velocity. Furthermore, taking advantage of the new cancellations, we go beyond to show the persistence of regularity for $C^{2+\gamma }$ patches.