Global well-posedness of the viscous Boussinesq equations

We prove the global well-posedness of the viscousincompressible Boussinesq equations in two spatial dimensions for generalinitial data in Hᵐ with m≥ 3.It is known that when both the velocity and thedensity equations have finite positive viscosity,the Boussinesq system does not develop finite time singularities.We consider here the challenging case when viscosity enters only in the velocityequation, but there is no viscosity in the density equation.Using sharp and delicate energy estimates, we prove global existenceand strong regularity of this viscous Boussinesq system for generalinitial data in Hᵐ with m ≥ 3.