Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains

We address the well-posedness for the two-dimensional Boussinesq equations with zero diffusivity in bounded domains. We prove global in time regularity for rough initial data: the initial velocity has ϵ fractional derivatives in Lq and the initial temperature is in Lq, for some q>2 and ϵ>0 arbitrarily small.