The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the L-tau-norm of the vertical velocity v for any 1 < r < infinity is globally bounded and that the L-infinity-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace (-Delta)(delta) for delta > 0 would guarantee the global regularity of classical solutions. (C) 2010 Elsevier Inc. All rights reserved.