Boussinesq System with Partial Viscous Diffusion or Partial Thermal Diffusion Forced by a Random Noise

We study the Boussinesq system in a two-dimensional domain. In case only the velocity equation is forced by a white-in-time multiplicative noise, we prove the global existence of a martingale solution in the following two cases: positive horizontal viscous diffusion but zero vertical viscous diffusion and zero thermal diffusion; positive horizontal thermal diffusion but zero viscous diffusion and zero vertical thermal diffusion. Finally, in case both velocity and temperature equations are forced by a white-in-time multiplicative noise, and the velocity equation has zero viscous diffusion while the temperature equation has full thermal diffusion, the global existence of a martingale solution was shown in Yamazaki (Stoch Anal Appl, 34:404–426, 2016); in this manuscript we additionally prove significantly higher regularity.