Global regularity results for the 2D Boussinesq equations with vertical dissipation

AbstractThis paper furthers the study of Adhikari et al. (2010) [2] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity v of any classical solution in the Lebesgue space Lq with 2⩽q<∞ is bounded by C1q for C1 independent of q. This bound significantly improves the previous exponential bound. In addition, we prove that, if v satisfies ∫0Tsupq⩾2‖v(⋅,t)‖Lq2qdt<∞, then the associated solution of the 2D Boussinesq equations preserve its smoothness on [0,T]. In particular, ‖v‖Lq⩽C2q implies global regularity