Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation

The present paper is devoted to the study of the global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. In the absence of horizontal dissipation, we establish agrowth estimate on vertical component of velocity, that is, supp≥,{norm of matrix}u2(t){norm of matrix}Lpplogp which is close to {norm of matrix}u2(t){norm of matrix}L∞ and is bounded via the low-high decomposition technique. This together with the smoothing effect in vertical direction enables us to obtain the H¹-estimate for velocity. Based on this, we prove the existence and uniqueness of classical solution without smallness assumptions. In addition, we also discuss the global well-posedness result for the rough initial data.