| Abstract: | We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data ({(u_0,theta_0)}) are required to be only in the space ({X={fin L^2(mathbb{R}^2),|,partial_{x} f in L^2(mathbb{R}^2)}}), and thus our result generalizes that of Cao and Wu (Arch Rational Mech Anal, 208:985–1004, 2013), where the initial data are assumed to be in ({H^2(mathbb{R}^2)}). The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the ({L^infty(mathbb{R}^2)}) norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one. |
