  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 $X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in
L^2(\mathbb R^2)\}$, and thus our result generalizes that in [C. Cao, J. Wu,
Global regularity for the two-dimensional anisotropic Boussinesq equations with
vertical dissipation, Arch. Rational Mech. Anal., Vol. 208 (2013), 985-1004],
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.
