  We prove the global well-posedness of the two-dimensional Boussinesq
equations with zero viscosity and positive diffusivity in bounded domains for
rough initial data [ $u_{0}\in L^{2}$, $\text{curl}\,u_{0}\in L^{\infty}$ and
$\theta_{0}\in B^{2-2/p}_{q,p}$ with $p\in (1,\infty)$, $q\in (2,\infty)$ ].
Our method is based on the maximal regularity for heat equation.
