  We prove local well-posedness for the inviscid surface quasigeostrophic (SQG)
equation in bounded domains of $\mathbb{R}^2$. When fractional Dirichlet
Laplacian dissipation is added, global existence of strong solutions is
obtained for small data for critical and supercritical cases. Global existence
of strong solutions with arbitrary data is obtained in the subcritical cases.
