Global regularity for critical SQG in bounded domains

We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in ${\mathbb{R}}^2$. This solves an open problem. We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.