  We consider the subcritical SQG equation in its natural scale invariant
Sobolev space and prove the existence of a global attractor of optimal
regularity. The proof is based on a new energy estimate in Sobolev spaces to
bootstrap the regularity to the optimal level, derived by means of nonlinear
lower bounds on the fractional laplacian. This estimate appears to be new in
the literature, and allows a sharp use of the subcritical nature of the
$L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for
weak solutions as well. Moreover, we study the critical limit of the attractors
and prove their stability and upper-semicontinuity with respect to the strength
of the diffusion.
