  We consider the evolution of weak vanishing viscosity solutions to the
critically dissipative surface quasi-geostrophic equation. Due to the possible
non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical
system and prove the existence of a global attractor of optimal Sobolev
regularity. To achieve this, we derive a new Sobolev estimate involving
H\"older norms, which complement the existing estimates based on commutator
analysis.
