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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.

Smooth attractors for weak solutions of the SQG equation with critical dissipation