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| We prove the existence of a compact global attractor in $H^1(\mathbb{T}^2)$ for the dynamics of the forced critical surface quasi-geostrophic equation (SQG). After a transient time, the solution is bounded in $C^\alpha$ and $H^1$ by higher regularity norms of the forcing term $f$ and independently of the initial data. The attractor also has finite fractal (box-counting) dimension. |
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