The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$

We prove that the critical surface quasi-geostrophic equation driven by a force $f$ possesses a compact global attractor in $L^2({\mathbb{T}}^2)$ provided $f\in L^p({\mathbb{T}}^2)$ for some $p>2$. First, the De Giorgi method is used to obtain uniform $L^{\mathrm{\infty }}$ estimates on viscosity solutions. Even though this does not provide a compact absorbing set, the existence of a compact global attractor follows from the continuity of solutions, which is obtained by estimating the energy flux using the Littlewood-Paley decomposition.