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

Abstract

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 $L^{\infty }$ 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.

14 pages. Details are added in the proof of Lemma 2.3. Minor corrections are made in the proof of Theorem 3.1

References in corpus (2)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• Absence of anomalous dissipation of energy in forced two dimensional fluid equations

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