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

Alexey Cheskidov; Mimi Dai

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arXiv · 1402.4801

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

Alexey Cheskidov·Mimi Dai

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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.

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Alexey Cheskidov, Mimi Dai. 2014-02-19. The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$. https://arxiv.org/abs/1402.4801

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