The existence of a global attractor for the forced critical surface quasi-geostrophic equation in L²

2014/02/19 by Alexey Cheskidov, Cheskidov, Alexey, Mimi Dai +1
Mathematics · #35Q35 #37L30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #math.AP #msc:35Q35 #msc:37L30

paper · pdf · doi:10.48550/arxiv.1402.4801

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

openalex publication_date 2014/02/19 · openalex created_date 2016/06/24 · arxiv created 2017/12/07 · arxiv updated 2017/12/08 · openalex updated_date 2026/07/28

Abstract

We prove that the critical surface quasi-geostrophic equation driven by a force f possesses a compact global attractor in L²(\mathbb T²) provided f∈ L^p(\mathbb T²) for some p>2. First, the De Giorgi method is used to obtain uniform L^∞ 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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