Quasi-geostrophic equation in ℝ²

2014/11/05 by Tomasz Dłotko, Tomasz Dlotko, Dlotko, Tomasz +4
Mathematics · Physics and Astronomy · #35B41 #35Q35 #35S10 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #math-ph #math.DS #math.MP #msc:35B41 #msc:35Q35 #msc:35S10

paper · pdf · doi:10.48550/arxiv.1411.1178

openalex publication_date 2014/11/05 · arxiv created 2014/11/07 · arxiv updated 2014/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Solvability of Cauchy's problem in ℝ² for subcritical quasi-geostrophic equation is discussed here in two phase spaces; L^p(ℝ²) with p> (2)/(2α-1) and H^s(ℝ²) with s>1. A solution to that equation in critical case is obtained next as a limit of the H^s-solutions to subcritical equations when the exponent α of (-Δ)^α tends to (1)/(2)⁺. Such idea seems to be new in the literature. Existence of the global attractor in subcritical case is discussed in the paper. In section 7 we also discuss solvability of the critical problem with Dirichlet boundary condition in bounded domain Ω⊂ ℝ², when ‖ θ₀ ‖_L^∞(Ω) is small.

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