Quasi-geostrophic equation in R2R2

Solvability of Cauchy's problem in R2R2 for subcritical quasi-geostrophic equation is discussed here in two phase spaces; Lp(R2)Lp(R2) with p>22α−1 and Hs(R2)Hs(R2) with s>1s>1. A solution to that equation in critical case   is obtained next as a limit of the HsHs-solutions to subcritical equations when the exponent α   of (−Δ)α(−Δ)α tends to 12+. Such idea seems to be new in the literature. Existence of the global attractor in subcritical case is also studied in the paper. In Section 7 we discuss solvability of the critical problem with Dirichlet boundary condition in a bounded domain Ω⊂R2Ω⊂R2, when ‖θ0‖L∞(Ω)‖θ0‖L∞(Ω) and ‖f‖L∞(Ω)‖f‖L∞(Ω) are small.