Global Solutions to the Two Dimensional Quasi-Geostrophic Equation with Critical or Super-Critical Dissipation

The two dimensional quasi-geostrophic (2D QG) equation with critical and super-critical dissipation is studied in Sobolev space Hs(ℝ2). For critical case (α=[inline-graphic not available: see fulltext]), existence of global (large) solutions in Hs is proved for s≥[inline-graphic not available: see fulltext] when [inline-graphic not available: see fulltext] is small. This generalizes and improves the results of Constantin, D. Cordoba and Wu [4] for s = 1, 2 and the result of A. Cordoba and D. Cordoba [8] for s=[inline-graphic not available: see fulltext]. For s≥1, these solutions are also unique. The improvement for pushing s down from 1 to [inline-graphic not available: see fulltext] is somewhat surprising and unexpected. For super-critical case (α ∈ (0,[inline-graphic not available: see fulltext])), existence and uniqueness of global (large) solution in Hs is proved when the product [inline-graphic not available: see fulltext] is small for suitable s≥2−2α, p ∈ [1,∞] and β ∈ (0,1].