Global Solutions to the Two Dimensional Quasi-Geostrophic Equation with Critical or Super-Critical Dissipation
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The two dimensional quasi-geostrophic (2D QG) equation with critical and super-critical dissipation is studied in Sobolev space H s (ℝ2). For critical case (α= Open image in new window ), existence of global (large) solutions in H s is proved for s≥ Open image in new window when Open image in new window is small. This generalizes and improves the results of Constantin, D. Cordoba and Wu  for s = 1, 2 and the result of A. Cordoba and D. Cordoba  for s= Open image in new window . For s≥1, these solutions are also unique. The improvement for pushing s down from 1 to Open image in new window is somewhat surprising and unexpected. For super-critical case (α ∈ (0, Open image in new window )), existence and uniqueness of global (large) solution in H s is proved when the product Open image in new window is small for suitable s≥2−2α, p ∈ [1,∞] and β ∈ (0,1].
KeywordsTwo dimensional dissipative quasi-geostrophic equations Existence Uniqueness Critical Super-critical Sobolev space
Mathematics Subject Classification (2000)76D 35Q
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