Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation

This paper concerns itself with Besov space solutions of the 2-D quasi-geostrophic (QG) equation with dissipation induced by a fractional Laplacian (−Δ)α. The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov space B2,qr [J. Wu, Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. Math. Anal. 36 (2004–2005) 1014–1030] to cover the case when r=2−2αr=2−2α; second, to establish the global existence of solutions in the homogeneous Besov space B̊p,qr with general indices pp and qq; and third, to determine the uniqueness of solutions in any one of the four spaces: B2,qs, B̊p,qr, Lq((0,T);B2,qs+2αq) and Lq((0,T);B̊p,qr+2αq), where s≥2−2αs≥2−2α and r=1−2α+2p.