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 (−1)α . The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov space Br 2,q [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 − 2α; second, to establish the global existence of solutions in the homogeneous Besov space B̊p,q with general indices p and q; and third, to determine the uniqueness of solutions in any one of the four spaces: Bs 2,q , B̊ r p,q , L q ((0, T ); B s+ 2α q 2,q ) and L q ((0, T ); B̊ r+ 2α q p,q ), where s ≥ 2− 2α and r = 1− 2α + 2 p . c © 2006 Elsevier Ltd. All rights reserved. MSC: 35Q35; 76B03