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

Abstract

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 B 2 , q r [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 − 2 α ; second, to establish the global existence of solutions in the homogeneous Besov space B ̊ p , q r with general indices p and q ; and third, to determine the uniqueness of solutions in any one of the four spaces: B 2 , q s , B ̊ p , q r , L q ( ( 0 , T ) ; B 2 , q s + 2 α q ) and L q ( ( 0 , T ) ; B ̊ p , q r + 2 α q ) , where s ≥ 2 − 2 α and r = 1 − 2 α + 2 p .