Global well-posedness for the critical dissipative quasi-geostrophic equations in L-infinity

Abstract

In this paper, we study the critical dissipative quasi-geostrophic equations in scaling invariant spaces. We prove that there exists a global-in-time small solution for small initial data theta(0) is an element of L-infinity boolean AND (H) over dot(1) such that R(theta(0)) is an element of L-infinity, where R is the Riesz transform. As a corollary, we prove that if in addition, theta(0) is an element of (B) over dot(infinity,q)(0), 1 <= q < 2, is small enough, then theta is an element of (L) over tilde (infinity)(t)(B) over dot(infinity,q)(0)boolean AND(L) over tilde (1)(t) (B) over dot(infinity,q)(1). (C) 2010 Elsevier Ltd. All rights reserved