On the regularity of a class of generalized quasi-geostrophic equations

In this article we consider the following generalized quasi-geostrophic equation partial derivative(t)theta + u . del theta + nu A(beta)theta = 0, u = A(alpha) R-perpendicular to theta, x is an element of R-2 where nu > 0, A := root-Delta . alpha is an element of]0, 1[ and beta is an element of ]0, 2[. We first show a general conditional criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case alpha is an element of ]0, 1[ and beta is an element of ]alpha + 1,2] we obtain the global well-posedness of the system with smooth initial data: and for the case alpha is an element of ]0, 1[ and beta is an element of ]2 alpha, alpha + 1] we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data. (C) 2011 Elsevier Inc. All rights reserved.