Regularity criterion to the axially symmetric Navier-Stokes equations

摘要

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle: parallel to ru(theta) (r, z, t)parallel to(L infinity) <= parallel to ru(theta) (r, z, 0)parallel to(L infinity). We first prove the global regularity of solutions if parallel to ru(theta) (r, z, 0)parallel to(L infinity) or parallel to ru(theta) (r, z, t)parallel to(L infinity(r <= r0)) is small compared with certain dimensionless quantity of the initial data. This result improves the one in Zhen Lei and Qi S. Zhang [10]. As a corollary, we also prove the global regularity under the assumption that} vertical bar ru(theta)(r,z,t)vertical bar <= vertical bar ln r vertical bar(-3/2), for all 0 < r <= delta(0) is an element of (0,1/2). (C) 2015 Elsevier Inc. All rights reserved.