Regularity Criterion to the axially symmetric Navier-Stokes Equations

2015/08/13 by Wei, Dongyi
#35Q30 #76D05 #76D07 #76N10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.03318

Abstract

Smooth solutions to the axially symmetric Navier-Stokes equations obey the following maximum principle:‖ru_θ(r,z,t)‖_L^∞≤‖ru_θ(r,z,0)‖_L^∞. We first prove the global regularity of solutions if ‖ru_θ(r,z,0)‖_L^∞ or ‖ru_θ(r,z,t)‖_{L^∞(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 \cite1. As a corollary, we also prove the global regularity under the assumption that |ru_θ(r,z,t)|≤ |ln r|^-3/2, ∀ 0

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