2015/05/04 by Qi S. Zhang, Zhang, Qi S.
Engineering · Mathematics · #35Q30 and 35B07 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1505.00528
openalex publication_date 2015/05/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let v be the velocity of Leray-Hopf solutions to the axially symmetric three-dimensional Navier-Stokes equations. It is shown that v is regular if the angular velocity vθ satisfies an integral condition which is critical under the standard scaling. This condition allows functions satisfying |vθ(x, t)| ≤ \fracCr |ln r|2+ε, rlt;1/2, where r is the distance from x to the axis, C and ε are any positive constants. Comparing with the critical a priori bound |vθ(x, t)| ≤ (C)/(r), 0lt; r ≤ 1/2,our condition is off by the log factor |ln r|2+ε at worst. This is inspired by the recent interesting paper \citeCFZ:1 where H. Chen, D. Y. Fang and T. Zhang establish, among other things, an almost critical regularity condition on the angular velocity. Previous regularity conditions are off by a factor r-1. The proof is based on the new observation that, when viewed differently, all the vortex stretching terms in the 3 dimensional axially symmetric Navier-Stokes equations are critical instead of supercritical as commonly believed.