Regularity criterion to the axially symmetric Navier–Stokes equations

Smooth solutions to the axially symmetric Navier–Stokes equations obey the following maximum principle: ‖ruθ(r,z,t)‖L∞≤‖ruθ(r,z,0)‖L∞‖ruθ(r,z,t)‖L∞≤‖ruθ(r,z,0)‖L∞. We first prove the global regularity of solutions if ‖ruθ(r,z,0)‖L∞‖ruθ(r,z,0)‖L∞ or ‖ruθ(r,z,t)‖L∞(r≤r0)‖ruθ(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 [10]. As a corollary, we also prove the global regularity under the assumption that |ruθ(r,z,t)|≤|ln⁡r|−3/2|ruθ(r,z,t)|≤|ln⁡r|−3/2, ∀0