Vol. 289, No. 1, 2017

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ISSN: 0030-8730
Criticality of the axially symmetric Navier–Stokes equations

Zhen Lei and Qi S. Zhang

Vol. 289 (2017), No. 1, 169–187
DOI: 10.2140/pjm.2017.289.169

Smooth solutions to the axisymmetric Navier–Stokes equations obey the following maximum principle:

supt0rvθ(t,) L rvθ(0,) L.

We prove that all solutions with initial data in H12 are smooth globally in time if rvθ satisfies a kind of form boundedness condition (FBC) which is invariant under the natural scaling of the Navier–Stokes equations. In particular, if rvθ satisfies

supt0|rvθ(t,r,z)| C |lnr|2, wherer δ 0 (0, 1 2),C < ,

then our FBC is satisfied. Here δ0 and C are independent of neither the profile nor the norm of the initial data. So the gap from regularity is logarithmic in nature. We also prove the global regularity of solutions if rvθ(0,)L or supt0rvθ(t,)L(rr0) is small but the smallness depends on a certain dimensionless quantity of the initial data.

axially symmetric Navier–Stokes equations, regularity condition, criticality
Mathematical Subject Classification 2010
Primary: 35K55
Received: 19 July 2016
Revised: 26 September 2016
Accepted: 12 October 2016
Published: 12 May 2017
Zhen Lei
School of Mathematical Sciences
Fudan University
Shanghai, 200433
Qi S. Zhang
Department of Mathematics
University of California, Riverside
Riverside, CA 92521
United States