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

Abstract

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

$${sup}_{t\ge 0}\parallel r{v}^{\theta }\left(t,\cdot \right){\parallel }_{L^{\infty }}\le \parallel r{v}^{\theta }\left(0,\cdot \right){\parallel }_{L^{\infty }}.$$

We prove that all solutions with initial data in $H^{1∕2}$ are smooth globally in time if $r{v}^{\theta }$ satisfies a kind of form boundedness condition (FBC) which is invariant under the natural scaling of the Navier–Stokes equations. In particular, if $r{v}^{\theta }$ satisfies

$${sup}_{t\ge 0}\left|r{v}^{\theta }\left(t,r,z\right)\right|\le C_{\ast }|lnr{|}^{-2},\mathrm{\text{ where}}r\le {\delta }_0\in \left(0,\frac{1}{2}\right),C_{\ast }<\infty ,$$

then our FBC is satisfied. Here ${\delta }_0$ and $C_{\ast }$ 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 $\parallel r{v}^{\theta }\left(0,\cdot \right){\parallel }_{L^{\infty }}$ or $\underset{t\ge 0}{sup}\parallel r{v}^{\theta }\left(t,\cdot \right){\parallel }_{L^{\infty }\left(r\le r_0\right)}$ is small but the smallness depends on a certain dimensionless quantity of the initial data.

Keywords

axially symmetric Navier–Stokes equations, regularity condition, criticality

Mathematical Subject Classification 2010

Primary: 35K55

Milestones

Received: 19 July 2016

Revised: 26 September 2016

Accepted: 12 October 2016

Published: 12 May 2017