MATHPH In the literature: open

Regularity or blowup of axisymmetric Navier-Stokes flows with swirl

In plain words

Flows that are symmetric around an axis but also spin around it are the leading candidates for viscous blowup. Without spin they are known to stay smooth; with spin the question is open even in this restricted class.

Precise statement

3D Navier-Stokes on $R^{3}$ with $f = 0$ and smooth, finite-energy, axisymmetric initial data with nonzero swirl $u_{\theta}$. Prove global regularity, or give smooth axisymmetric data whose solution has limsup_{t -> T} ||u(t)||_inf = infinity. Answer: yes or no with proof; this is a special case of mathph.fluid-singularities.ns-unforced.

What would settle it

A global regularity proof for all smooth axisymmetric data, or a verified (e.g. computer-assisted) blowup construction in this class without forcing.

Status in the literature

Hou (arXiv 2107.06509, Foundations of Computational Mathematics) reported numerical evidence of potentially singular axisymmetric flow with maximum vorticity growing by a factor of about $10^{7}$; regularity is proven for zero swirl (Ukhovskii and Yudovich; Ladyzhenskaya, 1968).