Revision 2

Accepted · core change · record

Changes against revision 1:

Title: Regularity or blowup of axisymmetric Navier-Stokes flows with swirl
Plain: 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 · changed · core: Revision 1: 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.; Revision 2: 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 $\operatorname{limsup}_{t \to T} \mid\mid u(t)\mid\mid_{\infty} = \infty$. Answer: yes or no with proof; this is a special case of mathph.fluid-singularities.ns-unforced.
Settled by: A global regularity proof for all smooth axisymmetric data, or a verified (e.g. computer-assisted) blowup construction in this class without forcing.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: open
Status note: 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).
Posed since: (empty)

Decisions

Accept · 2026-10-15 · record