Swirl ill-posedness for incompressible fluids in (sub)critical spaces

Abstract

For the 3D Navier-Stokes equations with sufficiently smooth initial data having zero swirl $u_{0θ}\equiv 0$, Ladyzhenskaya \cite{L1968} and Ukhovskii and Yudovich \cite{UY1968} established global regularity. It has since remained open as to whether well-posedness still holds for smooth axisymmetric initial data, with the initial swirl being smaller than an \textit{absolute} constant in a critical space. In this remark, we prove \textit{norm inflation} occurs for the Navier-Stokes scalar swirl component, with initial smallness in the critical space $\dot{H}^{\frac{1}{2}}$ or even subcritical Sobolev spaces. We also obtain norm inflation for the 3D Euler swirl \textit{vector} in the Euler subcritical space $\dot{H}^3$. Our results are achieved by introducing an anisotropic scale in Luo's supercritical norm inflation construction \cite{Luo24}, as well as performing an anisotropic bootstrap argument to get a sufficiently small swirl remainder term.