Classification of local $(-1)$-homogeneous axisymmetric solutions of the $3$D stationary Navier-Stokes equations with logarithmic singular-ray behavior

Abstract

We study the existence and classification of local $(-1)$-homogeneous axisymmetric solutions $(u, p)$ of the three-dimensional incompressible stationary Navier-Stokes equations near a singular ray. We focus on such solutions with Type II behavior, which satisfy $0<\limsup_{x\in\bS^2, x\to P}|u|/|\ln \text{dist}(x, P)|<\infty$, where $P$ is the south or north pole. Apart from the Landau solutions, these are the least singular nontrivial $(-1)$-homogeneous axisymmetric solutions. We construct a four-parameter family of local $(-1)$-homogeneous axisymmetric solutions with at most logarithmic growth near the singular ray and represent them as convergent series whose coefficients are determined recursively. Conversely, we prove that every local $(-1)$-homogeneous axisymmetric solution satisfying $|u|=o(\text{dist} (x, P)^{-1})$ as $x\to P$ on $\bS^2$ belongs to this family. In particular, this family contains Type II solutions with nonzero swirl, in contrast to the global $(-1)$-homogeneous axisymmetric Type II solutions in $\R^3\setminus\{x'=0\}$, which have no swirl. We also establish derivative estimates for the constructed solutions and identify the singular force generated by these solutions across the singular ray in the sense of distributions.