Removable singularity of (-1)-homogeneous solutions of stationary Navier-Stokes equations
Li Li, YanYan Li, Xukai Yan
TL;DR
The paper advances the understanding of removable singularities for $(-1)$-homogeneous, stationary Navier–Stokes flows in $\mathbb{R}^3$ by proving that a local solution extending across a potential singular ray exists under the sharp condition $|u|=o(\ln \text{dist})$ on the associated sphere, and that this rate is optimal through explicit logarithmic-growth examples. The authors reduce the NS system to equations on $\mathbb{S}^2$ and employ a Stokes-regularity bootstrap from weak formulations to obtain full smoothness away from the origin, with precise derivative decay. They provide a comprehensive discussion of isolated singularities, including a detailed taxonomy of singularity types for axisymmetric configurations, and survey explicit families of solutions (Landau, Serrin-type, Liouville-based, and Euler-compatible) that illuminate the possible local behaviors and multiplicities of singularities on $\mathbb{S}^2$. Collectively, the results connect classical classifications with new removability criteria, and establish existence results for $(-1)$-homogeneous NS solutions with any finite number of prescribed singular points on $\mathbb{S}^2$, enriching the landscape of homogeneous NS solutions and their singular structures.
Abstract
We study the removable singularity problem for $(-1)$-homogeneous solutions of the three-dimensional incompressible stationary Navier-Stokes equations with singular rays. We prove that any local $(-1)$-homogeneous solution $u$ near a potential singular ray from the origin, which passes through a point $P$ on the unit sphere $\mathbb{S}^2$, can be smoothly extended across $P$ on $\mathbb{S}^2$, provided that $u=o(\ln \text{dist} (x, P))$ on $\mathbb{S}^2$. The result is optimal in the sense that for any $α>0$, there exists a local $(-1)$-homogeneous solution near $P$ on $\mathbb{S}^2$, such that $\lim_{x\in \mathbb{S}^2, x\to P}|u(x)|/\ln |x'|=-α$. Furthermore, we discuss the behavior of isolated singularities of $(-1)$-homogeneous solutions and provide examples from the literature that exhibit varying behaviors. We also present an existence result of solutions with any finite number of singular points located anywhere on $\mathbb{S}^2$.