Removable singularity of (-1)-homogeneous solutions of stationary Navier-Stokes equations

TL;DR

This paper investigates the conditions under which singularities in (-1)-homogeneous solutions of stationary Navier-Stokes equations can be smoothly removed, establishing optimal criteria and exploring solutions with multiple singular points.

Contribution

It proves that certain local solutions near singular rays can be extended smoothly under specific growth conditions and constructs solutions with multiple singularities on the sphere.

Findings

Solutions can be extended smoothly if they grow slower than logarithm near singular points.

Existence of solutions with any finite number of singular points on the sphere is established.

Examples demonstrate various behaviors of singularities in these solutions.

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 $\alpha>0$, there exists a local $(-1)$-homogeneous solution near $P$ on $\mathbb{S}^2$, such that $\lim_{x\in \mathbb{S}^2, x\to P}|u(x)|/\ln |x'|=-\alpha$. 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 $\mathbb{S}^2$.