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

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 ${\lim }_{x\in {\mathbb{S}}^2,x\to P}|u(x)|/\ln |x^{\prime }|=-\alpha$. 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 ${\mathbb{S}}^2$.

Minor revisions. To appear in the Transactions of the American Mathematical Society