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

We study the removable singularity problem for ( − 1 ) (-1) -homogeneous solutions of the three-dimensional incompressible stationary Navier-Stokes equations with singular rays. We prove that any local ( − 1 ) (-1) -homogeneous solution u u near a potential singular ray from the origin, which passes through a point P P on the unit sphere S 2 S^2 , can be smoothly extended across P P on S 2 S^2 , provided that u = o ( ln ⁡ d i s t ( x , P ) ) u=o(ln {dist} (x, P)) on S 2 S^2 . The result is optimal in the sense that for any α > 0 α >0 , there exists a local ( − 1 ) (-1) -homogeneous solution near P P on S 2 S^2 , such that lim x ∈ S 2 , x → P | u ( x ) | / ln ⁡ d i s t ( x , P ) = − α lim x∈ S^2, x→ P|u(x)|/ln {dist}(x, P)=-α . Furthermore, we discuss the behavior of isolated singularities of ( − 1 ) (-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 S 2 S^2 .