Homogeneous solutions of stationary incompressible Navier-Stokes equations with singular rays

In 1944, Landau discovered a three-parameter family of explicit (-1)-homogeneous solutions of 3D stationary incompressible Navier-Stokes equations with precisely one singularity at the origin. These solutions, now called Landau solutions, are axisymmetric and has no swirl. In 1998 Tian and Xin proved that all (-1)-homogeneous axisymmetric solutions with one singularity are Landau solutions. In 2006 Sverak proved that all (-1)-homogeneous solutions smooth on the unit sphere are classified as Landau solutions. This talk is focused on (-1)-homogeneous solutions of 3D incompressible stationary NSE with finitely many singular rays. I will first discuss the existence and classification of such solutions that are axisymmetric with two singular rays passing through the north and south poles. We classify all such solutions with no swirl and then obtain existence of nonzero swirl solutions through perturbation methods. I will then talk about some isolated singularity behavior of homogeneous solutions to Navier-Stokes equations, and present some removable singularity result. I will also establish the asymptotic stability for some of the axisymmetric no-swirl solutions we obtained,  and talk about some anisotropic Caffarelli-Kohn-Nirenberg type inequalities we derived and applied in the study. This talk is based on joint works with Li Li and Yanyan Li.