Abstract: 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. In this talk, I will first talk about how Landau solutions are derived and the proofs of the above mentioned results. I will then discuss (-1)-homogeneous solutions of 3D incompressible stationary NSE with finitely many singular rays. I will 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 describe the asymptotic expansions of such solutions near a singular ray and the asymptotic stability of them.
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