Abstract: The singularity behavior and rigidity of solutions are important problems in the study of PDEs. One crucial problem is to understand under what conditions a potential singularity is removable. In this talk, we discuss such problems for steady states of the 3D incompressible Navier-Stokes equations. I will first introduce the problem through some classical results for Laplace equations, and describe some known results for steady states of Navier-Stokes equations with an isolated singularity. It known that an isolated singularity at the origin is removable if the solution satisfies u=o(1/|x|), and the optimality is given by the well-known Landau solutions. I will then talk about the singularity behavior and removable singularity problem for (-1)-homogeneous solutions of Navier-Stokes equations with singular rays. I will discuss the existence of such solutions that are axisymmetric with vertical singular rays, and describe the asymptotic expansions of such local solutions. I will then present an optimal removable singularity result for general (-1)-homogeneous solutions with singular rays without the axisymmetry assumption.
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