Event Detail

Event Type: 
Analysis Seminar
Date/Time: 
Monday, February 28, 2022 - 12:00 to 12:50
Location: 
BEXL 328

Speaker Info

Institution: 
Oklahoma State University
Abstract: 

In 1944, Landau discovered a three parameter family of explicit (-1)-homogeneous solutions of the 3D stationary incompressible Navier-Stokes equations (NSE) with precisely one singularity at the origin, which are axisymmetric with no swirl. These solutions are now called Landau solutions. 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 that are smooth on the unit sphere are Landau solutions. This talk focuses on (-1)-homogeneous solutions of the 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 describe the asymptotic expansions of such solutions near a singular ray. I will also discuss a recent result on the asymptotic stability for some of the solutions we obtained. This is a joint work with Li Li and Yanyan Li.