On bifurcation of self-similar solutions of the stationary Navier-Stokes equations

TL;DR

This paper investigates bifurcation phenomena of self-similar solutions to the stationary Navier-Stokes equations, showing that adding swirl does not induce bifurcation and providing numerical evidence supporting this.

Contribution

It extends the understanding of self-similar solutions by analyzing bifurcations in axisymmetric discretely self-similar solutions, including the effect of swirl.

Findings

Swirl component does not enhance bifurcation.

Numerical evidence indicates no bifurcation occurs.

Landau solutions are unique self-similar solutions under certain conditions.

Abstract

Landau solutions are special solutions to the stationary incompressible Navier-Stokes equations in the three dimensional space excluding the origin. They are self-similar and axisymmetric with no swirl. In fact, any self-similar smooth solution must be a Landau solution. In the effort of extending this result to the one for the solution class with the pointwise scale-invariant bound $|u(x)|\leq C_0|x|^{-1}$ for some $C_0>0$, we consider axisymmetric discretely self-similar solutions, and investigate the existence of such solution curve emanating from some Landau solution. We prove that the inclusion of the swirl component does not enhance the bifurcation and present numerical evidence of no bifurcation.