Communications in Mathematical Sciences

Volume 19 (2021)

Number 6

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

Pages: 1703 – 1733



Hyunju Kwon (School of Mathematics, Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Tai-Peng Tsai (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)


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 an effort of extending this result to the solution class with the pointwise scale-invariant bound given by an arbitrarily large constant divided by the distance to the origin, 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.


incompressible, stationary Navier–Stokes equations, discretely self-similar, axisymmetric, swirl, Landau solutions, bifurcation

2010 Mathematics Subject Classification

35B07, 35B32, 35Q30, 76D05

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Dedicated to Vladimir Šverák on the occasion of his 60th birthday.

The research of both Kwon and Tsai was partially supported by NSERC grant RGPIN-2018-04137 (Canada).

Received 6 November 2020

Accepted 1 March 2021

Published 2 August 2021