Steady-state bifurcations of the three-dimensional Kolmogorov problem

Abstract

This paper studies the spatially periodic incompressible fluid motion in $mathbb R^3$ excited by the external force $k^2(sin kz, 0,0)$ with $kgeq 2$ an integer. This driving force gives rise to the existence of the unidirectional basic steady flow $u_0=(sin kz,0, 0)$ for any Reynolds number. It is shown in Theorem 1.1 that there exist a number of critical Reynolds numbers such that $u_0$ bifurcates into either 4 or 8 or 16 different steady states, when the Reynolds number increases across each of such numbers.

Keywords

• 3D Navier-Stokes equations
• Kolmogorov flow
• multiple steady states
• supercritical pitchfork bifurcation
• continuous fractions.