Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 58, pp. 1-32.
Title: Steady-state bifurcations of the three-dimensional Kolmogorov problem
Authors: Zhi-Min Chen (Southampton University, UK)
Shouhong Wang (Indiana University, Bloomington, IN, USA)
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
$k\geq 2$ 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.
Thanks to the Rabinowitz global bifurcation theorem, all of the bifurcation
solutions are extended to global branches for $\lambda \in (0, \infty)$.
Moreover we prove that when $\lambda$ passes each critical value, a) all the
corresponding global branches do not intersect with the trivial branch
$(u_0,\lambda)$, and b) some of them never intersect each other;
see Theorem 1.2.
Submitted February 12, 1999. Revised May 26, 2000. Published August 30, 2000.
Math Subject Classifications: 35Q30, 76D05, 58J55, 35B32.
Key Words: 3D Navier-Stokes equations; Kolmogorov flow; multiple steady states;
supercritical pitchfork bifurcation; continuous fractions.