Electron. J. Diff. Eqns., Vol. 2000(2000), No. 58, pp. 1-32.

Steady-state bifurcations of the three-dimensional Kolmogorov problem

Zhi-Min Chen & Shouhong Wang

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 in this article.

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.

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Zhi-Min Chen
Department of Ship Science, Southampton University,
Southampton SO17 1BJ, UK
and: Department of Mathematics, Tianjin University, China
email: zhimin@ship.soton.ac.uk

Shouhong Wang
Department of Mathematics, Indiana University,
Bloomington, IN 47405, USA
email: showang@indiana.edu

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