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
excited by the external force
with
an integer.
This driving force gives rise to the existence of the
unidirectional basic steady flow
for any Reynolds number. It is shown in Theorem 1.1 that there exist a
number of critical Reynolds numbers such that
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
.
Moreover we prove that when
passes each critical value, a) all the
corresponding global branches do not intersect with the trivial branch
,
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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