×

Steady-state bifurcations of the three-dimensional Kolmogorov problem. (English) Zbl 0947.35105

Summary: 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 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.

MSC:

35Q30 Navier-Stokes equations
35B32 Bifurcations in context of PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
58J55 Bifurcation theory for PDEs on manifolds
PDFBibTeX XMLCite
Full Text: EuDML EMIS