Long wavelength asymptotics of self-oscillations of viscous incompressible fluid. (English) Zbl 1479.35631

Kusraev, Anatoly G. (ed.) et al., Operator theory and differential equations. Selected papers based on the presentations at the 15th conference on order analysis and related problems of mathematical modeling, Vladikavkaz, Russia, July 15–20, 2019. Cham: Birkhäuser. Trends Math., 185-203 (2021).
Summary: We obtain the long wavelength asymptotics of a secondary regime formed at stability loss of a stationary spatially periodic shear flow with non-zero average as one of the spatial periods tends to infinity (the wave number vanishes). It is known that if certain non-degeneracy conditions are satisfied, then from the basic solution a self-oscillatory regime branches. Recurrence formulas for kth term of the asymptotics of this secondary solution are obtained. To study the bifurcations of basic flow we obtain the scheme of Lyapunov-Schmidt method proposed by V. I. Yudovich [J. Appl. Math. Mech. 35, 587–603 (1971; Zbl 0247.76044); translation from Prikl. Mat. Mekh. 35, 638–655 (1971); ibid. 36, 424–432 (1972; Zbl 0257.34040); translation from Prikl. Mat. Mekh. 36, 450–459 (1972)]. At each step of the Lyapunov-Schmidt method series expansion in the small parameter \(\alpha\) is applied.
For the entire collection see [Zbl 1470.47003].


35Q30 Navier-Stokes equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic solutions to PDEs
35B32 Bifurcations in context of PDEs
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