On the origination of auto-oscillations in flows
In studying the origination of cycles in nonlinear differential equations of the form \(\omega(dv/dt)+Av= K(v,\delta)\) in Banach spaces one assumed that the operator \(A\) has a bounded inverse. At the same time, in certain problems (parabolic equations in unbounded domains, the flow problem for the Navier-Stokes system) the point \(\lambda=0\) is at least a point of the continuous spectrum of the operator \(A\). We establish that the methods of the article [\textit{V. I. Yudovich}, J. Appl. Math. Mech. 35, 587-603 (1971); translation from Priklad. Math. Mekh. 35, 638-655 (1971; Zbl 0247.76044)] can be transferred to this critical case as well as under a certain subordination of nonlinearity to the operator \(A\). Furthermore, we apply the proposed scheme to the Navier-Stokes system in an exterior domain. In particular, the scheme covers the flow problem which was investigated by other methods in [\textit{K. I. Babenko}, Sov. Math. Dokl. 25, 211-216 (1982); translation from Dokl. Akad. Nauk SSSR 262, 1293-1298 (1982; Zbl 0504.76049)].
Navier-Stokes system in an exterior domain
scientific article; zbMATH DE number 917142