Operator-valued Laplace's integrals and stability of the open flows of inviscid incompressible fluid

We study the spectra of boundary value problems arising upon the linearization of the Euler equations of an ideal incompressible fluid near stationary solutions, describing the flows in which the fluid is entering the flow region and leaving it through some parts of the boundary. It is natural to refer to such flows as the open ones. The spectra of open flows have been explored in less details than in the case of completely impermeable boundaries or conditions of periodicity. In this paper, we discover a class of open flows the spectra of which consists of ‘zeros’ of an entire operator-valued function represented by kind of Laplace's integral. The localizing of the spectra of such flows reduces, therefore, to an operator-valued Routh–Hurwitz's problem for this integral. In a number of interesting special cases, this operator function can be expressed as a multiplier transformation of Fourier series, and then the above Routh–Hurwitz's problem turns to be scalar, and moreover, it can be solved with the help of Polias' theorem on zeros of the Laplace integrals. On this base, we proved the localization of the spectra inside the open left complex half-plane for a number of specific flows for which such proofs have not been known earlier.