Laplace’s Integrals and Stability of the Open Flows of Inviscid Incompressible Fluid

In this article, we study the spectra of the boundary value problems which arise from linearizing the Euler equations of incompressible hydrodynamics near a stationary solution describing a steady flow through a given domain, in the case when the fluid enters the domain and leave it through some parts of the boundary. It’s natural to call such flows as open. The spectra of the open flows are not widely studied compared to the classical case of the fully impermeable boundaries. Moreover, the methods widely used for the latter fail to cover the former. We propose a new reduction of finding the eigenmodes of an open flow to finding ‘zeroes’ of an entire operator-valued function which is a kind of Laplace’s integral. Here, by zeroes, we mean the values of the complex variable which deliver degenerations to the mentioned integral. Correspondingly, studying the flow stability reduces itself to Routh–Hurwitz problem for this integral. For several particular flows, this problem is solved explicitly with the use of the Polia theorem on the zeroes of the Laplace integral. As a result, we prove the stability of spectra for several concrete flows for which were unknown with such proofs before.