Operator-Valued Laplace's Integrals and Stability of the Open Flows of Inviscid Incompressible Fluid

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.23671/VNC.2019.3.36460 Operator-Valued Laplace's Integrals and Stability of the Open Flows of Inviscid Incompressible Fluid Ilin, K. I. , Morgulis, A. B. , Chernish, A. S. Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 3. Abstract: 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. Keywords: Euler equations, inviscid incompressible fluid, stability, spectra, entire functions, Routh-Gurwitz's problem. Language: Russian Download the full text For citation: Ilin, K. I., Morgulis, A. B. and Chernish, A. S. Operator-Valued Laplace's Integrals and Stability of the Open Flows of Inviscid Incompressible Fluid, Vladikavkaz Math. J., 2019, vol. 21, no. 2, pp. 31-49 (in Russian). DOI 10.23671/VNC.2019.3.36460 + References 1. Iooss, G. and Joseph, D. D. Elementary Stability and Bifurcation Theory, Springer Science & Business Media, 2012. 2. Arnold, V. I. et al. Dynamical Systems V: Bifurcation Theory and Catastrophe Theory, Springer Science & Business Media, 2013, vol. 5. 3. Yudovich, V. I. The Linearization Method in Hydrodynamical Stability Theory, Transl. of Math. Monogr., vol. 74, American Math. Soc., Providence, RI, 1989, iv+170 pp. 4. Yudovich, V. I. 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