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. The Onset of Auto-Oscillations in a Fluid, Journal of Applied Mathematics and Mechanics, 1971, vol. 35, no. 4, pp. 587-603. 5. Haragus, M. and Iooss, G. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, London, Springer, 2011. DOI: 10.1007/978-0-85729-112-7. 6. Dikiy, L. A. Hydrodynamic Stability and Atmosphere Dynamics, Gidrometeoizdat, Leningrad, 1976 (in Russian). 7. Shvydkoy, R. and Latushkin, Y. Operator Algebras and the Fredholm Spectrum of Advective Equations of Linear Hydrodynamics, Journal of Functional Analysis, 2009, vol. 257, no. 10, pp. 3309-3328. 8. Arnold, V. Sur la Geometrie Differentielle des Groupes de Lie de Dimension Infinie et ses Applications а l'Hydrodynamique des Fluides Parfaits, Annales de l'institut Fourier, 1966, vol. 16, no. 1, pp. 319-361. DOI: 10.5802/aif.233. 9. Morgulis, A. B. and Yudovich, V. I. Arnold's Method for Asymptotic Stability of Steady Inviscid Incompressible Flow Through a Fixed Domain with Permeable Boundary, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2002, vol. 12, no. 2, pp. 356-371. DOI: 10.1063/1.1480443. 10. Morgulis, A. B. Variational Principles and Stability of the Inviscid Open Flows, Sibirskie elektronnye matematicheskie izvestiya [Siberian Electronic Mathematical Reports], 2017, vol. 14, pp. 218-251 (in Russian). 11. Ostrovskii, I. V. M. G. Krein's Investigations in the Theory of Entire and Meromorphic Functions and their Further Development, Ukrainian Mathematical Journal, 1994, vol. 46, no. 1-2, pp. 87-100. DOI: 10.1007/BF01057003. 12. Sedletskii, A. M. On the Zeros of Laplace Transforms, Mathematical Notes, 2004, vol. 76, no. 5-6, pp. 824-833. DOI: 10.1023/B:MATN.0000049682.65990.e7. 13. Chemin, J. Y. Fluides Parfaits Incompressibles, Paris, 1995. Ser. Asterisque, vol. 230. 14. Yudovich, V. I. A Two-Dimensional Non-Stationary Problem on the Flow of an Ideal Incompressible Fluid Through a Given Region, Matematicheskii sbornik (N.S.) [Sbornik: Mathematics], 1964, vol. 64, no. 106, pp. 562-588 (in Russian). 15. Alekseev, G. V. On Solvability of the Nonhomogeneous Boundary Value Problem for Two-Dimensional Nonsteady Equations of Ideal Fluid Dynamics, Dinamika Sploshnoy Sredy, 1976, vol. 24, pp. 15-35 (in Russian). 16. Bardos, C. Existence et Unicite de la Solution de l'Equation d'Euler en Dimension Deux, Journal of Mathematical Analysis and Applications, 1972, vol. 40, no. 3, pp. 769-790. 17. Kazhikhov, A. V. Note on the Formulation of the Problem of Flow Through a Bounded Region Using Equations of Perfect Fluid, Journal of Applied Mathematics and Mechanics, 1980, vol. 44, no. 5, pp. 672-674. 18. Antontsev, S. N., Kazhikhov, A. V. and Monakhov, V. N. Boundary-Value Problems in the Mechanics of Nonuniform Fluids, Amsterdam, North-Holland, 1990, Studies in Mathematics and its Applications, vol. 22. 19. Temam, R. and Wang, X. Boundary Layers Associated with Incompressible Navier-Stokes Equations: the Noncharacteristic Boundary Case, Journal of Differential Equations, 2002, vol. 179, no. 2, pp. 647-686. DOI: 10.1006/jdeq.2001.4038. 20. Ilin, K. Viscous Boundary Layers in Flows Through a Domain with Permeable Boundary, European Journal of Mechanics-B/Fluids, 2008, vol. 27, no. 5, pp. 514-538. DOI: 10.1016/j.euromechflu.2007.10.003. 21. Beavers, G. S. and Joseph, D. D. Boundary Conditions at a Naturally Permeable Wall, Journal of Fluid Mechanics, 1967, vol. 30, no 1, pp. 197-207. DOI: 10.1017/S0022112067001375. 22. Chetaev N. G. The Stability of Motion, Pergamon Press, 1961. ← Contents of issue


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