10 МАТЕМАТИКА MATHEMATICS C i t a t i o n: Gutlyanskiĭ V.Ya., Ryazanov V.I., Sevost’yanov E.A., Yakubov E. Hydrodynamic normalization conditions in the theory of degenerate Beltrami equations. Dopov. Nac. akad. nauk Ukr. 2023. No 2. P. 10—17. https://doi.org/10.15407/dopovidi2023.02.010 © Видавець ВД «Академперіодика» НАН України, 2023. Стаття опублікована за умовами відкритого до- ступу за ліцензією CC BY-NC-ND (https://creativecommons.org/licenses/by-nc-nd/4.0/) ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. № 2: 10—17 https://doi.org/10.15407/dopovidi2023.02.010 UDC 517.5 V.Ya. Gutlyanskiĭ 1,2 , https://orcid.org/0000-0002-8691-4617 V.I. Ryazanov 1, 2 , https://orcid.org/0000-0002-4503-4939 E.A. Sevost’yanov 1,3 , https://orcid.org/0000-0001-7892-6186 E. Yakubov 4 , https://orcid.org/0000-0002-2744-1338 1 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk 2 Institute of Mathematics of the NAS of Ukraine, Kyiv 3 Zhytomyr Ivan Fanko State University, Zhytomyr 4 Holon Institute of Technology, Israel E-mail: vgutlyanskii@gmail.com, vl.ryazanov1@gmail.com, esevostyanov2009@gmail.com, eduardyakubov@gmail.com Hydrodynamic normalization conditions in the theory of degenerate Beltrami equations Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskiĭ We study the existence of normalized homeomorphic solutions for the degenerate Beltrami equation () z z f zf =μ in the whole complex plane  , assuming that its measurable coefficient ( ), | ( )| 1 z z μ μ < a. e., has compact support and the degeneration of the equation is controlled by the tangential dilatation quotient 0 (, ) T K zz μ . We show that if 0 (, ) T K zz μ has bounded or finite mean oscillation dominants, or satisfies the Lehto type integral divergence condi- tion, then the Beltrami equation admits a regular homeomorphic 1, 1 loc W solution f with the hydrodynamic normaliza- tion at infinity. We also give integral criteria of Calderon-Zygmund or Orlicz types for the existence of the normalized solutions in terms of 0 (, ) T K zz μ and the maximal dilatation () K z μ . Keywords: BMO, bounded mean oscillation, FMO, finite mean oscillation, degenerate Beltrami equations, hydro- dynamic normalization. 1. Introduction. It is well known that quasiconformal mappings and functions and their generaliza- tions, the mathematical basis for the study of which is the analytic and geometric theory of linear and quasilinear partial differential equations of elliptic type, are a powerful tool in the theory of two- dimensional subsonic compressible flows (see, e. g., [1, Ch. 2]). The Beltrami PDE, that generates quasiconformal mappings, plays here a cruсial role. Among the variety of approaches related to the study of such flows, special attention is paid to the proof of existence theorems for homeomorphisms of the whole complex plane that satisfy the degenerate Beltrami equation, i. e. when the condition of