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## Archive of Issues

Russia Nizhni Novgorod
Year
2020
Volume
30
Issue
1
Pages
92-111
 Section Mathematics Title On totally global solvability of controlled second kind operator equation Author(-s) Chernov A.V.ab Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb Abstract We consider the nonlinear evolutionary operator equation of the second kind as follows $\varphi=\mathcal{F}\bigl[f[u]\varphi\bigr]$, $\varphi\in W[0;T]\subset L_q\bigl([0;T];X\bigr)$, with Volterra type operators $\mathcal{F}\colon L_p\bigl([0;\tau];Y\bigr)\to W[0;T]$, $f[u]$: $W[0;T]\to L_p\bigl([0;T];Y\bigr)$ of the general form, a control $u\in\mathcal{D}$ and arbitrary Banach spaces $X$, $Y$. For this equation we prove theorems on solution uniqueness and sufficient conditions for totally (with respect to set $\mathcal{D}$) global solvability. Under natural hypotheses associated with pointwise in $t\in[0;T]$ estimates the conclusion on univalent totally global solvability is made provided global solvability for a comparison system which is some system of functional integral equations (it could be replaced by a system of equations of analogous type, and in some cases, of ordinary differential equations) with respect to unknown functions $[0;T]\to\mathbb{R}$. As an example we establish sufficient conditions of univalent totally global solvability for a controlled nonlinear nonstationary Navier-Stokes system. Keywords nonlinear evolutionary operator equation of the second kind, totally global solvability, Navier-Stokes system UDC 517.957, 517.988, 517.977.56 MSC 47J05, 47J35, 47N10 DOI 10.35634/vm200107 Received 23 August 2019 Language Russian Citation Chernov A.V. On totally global solvability of controlled second kind operator equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 92-111. References Chernov A.V. A majorant criterion for the total preservation of global solvability of controlled functional operator equation, Russian Mathematics, 2011, vol. 55, no. 3, pp. 85-95. https://doi.org/10.3103/S1066369X11030108 Kalantarov V.K., Ladyzhenskaya O.A. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, Journal of Soviet Mathematics, 1978, vol. 10, issue 1, pp. 53-70. https://doi.org/10.1007/BF01109723 Sumin V.I. The features of gradient methods for distributed optimal control problems, USSR Computational Mathematics and Mathematical Physics, 1990, vol. 30, no. 1, pp. 1-15. https://doi.org/10.1016/0041-5553(90)90002-A Sumin V.I. Funktsional'nye vol'terrovy uravneniya v teorii optimal'nogo upravleniya raspredelennymi sistemami. Chast' I. Vol'terrovy uravneniya i upravlyaemye nachal'no-kraevye zadachi (Functional Volterra equations in the theory of optimal control of distributed systems. Part I. Volterra equations and controlled initial boundary value problems), Nizhny Novgorod: Nizhny Novgorod State University, 1992. Korpusov M.O., Sveshnikov A.G. Blow-up of solutions to strongly nonlinear equations of the pseudoparabolic type, Sovremennaya matematika i ee prilozheniya, 2006, vol. 40, Differential Equations, pp. 3-138 (in Russian). Chernov A.V. On total preservation of solvability of controlled Hammerstein-type equation with non-isotone and non-majorizable operator, Russian Mathematics, 2017, vol. 61, no. 6, pp. 72-81. https://doi.org/10.3103/S1066369X1706010X Chernov A.V. Preservation of the solvability of a semilinear global electric circuit equation, Computational Mathematics and Mathematical Physics, 2018, vol. 58, no. 12, pp. 2018-2030. https://doi.org/10.1134/S0965542518120096 Sumin V.I. The problem of global solutions existence-stability for controllable boundary-value problems and Volterra functional equations, Vestnik Nizhegorodskogo Universiteta imeni N.I. Lobachevskogo. Matematika, 2003, issue 1, pp. 91-107 (in Russian). http://www.vestnik.unn.ru/ru/nomera?anum=1411 Sumin V.I., Chernov A.V. Volterra functional-operator equations in the theory of optimization of distributed systems, Systems Dinamics and Control Processes (SDCP-2014): Proceedings of Int. Conf. Dedicated to the 90th Anniversary of the birth of Acad. N.N. Krasovskii, Ekaterinburg: “UMTS UPI” Publ., 2015, pp. 293-300 (in Russian). Sumin V.I. Controlled Volterra functional equations and the contraction mapping principle, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 1, pp. 262-278 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-1-262-278 Chernov A.V. A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation, Russian Mathematics, 2012, vol. 56, no. 3, pp. 55-65. https://doi.org/10.3103/S1066369X12030085 Chernov A.V. On the totally global solvability of a controlled Hammerstein type equation with a varied linear operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 230-243 (in Russian) https://doi.org/10.20537/vm150207 Chernov A.V. Majorant sign of the first order for totally global solvability of a controlled functional operator equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 531-548 (in Russian). https://doi.org/10.20537/vm180407 Chernov A.V. The total preservation of unique global solvability of the first kind operator equation with additional controlled nonlinearity, Russian Mathematics, 2018, vol. 62, no. 11, pp. 53-66. https://doi.org/10.3103/S1066369X18110063 Solonnikov V.A. Estimates for solutions of nonstationary Navier-Stokes equations, Zapiski Nauchnykh Seminarov POMI, 1973, vol. 38, pp. 153-231 (in Russian). http://mi.mathnet.ru/eng/znsl2649 Tikhonov A.N. On functional equations of Volterra type and their applications to some problems of mathematical physics, Bull. Mos. Univ. Sec. A., 1938, vol. 1, issue 8, pp. 1-25 (in Russian). https://zbmath.org/?q=an:0021.23404 Seregin G.A. Necessary conditions of potential blow-up for Navier-Stokes equations, Journal of Mathematical Sciences, 2011, vol. 178, no. 3, pp. 345-352. https://doi.org/10.1007/s10958-011-0551-z Leray J. Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Mathematica, 1934, vol. 63, no. 1, pp. 193-248. https://projecteuclid.org/euclid.acta/1485888078 Prodi G. Un teorema di unicita per le equazioni di Navier-Stokes, Annali di Matematica Pura ed Applicata, 1959, vol. 48, no. 4, pp. 173-182. https://doi.org/10.1007/BF02410664 Kozono H., Sohr H. Remarks on uniqueness of weak solutions to the Navier-Stokes equations, Analysis, 1996, vol. 16, no. 3, pp. 255-271. https://doi.org/10.1524/anly.1996.16.3.255 Ladyzhenskaya O.A. Sixth problem of the millenium: Navier-Stokes equations, existence and smoothness, Russian Mathematical Surveys, 2003, vol. 58, no. 2, pp. 251-286. https://doi.org/10.1070/RM2003v058n02ABEH000610 Saito H. Global solvability of the Navier-Stokes equations with a free surface in the maximal regularity $L_p$-$L_q$ class, Journal of Differential Equations, 2018, vol. 264, no. 3, pp. 1475-1520. https://doi.org/10.1016/j.jde.2017.09.045 Seregin G.A., Shilkin T.N. Liouville-type theorems for the Navier-Stokes equations, Russian Mathematical Surveys, 2018, vol. 73, no. 4, pp. 661-724. https://doi.org/10.1070/RM9822 Fursikov A.V. Optimal control of distributed systems. Theory and applications, Providence, RI: AMS, 2000. https://zbmath.org/?q=an:1027.93500. Full text