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This article is cited in 7 scientific papers (total in 7 papers)
Notion of blowup of the solution set of differential equations and averaging of random semigroups
L. S. Efremovaa, V. Zh. Sakbaevb a Lobachevsky State University of Nizhny Novgorod, Nizhny
Novgorod, Russia
b Moscow Institute of Physics and Technology (State
University), Dolgoprudny, Moscow Oblast, Russia
Abstract:
We propose a unique approach to studying the violation of the well-posedness of initial boundary-value problems for differential equations. The blowup of the set of solutions of a problem for a differential equation is defined as a discontinuity of a multivalued map associating an initial boundary-value problem with the set of solutions of this problem. We show that such a definition not only describes effects of the solution destruction or its nonuniqueness but also permits prescribing a procedure for extending the solution through the singularity origination instant by using an appropriate random process. Considering the initial boundary-value problems whose solution sets admit singularities of the blowup type and a neighborhood of these problems in the space of problems permits associating the initial problem with the set of limit points of a sequence of solutions of the approximating problems. Endowing the space of problems with the structure of a space with measure, we obtain a random semigroup generated by the initial problem. We study the properties of the mathematical expectations (means) of a random semigroup and their equivalence in the sense of Chernoff to semigroups with averaged generators.
Keywords:
boundary-value problem, blowup, dynamical system, $\Omega$-explosion, semigroup, random dynamical system, Chernoff's theorem, averaging
DOI:
https://doi.org/10.4213/tmf8835
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English version:
Theoretical and Mathematical Physics, 2015, 185:2, 1582–1598
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Received: 05.12.2014 Revised: 13.04.2015
Citation:
L. S. Efremova, V. Zh. Sakbaev, “Notion of blowup of the solution set of differential equations and averaging of random semigroups”, TMF, 185:2 (2015), 252–271; Theoret. and Math. Phys., 185:2 (2015), 1582–1598
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http://mi.mathnet.ru/eng/tmf8835https://doi.org/10.4213/tmf8835 http://mi.mathnet.ru/eng/tmf/v185/i2/p252
Citing articles on Google Scholar:
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This publication is cited in the following articles:
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V. Zh. Sakbaev, “On the law of large numbers for compositions of independent random semigroups”, Russian Math. (Iz. VUZ), 60:10 (2016), 72–76
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L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178
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V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space”, Theoret. and Math. Phys., 191:3 (2017), 886–909
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V. Zh. Sakbaev, “Averaging of random flows of linear and nonlinear maps”, European Conference - Workshop Nonlinear Maps and Applications, Journal of Physics Conference Series, 990, IOP Publishing Ltd, 2018, UNSP 012012
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V. Zh. Sakbaev, “Polugruppy preobrazovanii prostranstva funktsii, kvadratichno integriruemykh po translyatsionno invariantnoi mere na banakhovom prostranstve”, Kvantovaya veroyatnost, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 151, VINITI RAN, M., 2018, 73–90
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Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups”, Proc. Steklov Inst. Math., 306 (2019), 196–211
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V. M. Busovikov, V. Zh. Sakbaev, “Sobolev spaces of functions on a Hilbert space endowed with a translation-invariant measure and approximations of semigroups”, Izv. Math., 84:4 (2020), 694–721
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