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TMF, 2015, Volume 185, Number 2, Pages 252–271 (Mi tmf8835)  

This article is cited in 8 scientific papers (total in 8 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

Funding Agency Grant Number
Russian Science Foundation 14-11-00687
Ministry of Education and Science of the Russian Federation 10-14


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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

Citation in format AMSBIB
\by L.~S.~Efremova, V.~Zh.~Sakbaev
\paper Notion of blowup of the~solution set of differential equations and averaging of random semigroups
\jour TMF
\yr 2015
\vol 185
\issue 2
\pages 252--271
\jour Theoret. and Math. Phys.
\yr 2015
\vol 185
\issue 2
\pages 1582--1598

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    This publication is cited in the following articles:
    1. V. Zh. Sakbaev, “On the law of large numbers for compositions of independent random semigroups”, Russian Math. (Iz. VUZ), 60:10 (2016), 72–76  mathnet  crossref  mathscinet  isi  elib  elib
    2. L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space”, Theoret. and Math. Phys., 191:3 (2017), 886–909  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. 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  crossref  isi  scopus
    5. 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  mathnet  mathscinet
    6. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    7. Efremova L.S., Grekhneva A.D., Sakbaev V.Zh., “Phase Flows Generated By Cauchy Problem For Nonlinear Schrodinger Equation and Dynamical Mappings of Quantum States”, Lobachevskii J. Math., 40:10, SI (2019), 1455–1469  crossref  isi
    8. 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  mathnet  crossref  crossref  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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