RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 TMF: Year: Volume: Issue: Page: Find

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

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

DOI: https://doi.org/10.4213/tmf8835

Full text: PDF file (548 kB)
References: PDF file   HTML file

English version:
Theoretical and Mathematical Physics, 2015, 185:2, 1582–1598

Bibliographic databases:

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
\Bibitem{EfrSak15} \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 \mathnet{http://mi.mathnet.ru/tmf8835} \crossref{https://doi.org/10.4213/tmf8835} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3438619} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2015TMP...185.1582E} \elib{https://elibrary.ru/item.asp?id=24850721} \transl \jour Theoret. and Math. Phys. \yr 2015 \vol 185 \issue 2 \pages 1582--1598 \crossref{https://doi.org/10.1007/s11232-015-0366-z} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000366113400002} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84949238996} 

• http://mi.mathnet.ru/eng/tmf8835
• https://doi.org/10.4213/tmf8835
• http://mi.mathnet.ru/eng/tmf/v185/i2/p252

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178
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
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
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
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
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
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
•  Number of views: This page: 785 Full text: 107 References: 109 First page: 177