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.
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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Linking options:
https://www.mathnet.ru/eng/tmf8835
https://doi.org/10.4213/tmf8835
https://www.mathnet.ru/eng/tmf/v185/i2/p252
This publication is cited in the following 15 articles:
V. Zh. Sakbaev, “Application of Banach limits to invariant measures of infinite-dimensional Hamiltonian flows”, Ann. Funct. Anal., 15:2 (2024)
René Lozi, Vladimir Belykh, Jim Michael Cushing, Lyudmila Efremova, Saber Elaydi, Laura Gardini, Michał Misiurewicz, Eckehard Schöll, Galina Strelkova, “The paths of nine mathematicians to the realm of dynamical systems”, Journal of Difference Equations and Applications, 30:1 (2024), 1
V. A. Glazatov, V. Zh. Sakbaev, “Measures on Hilbert space invariant with respect to Hamiltonian flows”, Ufa Math. J., 14:2 (2022), 3–21
V. Zh. Sakbaev, A. D. Shiryaeva, “Blow-Up of States in the Dynamics Given by the Schrödinger Equation with a Power-Law Nonlinearity in the Potential”, Diff Equat, 58:4 (2022), 497
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
V. Zh. Sakbaev, N. V. Tsoi, “Analogue of Chernoff Theorem For Cylindrical Pseudomeasures”, Lobachevskii J. Math., 41:12, SI (2020), 2369–2382
A. D. Grekhneva, V. Zh. Sakbaev, “Dynamics of a set of quantum states generated by a nonlinear Liouville–von Neumann equation”, Comput. Math. Math. Phys., 60:8 (2020), 1337–1347
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
L. S. Efremova, A. D. Grekhneva, V. Zh. Sakbaev, “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
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
V. Zh. Sakbaev, “Transformation Semigroups of the Space of Functions That Are Square Integrable with respect to a Translation-Invariant Measure on a Banach Space”, J. Math. Sci. (N. Y.), 252:1 (2021), 72–89
L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178
V. Zh. Sakbaev, “Averaging of random walks and shift-invariant measures on a Hilbert space”, Theoret. and Math. Phys., 191:3 (2017), 886–909
V. Zh. Sakbaev, “Random walks and measures on Hilbert space that are invariant with respect to shifts and rotations”, J. Math. Sci. (N. Y.), 241:4 (2019), 469–500
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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