Abstract:
We investigate stability of solutions in
linear stochastic Sobolev type models with the relatively bounded operator in spaces of smooth differential forms
defined on smooth compact oriented Riemannian manifolds without boundary. To this end, in the space of differential forms,
we use the pseudo-differential Laplace–Beltrami operator instead of the usual Laplace operator.
The Cauchy condition and the Showalter–Sidorov condition are used as the initial conditions. Since “white noise”
of the model is non-differentiable in the usual sense, we use the derivative of stochastic process in the sense of Nelson–Gliklikh.
In order to investigate stability of solutions, we establish existence of exponential dichotomies dividing the space of solutions into stable
and unstable invariant subspaces. As an example, we use a stochastic version of the Barenblatt–Zheltov–Kochina equation in the space of differential
forms defined on a smooth compact oriented Riemannian manifold without boundary.
Keywords:
Sobolev type equations, differential forms, stochastic equations, Nelson–Gliklikh derivative.
Citation:
O. G. Kitaeva, D. E. Shafranov, G. A. Sviridyuk, “Exponential dichotomies in Barenblatt– Zheltov–Kochina model in spaces of differential forms with “noise””, Vestnik YuUrGU. Ser. Mat. Model. Progr., 12:2 (2019), 47–57
\Bibitem{KitShaSvi19}
\by O.~G.~Kitaeva, D.~E.~Shafranov, G.~A.~Sviridyuk
\paper Exponential dichotomies in Barenblatt-- Zheltov--Kochina model in spaces of differential forms with ``noise''
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2019
\vol 12
\issue 2
\pages 47--57
\mathnet{http://mi.mathnet.ru/vyuru487}
\crossref{https://doi.org/10.14529/mmp190204}
\elib{https://elibrary.ru/item.asp?id=38225236}
Linking options:
https://www.mathnet.ru/eng/vyuru487
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This publication is cited in the following 21 articles:
A. V. Keller, “O napravleniyakh issledovanii uravnenii sobolevskogo tipa”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 16:4 (2023), 5–32
N. S. Goncharov, G. A. Sviridyuk, “Analiz stokhasticheskoi sistemy Venttselya, sostavlennoi iz uravnenii filtratsii vlagi v share i na ego granitse”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 16:4 (2023), 84–92
N. S. Goncharov, G. A. Sviridyuk, “Analysis of the stochastic Wentzell system of fluid filtration equations in a circle and on its boundary”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 15:3 (2023), 15–22
D. E. Shafranov, “Chislennye resheniya neklassicheskikh uravnenii v prostranstvakh differentsialnykh form”, J. Comp. Eng. Math., 9:4 (2022), 3–17
Alexander Aleksandrovich Zheleznyak, “Analyzing mathematical model of random processes in automatic control system of ship electric power unit”, Vestnik of Astrakhan State Technical University. Series: Marine engineering and technologies, 2022:1 (2022), 90
O. G. Kitaeva, “Eksponentsialnye dikhotomii stokhasticheskikh uravnenii sobolevskogo tipa”, J. Comp. Eng. Math., 9:3 (2022), 3–19
A. A. Zamyshlyaeva, O. N. Tsyplenkova, “Reconstruction of dynamically distorted signals based on the theory of optimal control of solutions for Sobolev type equations in the spaces of stochastic processes”, Vestn. Yuzhno-Ural. Gos. Un-ta. Ser. Matem. Mekh. Fiz., 14:3 (2022), 38–44
D. E. Shafranov, “Uravneniya sobolevskogo tipa v prostranstvakh differentsialnykh form na rimanovykh mnogoobraziyakh bez kraya”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 15:1 (2022), 112–122
O. G. Kitaeva, “Invariant spaces of Oskolkov stochastic linear equations on the manifold”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 13:2 (2021), 5–10
E. A. Soldatova, A. V. Keller, “Algoritmy i obrabotka informatsii v chislennom issledovanii stokhasticheskoi modeli Barenblatta-Zheltova-Kochinoi”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 13:4 (2021), 29–36
O. G. Kitaeva, “Eksponentsialnye dikhotomii odnogo stokhasticheskogo neklassicheskogo uravneniya na dvumernoi sfere”, J. Comp. Eng. Math., 8:1 (2021), 60–67
E. V. Bychkov, A. V. Bogomolov, K. Yu. Kotlovanov, “Stochastic mathematical model of internal waves”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 13:2 (2020), 33–42
Sophiya A. Zagrebina, Natalya N. Solovyova, Springer Proceedings in Mathematics & Statistics, 325, Semigroups of Operators – Theory and Applications, 2020, 95
D. E. Shafranov, “O chislennom reshenii v prostranstve differentsialnykh form dlya odnogo stokhasticheskogo uravneniya sobolevskogo tipa s otnositelno radialnym operatorom”, J. Comp. Eng. Math., 7:4 (2020), 48–55
Olga G. Kitaeva, Dmitriy E. Shafranov, Georgy A. Sviridyuk, Springer Proceedings in Mathematics & Statistics, 325, Semigroups of Operators – Theory and Applications, 2020, 279
O. G. Kitaeva, “Dikhotomii reshenii stokhasticheskogo uravneniya Ginzburga – Landau na tore”, J. Comp. Eng. Math., 7:4 (2020), 17–25
D. E. Shafranov, “Chiclennoe reshenie uravneniya Dzektsera s “belym shumom” v prostranstve gladkikh differentsialnykh form, opredelennykh na tore”, J. Comp. Eng. Math., 7:2 (2020), 58–65
E. V. Bychkov, N. N. Soloveva, G. A. Sviridyuk, “Matematicheskaya model akusticheskikh voln v ogranichennoi oblasti s «belym shumom»”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 11:3 (2019), 12–19
O. G. Kitaeva, “Eksponentsialnye dikhotomii odnogo neklassicheskogo uravneniya v prostranstvakh differentsialnykh form na dvumernom tore s “shumami””, J. Comp. Eng. Math., 6:3 (2019), 26–38
D. E. Shafranov, “Chislennoe reshenie uravneniya Barenblatta – Zheltova – Kochinoi s additivnym “belym shumom” v prostranstvakh differentsialnykh form na tore”, J. Comp. Eng. Math., 6:4 (2019), 31–43
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