Volume 13, no. 2Pages 33 - 42
Stochastic Mathematical Model of Internal WavesE.V. Bychkov, A.V. Bogomolov, K.Yu. Kotlovanov
The paper studies a mathematical model of internal gravitational waves with additive "white noise", which models the fluctuations and random heterogeneity of the medium. The mathematical model is based on the Sobolev stochastic equation, Dirichlet boundary conditions and the initial Cauchy condition. The Sobolev equation is obtained from the assumption of the propagation of waves in a uniform incompressible rotation with a constant angular velocity of the fluid. The solution to this problem is called the inertial (gyroscopic) wave, since it arises due to the Archimedes's law and under the influence of inertia forces. By ''white noise'' we mean the Nelson-Gliklikh derivative of the Wiener process. The study was conducted in the framework of the theory of relatively bounded operators, the theory of stochastic equations of Sobolev type and the theory of (semi) groups of operators. It is shown that the relative spectrum of the operator is bounded, and the solution of the Cauchy-Dirichlet problem for the Sobolev stochastic equation is constructed in the operator form.Full text
- relatively bounded operator; Sobolev equation; propagators; “white noise”; Nelson–Gliklikh derivative.
- 1. Brekhovskikh L.M., Goncharov V.V. Vvedenie v mekhaniku sploshnykh sred (v prilozhenii k teorii voln) [Introduction to the Mechanics of Continuous Media (as Applied to the Theory of Waves)]. Moscow, Nauka, 1982. (in Russian)
2. Sobolev S.L. [On a New Problem of Mathematical Physics]. Izvestiya Akademii Nauk SSSR, 1954, vol. 18, no. 1, pp. 3-50. (in Russian)
3. Fokin M.V. Hamiltonian Systems in the Theory of Small Oscillations of a Rotating Ideal Fluid I. Siberian Advances in Mathematics, 2002, vol. 12, no. 1, pp. 1-50.
4. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer, 2011. DOI: 10.1007/978-0-85729-163-9
5. Gliklikh Yu.E. The Study of Leontieff Type Equations with White Noise by the Methods of Mean Derivatives of Stochastic Processes. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 27, pp. 24-34. (in Russian)
6. Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press, 1967.
7. Kovacs M., Larsson S. Introduction to Stochastic Partial Differential Equations. Proceedings of New Directions in the Mathematical and Computer Sciences, 2007, vol. 4, pp. 159-232.
8. Melnikova I.V., Alshanskiy M.A. The Generalized Well-Posedness of the Cauchy Problem for an Abstract Stochastic Equation with Multiplicative Noise. Proceedings of the Steklov Institute of Mathematics (Supplementary Issues), 2013, vol. 280, pp. 134-150. DOI: 10.1007/978-3-0348-0585-8
9. Sagadeeva M.A. Reconstruction of Observation from Distorted Data for the Optimal Dynamic Measurement Problem. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2019, vol. 12, no. 2, pp. 82-96. (in Russian) DOI: 10.14529/mmp190207
10. Sviridyuk G.A., Manakova N.A. The Dynamical Models of Sobolev Type with Showalter-Sidorov Condition and Additive "Noise". Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 90-103. (in Russian) DOI: 10.14529/mmp140108
11. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of "Noises". Abstract and Applied Analysis, 2015, vol. 2015, pp. 1-8. DOI: 10.1155/2015/697410
12. Zamyshlyaeva A.A., Sviridyuk G.A. The Linearized Benney-Luke Mathematical Model with Additive White Noise. Semigroups of Operators. Theory and Applications, 2015, vol. 113, pp. 327-337. DOI: 10.1007/978-3-319-12145-1_21
13. Favini A., Sviridyuk G.A., Zamyshlyaeva A.A. One Class of Sobolev Type Equations of Higher Order with Additive "White Noise". Communications on Pure and Applied Analysis, 2016, vol. 15, no. 1, pp. 185-196. DOI: 10.3934/cpaa.2016.15.185
14. Favini A., Zagrebina,S.A., Sviridyuk G.A. Multipoint Initial-Final Value Problems for Dynamical Sobolev-Type Equations in the Space of "Noises". Electronic Journal of Differential Equations, 2018, vol. 2018, p. 128.
15. Sviridyuk G.A., Zamyshlyaeva A.A., Zagrebina S.A. Multipoint Initial-Final Problem for One Class of Sobolev Type Models of Higher Order with Additive "White Noise". Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2018, vol. 11, no. 3, pp. 103-117. DOI: 10.14529/mmp180308
16. Kitaeva O.G., Shafranov D.E., Sviridyuk G.A. Exponential Dichotomies in Barenblatt-Zheltov-Kochina Model in Spaces of Differential Forms with "Noise". Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2019, vol. 12, no. 2, pp. 47-57. DOI: 10.14529/mmp190204