Abstract
Group analysis is used to study stochastic equations of fluid dynamics. Determining equations for admitted Lie groups of transformation involving independent and dependent variables and Wiener processes are obtained. It is shown that, as in the case of deterministic differential equations, admitted groups make it possible to reduce invariant solutions of stochastic differential equations to solutions with a smaller number of independent variables.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. B. Pope, Turbulent Flows (Cambridge Univ. Press, Cambridge, 2000).
C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences (Springer, New York, 1997).
N. G. van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier, Amsterdam, 2007).
A. N. Shiryaev, Essentials of Stochastic Finance. Facts, Models, Theory (World Sci., Hong Kong, 1999).
A. Bensoussan and R. Temam, “Equatios Stochastique du Type Navier-Stokes,” J. Funct. Anal. 13, 195–222 (1973).
R. Mikulevicius and B. L. Rozovskii, “Stochastic Navier-Stokes Equations for Turbulent Flows,” SIAM J. Math. Anal. 35(5), 1250–1310 (2004).
F. S. Nasyrov, Local Times, Symmetric Integrals, and Stochastic Analysis (Fizmatlit, Moscow, 2011) [in Russian].
L. V. Ovsyannikov, Group Analysis of Differential Equations (Nauka, Moscow, 1978; Academic Press, New York, 1982).
CRC Handbook of Lie Group Analysis of Differential Equations, Ed. by N. H. Ibragimov, Vol. 1 (CRC Press, Boca Raton, 1994).
CRC Handbook of Lie Group Analysis of Differential Equations Ed. by N. H. Ibragimov, Vol. 2 (CRC Press, Boca Raton, 1995).
CRC Handbook of Lie Group Analysis of Differential Equations Ed. by N. H. Ibragimov, Vol. 3 (CRC Press, Boca Raton, 1996).
L. V. Ovsyannikov, “Group and Group-Invariant Solutions of Differential Equations,” Dokl. Akad. Nauk SSSR 118,(3), 439–442 (1958).
L. V. Ovsyannikov, “The SUBMODELS Program. Gas dynamics,” Prikl. Mat. Mekh. 58(4), 30–55 (1994).
L. V. Ovsyannikov, “Some Results of the SUBMODELS Program for the Equations of Gas Dynamics,” Prikl. Mat. Mekh. 63(3), 362–372 (1999).
V. V. Pukhnachev, “Group Properties of the Navier-Stokes Equations in the Two-Dimensional Case,” Prikl. Mekh. Tekh. Fiz., No. 1, 83–90 (1960).
V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachev, and A. A. Rodionov, Use of Group-Theoretic Methods in Hydrodynamics (Nauka, Novosibirsk, 1994) [in Russian].
V. V. Pukhnachev, “Symmetry in the Navier-Stokes Equations,” Usp. Mekh. 4(1), 6–76 (2006).
S. V. Meleshko, “Methods for Constructing Exact Solutions of Partial Differential Equations,” in Mathematical and Analytical Techniques with Applications to Engineering (Springer, New York, 2005).
Yu. N. Grigoriev, N. H. Ibragimov, V. F. Kovalev, and S. V. Meleshko, Symmetries of Integro-Differential Equations and Their Applications in Mechanics and Plasma Physics (Springer, Berlin, Heidelberg, 2010). (Lecture Notes Phys., Vol. 806.)
T. Misawa, “New Conserved Quantities Form Symmetry for Stochastic Dynamical Systems,” J. Phys. A. 27, 177–192 (1994).
S. Albeverio and S. Fei, “Remark on Symmetry of Stochastic Dynamical Systems and Their Conserved Quantities,” J. Phys. A. 28, 6363–6371.
O. V. Aleksandrova, “Group Analysis of the Two-Dimensional Itô Stochastic Equation,” Vestn. Donbas. Nats. Akad. Str. Arkh. 1, 140–145 (2005).
O. V. Alexsandrova, “Group Analysis of the Itô Stochastic System,” Differ. Eq. Dyn. Syst. 14(3/4), 255–280 (2006).
G. Gaeta and N. R. Quinter, “Lie-Point Symmetries and Differential Equations,” J. Phys. A. 32, 8485–8505 (1999).
G. Gaeta, “Symmetry of Stochastic Equations,” in Symmetry in Nonlinear Mathematical Physics (Inst. of Math. of NAS of Ukraine, Kyiv, 2004), pp. 98–109.
G. Ünal, “Symmetries of Itô and Stratonovich Dynamical Systems and Their Conserved Quantities,” Nonlinear Dyn. 32, 417–426 (2003).
G. Ünal, and J. Q. Sun, “Symmetries Conserved Quantities of Stochastic Dynamical Control Systems,” Nonlinear Dyn. 36, 107–122 (2004).
N. H. Ibragimov, G. Ünal, and C. Jogréus, “Approximate Symmetries and Conservation Laws for Itô and Stratonovich Dynamical Systems,” J. Math. Anal. Appl. 297, 152–168 (2004).
F. M. Mahomed and C. Wafo Soh, “Integration of Stochastic Ordinary Differential Equations from a Symmetry Standpoint,” J. Phys. A. 34, 777–782 (2001).
S. A. Melnick, “The Group Analysis of Stochastic Partial Differential Equations,” Theory Stochast. Proc. 9(1/2), 99–107 (2003).
B. Srihirun, S. V. Meleshko, and E. Schulz, “On the Definition of an Admitted Lie Group for Stochastic Differential Equations,” Comm. Nonlinear Sci. Numer. Simulat. 12(8), 1379–1389 (2007).
B. Srihirun, S. V. Meleshko, and E. Schulz, “On the Definition of an Admitted Lie Group for Stochastic Differential Equations with Multi-Brownian Motion,” J. Phys. A. 39, 13951–13966 (2006).
B. Øksendal, Stochastic Ordinary Differential Equations. An Introduction with Applications (Springer, Berlin, 1998).
D. Nualart, Malliavin Calculus and Related Topics (Springer, New York, 2006).
A. C. Hearn, Reduce Users Manual, Ver. 3.3 (Rand Corp. CP 78, Santa Monica, 1987).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S.V. Meleshko, O. Samrum, E. Schulz.
__________
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 54, No. 1, pp. 25–39, January–February, 2013.
Rights and permissions
About this article
Cite this article
Meleshko, S.V., Sumrum, O. & Schulz, E. Application of group analysis to stochastic equations of fluid dynamics. J Appl Mech Tech Phy 54, 21–33 (2013). https://doi.org/10.1134/S0021894413010033
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0021894413010033