×

Random symplectic geometry and the realizations of the random representations of the Navier-Stokes equations by ordinary differential equations. (English) Zbl 1094.37042

The author considers two important aspects of fluid-dynamics. Firstly, the realization of Navier-Stokes equations by an approximation of its representations by almost everywhere ordinary differential equations. This approach comes from the extension of the classical development method due to E. Cartan by an approximation of the development of a Wiener process with Navier-Stokes connection by a sequence of polygonal paths. The author also treats the problem of construction of a symplectic theory for the Navier-Stokes equations. There is an important chapter of fluid-dynamics and of symplectic geometry which was elaborated by the AEM program, which deals with the construction of a symplectic approach for inviscid fluids described by the Euler equations.
In this approach, the author encountered the infinite-dimensional Lie group of the Riemannian volume-preserving diffeomorphisms of \(M\), and the symplectic structure is constructed in terms of the minimal action principle given by extremizing the energy function assotiated to the right-invariant metric in the dual of the Lie-algebra of divergence-free vector fields on \(M\), which the author derived from his constructions for the viscous case.

MSC:

37K65 Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
58D30 Applications of manifolds of mappings to the sciences
58J65 Diffusion processes and stochastic analysis on manifolds
60F99 Limit theorems in probability theory
76D05 Navier-Stokes equations for incompressible viscous fluids
76M35 Stochastic analysis applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Taylor. Partial DiR erential Equations, vols. I and III. Springer-Verlag, Berlin (1995).
[2] V. I. Arnold and B. A. Khesin. Topological Methods in Hydrodynamics, Springer, New York (1999). · Zbl 0743.76019
[3] Arnold V. I, Ann. Inst. Fourier 16 pp 361– (1966)
[4] Ebin D., Annals of Maths. 92 pp 163– (1971)
[5] DOI: 10.1016/S0034-4877(02)80002-7 · Zbl 1004.60060 · doi:10.1016/S0034-4877(02)80002-7
[6] D. L. Rapoport. Random representations for viscous Ruids and the passive magnetic - elds transported by them, in Proceedings, Third International Conference on Dynamical Systems and DiR erential Equations, Kennesaw, May (2000), Discrete & Cont.Dyn.Sys. special issue, S.Hu (ed.) (2000).
[7] Rapoport D. L., Rep.Math.Phys. 50 pp 2– (2002)
[8] Rapoport D. L., Rand. Oper. Stoch. Eqts. 11 pp 2– (2003)
[9] P. Malliavin. Stochastic Analysis. Springer Verlag (1998). · Zbl 0878.60001
[10] D. L. Rapoport. Torsion, Brownian motion, spinor - elds, quantum mechanics and Ruid-dynamics, I & II, In IX th. Marcel Grossmann Meeting in Relativity, Gravitation and Field Theories, Rome (2000), V. Gurzadyan et al (eds), World Sc., Singapore (2003) & www.icra.it/MG/mg9/Proceedings/Proceedings.html.
[11] Baxendale P., Gebiete 65 pp 267– (1983)
[12] C. Foias, O. Manley, R. Rosa & R.Temam. The Navier-Stokes equations and Turbulence, Encyclopedia of Mathematics, Cambridge Univ. Press (2001). · Zbl 0994.35002
[13] J. M. Bismut. M ecanique Analytique. Springer LNM 866, (1982).
[14] D. Rapoport. Torsion and Nonlinear Quantum Mechanics, in Group XXI, Physical Applications and Mathematical Aspects of Algebras, Groups and Geometries, Clausthal (Germany, June 1995), H.Doebner et al (edts.), World Scienti- c Publisher, 1996.
[15] DOI: 10.1016/0003-4916(84)90128-3 · Zbl 0564.53046 · doi:10.1016/0003-4916(84)90128-3
[16] D. Rapoport. Non-riemannian geometry and random dynamics of in- nite-particle systems, in Instabilities and Nonequilibrium Structures IX, Proceedings, Vina del Mar, December 2001, O. Descalzi, J. Martinez and S. Rica (edts.), Kluwer (2003).
[17] Rapoport D., Intern. J. Theor. Phys. 36 pp 10– (1997)
[18] R. Temam. In-nite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, New York (1988). · Zbl 0662.35001
[19] S. Pope. Turbulence. Cambridge Univ. Press, (2000).
[20] A. Chorin. Turbulence and Vorticity. Springer Verlag Series in Applied Mathematics, New York (1995). · Zbl 0795.76002
[21] C. Marchioro & M Pulvirenti. Mathematical Theory of Incompressible Nonviscous Fluids. Springer Verlag (1994). · Zbl 0789.76002
[22] K. Gustafson & J. Sethian (edts.):Vortex Methods and Vortex Motions, SIAM, Philadelphia (1991). · Zbl 0748.76010
[23] K. D. Elworthy. Stochastic Flows on Riemannian Manifolds, in DiR usion Processes and Related Problems in Analysis, M.Pinsky et al (edts.), vol. II, Birkhauser (1992). · Zbl 0758.58035
[24] G. N. Milstein. Numerical Integration of Stochastic Di erential Equations. Kluwer Series in Mathematics and its applications, (1995).
[25] P. Kloeden & E. Platen. Numerical Solutions of Stochastic DiR erential Equations, Applications of Mathematics (Stochastic Modelling and Applied Probability) vol. 23, Springer Verlag (1995).
[26] C. Doering & J.D. Gibbon. Applied Analysis of the Navier-Stokes equations. Cambridge Univ. Press, Cambridge (1996). · Zbl 0838.76016
[27] S. Shkoller. J.Funct. Anal. 160, 337{375 (1998).}
[28] K. D. Elworthy. Stochastic Di erential Equations on Manifolds. Cambridge Univ. Press of the London Mathematical Society, Cambridge (U.K.) (1982). · Zbl 0514.58001
[29] K. Kunita. Stochastic Flows and Stochastic DiR erential Equations. Cambridge Univ. Press, Cambridge (1994).
[30] N. Ikeda and S. Watanabe. Stochastic Di erential Equations and DiR usion Processes. North-Holland/Kodansha, (1981).
[31] K. Ito. The Brownian motion and tensor - elds on Riemannian manifolds, in Proc. Inter. Cong.Math., Stockholm 536{539 (1963).}
[32] P. Malliavin. G eom etrie Di- erentielle Stochastique, Les Presses Univ., Montreal, (1978).
[33] V. Arnold. Mathematical methods of Classical Mechanics, Springer Verlag (1986).
[34] R. Abraham & J. Marsden, Foundations of Mechanics,second edition, Wiley Interscience (1978). · Zbl 0393.70001
[35] T. Hida. Brownian Motion, Springer Verlag (1980). · Zbl 0423.60063
[36] Glimm and Ja-e. Quantum Physics, a Functional Integral Approach, Springer Verlag (1982).
[37] I. Prigogine. From Being to Becoming, Freeman, New York (1980).
[38] Petrosky T., Solitons and Fractals 4 pp 1– (1984)
[39] Yu. Gliklikh. Global analysis in mathematical-physics. Applied Mathematics Sciences 122, Springer, (1997). · Zbl 0868.58001
[40] D. Rapoport. Realizations of the random representations of the Navier-Stokes equations by ordinary di-erential equations, in Instabilities and Nonequilibrium Structures vol. VII & VIII, Valparaiso (Chile), E. Tirapegui et al (edts.), Kluwer Series in Complex Phenomenae and Nonlinear Systems, in press (2003).
[41] C. Gardiner. Handbook of Stochastic Processes. Springer Series in Synergetics (1991).
[42] L. Arnold. Stochastic Di erential Equations, Wiley (Interscience) (1971).
[43] Taylor T., Stochastics and Stochastic Reports 43 pp 197– (1993)
[44] Z. Schuss. Stochastic DiR erential Equations: Theory and Application, Wiley-Interscience (1980).
[45] N. van Kampen. Stochastic Processes in Physics and Chemistry, North-Holland, Berlin (1981). · Zbl 0511.60038
[46] DOI: 10.1016/0022-1236(83)90021-6 · Zbl 0533.35075 · doi:10.1016/0022-1236(83)90021-6
[47] P. Constantin & C. Foias. Navier- Stokes equations, Chicago Lectures in Mathematics (1988). · Zbl 0687.35071
[48] D. Ruelle. Turbulence, Strange Attractors and Chaos, Series A on Nonlinear Sciences vol. 16, World Scienti- c, Singapore (1995). · Zbl 0922.76009
[49] Hehl F., Rep 258 pp 157– (1995)
[50] DOI: 10.1007/BF00675614 · Zbl 0738.53055 · doi:10.1007/BF00675614
[51] Rapoport D., Groups and Geometries 11 pp 35– (1994)
[52] D. Rapoport. Scale - elds as a simplicity principle, Hadronic J. Suppl. 2, Proceedings of the Third International Workshop on Lie-isotopic theories, Patras, Greece, A. Jannusis (ed.), (1996).
[53] P. Meyer. G’eometrie stochastique sans larmes, in S eminaire des Probabilites XVI, Supplement, Lecture Notes in Mathematics 921, Springer, Berlin, 165{207 (1982).}
[54] Ya. Belopolskaya & Yu. Dalecky. Stochastic Processes and DiR erential Geometry, Kluwer Academic Press, Dordrecht (1989).
[55] P. Holmes, J. Lumley & G. Berkooz. Turbulence, Coherent Structures, Dynamical Systems and Symmetries, Cambridge Univ. Press, Cambridge (U.K.), (1996). · Zbl 0890.76001
[56] E. Nelson. Quantum Fluctuations, Princeton University Press, Princeton (New Jersey), (1985). · Zbl 0563.60001
[57] D. Rapoport, The geometry of quantum Ructuations, their quantum Lyapunov exponents and the stochastic Perron-Frobenius semigroups, in Dynamical Systems and Chaos vol. 2, Proceedings of the International Conference (Tokyo, May 1994), Y. Aizawa et al (edts.), World Scienti- c (1995).
[58] D. Rapoport. On the geometry of Ructuations I & II, in New Frontiers of Algebras, Groups and Geometries, Proceedings (Monteroduni, Italy, 1995), G. Tsagas (ed.), Hadronic Press , Palm Harbor (Florida) (1996).
[59] D. Rapoport. The classical Cartan geometry of classical and quantum gravity, in Relativity, the Space-Time Structure, Proceedings of the Eighth Latinoamerican Symposium in Relativity and Gravitation, Aguas de Lindoia (Brazil, July 1993), World Scienti- c 1994. ibid. Riemann-Cartan-Weyl di-usions and the equivalence of the Free Maxwell and Dirac-Hestenes equations, Advances in Applied Cli-ord Algebras vol. 8, 1, 129{146 (1998).}
[60] Rapoport D., Advances in Applied Cli-ord Algebras 8 pp 1– (1998)
[61] D. Rapoport. Torsion and quantum, thermodynamical and hydrodynamical Ructuations, Proceedings of the Eighth Marcel Grossmann Meeting in Relativity, Gravitation and Field Theories, Jerusalem, August 1997, T. Piran et al (edts.), 73{76, World Scienti- c (2000).}
[62] U. Frisch. Turbulence. Cambridge Univ. Press (1999).
[63] A. S. Monin & A. M. Yaglom. Statistical Fluid Dynamics, vol. II, J. Lumley (ed.), M.I.T. Press, Cambridge (MA) (1975).
[64] J. Lumley. Tools in Turbulence, Academic Press, New York (1970). · Zbl 0273.76035
[65] S. A. Orszag. Statistical Theory of Turbulence, in Fluid Dynamics, Les Houches 1973, 237{374, R. Balian and J. Peube (eds.), Gordon and Breach, New York (1977).}
[66] Reynolds O., Phil. Trans. Royal Soc. London A 186 pp 161– ((1894))
[67] Yu. Gliklikh. Ordinary and Stochastic diR erential Equations as Tools for Mathematical Physics, Kluwer, Dordrecht (1996).
[68] Yu. Gliklikh. Viscous hydrodynamics through stochastic perturbations of Row of perfect Ruids on grups of di-eomorphisms, Proceedings of the Voronezh State University, Russia, No. 1, 83{91 (2001). bibitem K. Kunita. Stochastic Flows and Stochastic DiR erential Equations, Cambridge Univ. Press, Cambridge (U.K.) (1994).}
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.