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April  2020, 13(4): 1229-1242. doi: 10.3934/dcdss.2020071

Predicting uncertainty in geometric fluid mechanics

François Gay-Balmaz 1, and  Darryl D. Holm 2,,

1. 

CNRS - LMD - IPSL, École Normale Supérieure de Paris, 24 Rue Lhomond, 75005, Paris, France
2. 

Department of Mathematics, Imperial College, London SW7 2AZ, UK, Springfield, MO 65810, USA

* Corresponding author: Darryl D. Holm

Received  December 2017 Revised  August 2018 Published  April 2019

We review opportunities for stochastic geometric mechanics to incorporate observed data into variational principles, in order to derive data-driven nonlinear dynamical models of effects on the variability of computationally resolvable scales of fluid motion, due to unresolvable, small, rapid scales of fluid motion.

Keywords: Dimension theory, Poincaré recurrences, multifractal analysis.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: François Gay-Balmaz, Darryl D. Holm. Predicting uncertainty in geometric fluid mechanics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1229-1242. doi: 10.3934/dcdss.2020071
References:
[1]

S. Albeverio, A. B. Cruzeiro and D. D. Holm, Stochastic Geometric Mechanics, Springer, 2017.  Google Scholar

[2]

A. Arnaudon, A. L. de Castro and D. D. Holm, Noise and dissipation on coadjoint orbits, J. Nonlin. Sci., 28 (2018), 91-145.  doi: 10.1007/s00332-017-9404-3.  Google Scholar

[3]

V. I. Arnol'd, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'institut Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

[4]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics, Springer-Verlag, New York  Google Scholar

[5]

J.-M. Bismut, Mécanique aléatoire, In Tenth Saint Flour Probability Summer School—1980 (Saint Flour, 1980), volume 929 of Lecture Notes in Math., pages 1–100, Springer, Berlin-New York, 1982.  Google Scholar

[6]

N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.  doi: 10.1093/imanum/drn018.  Google Scholar

[7]

C. J. Cotter, G. A. Gottwald and D. D. Holm, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proc. Roy. Soc. A, 473 (2017), 20170388, 10pp. doi: 10.1098/rspa.2017.0388.  Google Scholar

[8]

C. J. Cotter, D. Crisan, D. D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model, arXiv preprint, arXiv: 1802.05711. Google Scholar

[9]

D. Crisan, F. Flandoli and D. Holm, Solution properties of a 3d stochastic Euler fluid equation, arXiv: 1704.06989, [math-ph], 2017. doi: 10.1007/s00332-018-9506-6.  Google Scholar

[10]

A. B. Cruzeiro, D. D. Holm and T. S. Ratiu, Momentum maps and stochastic Clebsch action principles, Commun. in Math. Phys., 357 (2018), 873-912.  doi: 10.1007/s00220-017-3048-x.  Google Scholar

[11]

F. Gay-Balmaz and D. D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlin. Sci., 28 (2018), 873-904.  doi: 10.1007/s00332-017-9431-0.  Google Scholar

[12]

F. Gay-Balmaz and V. Putkaradze, On noisy extensions of nonholonomic constraints, J. Nonlin. Sci., 26 (2016), 1571-1613.  doi: 10.1007/s00332-016-9313-x.  Google Scholar

[13]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[14]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[15]

J. McWilliams, A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 1 (1977), 427-441.  doi: 10.1016/0377-0265(77)90002-1.  Google Scholar

[16]

H. Yoshimura and F. Gay-Balmaz, Hamilton–Pontryagin principle for incompressible ideal fluids, AIP Conference Proceedings, 1376 (2011), 645-647.  doi: 10.1063/1.3652002.  Google Scholar

[17]

H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.  doi: 10.1016/j.geomphys.2006.02.009.  Google Scholar

show all references

References:
[1]

S. Albeverio, A. B. Cruzeiro and D. D. Holm, Stochastic Geometric Mechanics, Springer, 2017.  Google Scholar

[2]

A. Arnaudon, A. L. de Castro and D. D. Holm, Noise and dissipation on coadjoint orbits, J. Nonlin. Sci., 28 (2018), 91-145.  doi: 10.1007/s00332-017-9404-3.  Google Scholar

[3]

V. I. Arnol'd, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'institut Fourier, 16 (1966), 319-361.  doi: 10.5802/aif.233.  Google Scholar

[4]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics, Springer-Verlag, New York  Google Scholar

[5]

J.-M. Bismut, Mécanique aléatoire, In Tenth Saint Flour Probability Summer School—1980 (Saint Flour, 1980), volume 929 of Lecture Notes in Math., pages 1–100, Springer, Berlin-New York, 1982.  Google Scholar

[6]

N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), 421-443.  doi: 10.1093/imanum/drn018.  Google Scholar

[7]

C. J. Cotter, G. A. Gottwald and D. D. Holm, Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics, Proc. Roy. Soc. A, 473 (2017), 20170388, 10pp. doi: 10.1098/rspa.2017.0388.  Google Scholar

[8]

C. J. Cotter, D. Crisan, D. D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model, arXiv preprint, arXiv: 1802.05711. Google Scholar

[9]

D. Crisan, F. Flandoli and D. Holm, Solution properties of a 3d stochastic Euler fluid equation, arXiv: 1704.06989, [math-ph], 2017. doi: 10.1007/s00332-018-9506-6.  Google Scholar

[10]

A. B. Cruzeiro, D. D. Holm and T. S. Ratiu, Momentum maps and stochastic Clebsch action principles, Commun. in Math. Phys., 357 (2018), 873-912.  doi: 10.1007/s00220-017-3048-x.  Google Scholar

[11]

F. Gay-Balmaz and D. D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlin. Sci., 28 (2018), 873-904.  doi: 10.1007/s00332-017-9431-0.  Google Scholar

[12]

F. Gay-Balmaz and V. Putkaradze, On noisy extensions of nonholonomic constraints, J. Nonlin. Sci., 26 (2016), 1571-1613.  doi: 10.1007/s00332-016-9313-x.  Google Scholar

[13]

D. D. Holm, Variational principles for stochastic fluid dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963.  Google Scholar

[14]

J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3.  Google Scholar

[15]

J. McWilliams, A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 1 (1977), 427-441.  doi: 10.1016/0377-0265(77)90002-1.  Google Scholar

[16]

H. Yoshimura and F. Gay-Balmaz, Hamilton–Pontryagin principle for incompressible ideal fluids, AIP Conference Proceedings, 1376 (2011), 645-647.  doi: 10.1063/1.3652002.  Google Scholar

[17]

H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.  doi: 10.1016/j.geomphys.2006.02.009.  Google Scholar

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