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Circulation and Energy Theorem Preserving Stochastic Fluids

Published online by Cambridge University Press:  23 July 2019

Theodore D. Drivas  and
Darryl D. Holm
Theodore D. Drivas
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ08544, United States (tdrivas@math.princeton.edu)
Darryl D. Holm
Affiliation:
Department of Mathematics, Imperial College, London SW7 2AZ, UK (d.holm@imperial.ac.uk)
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Abstract

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincaré and stochastic Navier–Stokes–Poincaré equations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Keywords

stochastic fluid equationsvariational principleKelvin theorem
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2776 - 2814
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Arnaudon, M., Chen, X. and Cruzeiro, A. B.. Stochastic Euler–Poincaré reduction. J. Math. Physics 55 (2014), 081507.CrossRefGoogle Scholar
2Arnold, V. I. and Khesin, B. A.. Topological methods in hydrodynamics (New York: Springer, 1998).CrossRefGoogle Scholar
3Arnold, V. I. and Khesin, B. A.. Topological methods in hydrodynamics,vol. 125 (Springer Science & Business Media, 1999).Google Scholar
4Bernard, D., Gawedzki, K. and Kupiainen, A.. Slow modes in passive advection. J. Stat, Phys. 90 (1998), 519–569.CrossRefGoogle Scholar
5Chen, S., Foias, C., Holm, D. D., Olson, E. J., Titi, E. S. and Wynne, S.. The Camassa-Holm equations as a closure model for turbulent channel and pipe flows. Phys. Rev. Lett. 81 (1998), 5338–5341.CrossRefGoogle Scholar
6Cipriano, F. and Cruzeiro, A. B.. Navier–Stokes equation and diffusions on the group of homeomorphisms of the torus. Comm. Math. Phys. 275 (2007), 255–269.CrossRefGoogle Scholar
7Constantin, P.. An Euler–Lagrangian approach for incompressible fluids: local theory. J. Amer. Math. Soc. 14 (2001), 263–27.CrossRefGoogle Scholar
8Constantin, P. and Foias, C.. Navier-stokes equations (University of Chicago Press, 1988).CrossRefGoogle Scholar
9Constantin, P. and Iyer, G.. A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations. Commun. Pure Appl. Math 61 (2008), 330–345.CrossRefGoogle Scholar
10Constantin, P. and Iyer, G.. A stochastic-Lagrangian approach to the Navier–Stokes equations in domains with boundary. The Annals of Applied Probability 21 (2011), 1466–1492.CrossRefGoogle Scholar
11Cotter, C. J., Gottwald, G. A. and Holm, D. D.. Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics. Proc. R. Soc. A 473 (2017), 20170388.CrossRefGoogle ScholarPubMed
12Cotter, C., Crisan, D., Holm, D. D., Pan, W. and Shevchenko, I.. Numerically modelling stochastic Lie transport in fluid dynamics. arXiv preprint arXiv:1801.09729 (2018).CrossRefGoogle Scholar
13Cotter, C., Crisan, D., Holm, D. D., Pan, W. and Shevchenko, I.. Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model. arXiv preprint arXiv:1802.05711 (2018).Google Scholar
14Crisan, D., Flandoli, F. and Holm, D. D.. Solution properties of a 3D stochastic Euler fluid equation. J Nonlinear Sci 29 (2018), 813–870.CrossRefGoogle Scholar
15Drivas, T. D.. Anomalous Dissipation, Spontaneous Stochasticity & Onsager's Conjecture. Diss. Johns Hopkins University (2017).Google Scholar
16Drivas, T. D. and Eyink, G. L.. A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I. Flows with no bounding walls. J. Fluid Mech. 829 (2017), 153–189.CrossRefGoogle Scholar
17Drivas, T. D. and Eyink, G. L.. A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part II. Wall-bounded flows. J. Fluid Mech. 829 (2017), 236–279.CrossRefGoogle Scholar
18Eyink, G. L.. Turbulent diffusion of lines and circulations. Phys. Lett A 368 (2007), 486–490.CrossRefGoogle Scholar
19Eyink, G. L.. Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models. J. Math. Phys. 50 (2009), 083102.CrossRefGoogle Scholar
20Eyink, G. L.. Stochastic least-action principle for the incompressible Navier–Stokes equation. Phys. D: Nonlinear Phenomena 239 (2010), 1236–1240.CrossRefGoogle Scholar
21Eyink, G. L. and Drivas, T. D.. Spontaneous stochasticity and anomalous dissipation for Burgers equation. J. Stat. Phys. 158 (2015), 386–432.CrossRefGoogle Scholar
22Flandoli, F.. Random Perturbation of PDEs and Fluid Dynamic Models: École d'été de Probabilités de Saint-Flour XL 2010,vol. 2015 (Springer Science & Business Media, 2011).CrossRefGoogle Scholar
23Flandoli, F.. Random Perturbation of PDEs and Fluid Dynamic Models, Saint Flour summer school lectures 2010, Lecture Notes in Mathematics n. 2015 (Berlin: Springer, 2011).Google Scholar
24Flandoli, F. and Gatarek, D.. Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Related Fields 102 (1995), 367–391.CrossRefGoogle Scholar
25Flandoli, F. and Luo, D.. Euler–Lagrangian approach to 3D stochastic Euler equations. arXiv preprint arXiv:1803.05319 (2018).Google Scholar
26Flandoli, F., Maurelli, M. and Neklyudov, M.. Noise prevents infinite stretching of the passive field in a stochastic vector advection equation. J. Math. Fluid Mech. 16 (2014), 805–822.CrossRefGoogle Scholar
27Foias, C., Holm, D. D. and Titi, E. S.. The Navier–Stokes–alpha model of fluid turbulence. Phys. D 152–153 (2001), 505–519.CrossRefGoogle Scholar
28Foias, C., Holm, D. D. and Titi, E. S.. The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory. J. Dyn. Diff. Eqns. 14 (2002), 1–35.Google Scholar
29Friedman, A.. Stochastic Differential Equations and Applications (Mineola, NY: Dover, 2006).Google Scholar
30Gay–Balmaz, F. and Holm, D. D.. Predicting Uncertainty in Geometric Fluid Mechanics. arXiv preprint arXiv:1806.10470 (2018).Google Scholar
31Gay–Balmaz, F. and Holm, D. D.. Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows. J Nonlinear Sci 28 (2018), 873–904.CrossRefGoogle ScholarPubMed
32Geurts, B. J. and Holm, D. D.. Alpha-modeling strategy for LES of turbulent mixing. In Turbulent Flow Computation (ed. Drikakis, D. and Geurts, B. G.), pp. 237–278 (London: Kluwer, 2002).Google Scholar
33Geurts, B. J. and Holm, D. D.. Regularization modeling for large-eddy simulation. Phys. Fluids 15 (2003), L13–L16.CrossRefGoogle Scholar
34Gliklikh, Y. E.. Solutions of Burgers, Reynolds, and Navier–Stokes equations via stochastic perturbations of inviscid flows. J. Nonlin. Math. Phys. 17 (sup1 (2010), 15–29.CrossRefGoogle Scholar
35Gomes, D. A.. A variational formulation for the Navier–Stokes equation. Comm. Math. Phys. 257 (2005), 227–234.Google Scholar
36Helmholtz, H.. Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen. Crelles J. 55 (1858), 25–55.Google Scholar
37Holm, D. D.. Variational principles for stochastic fluid dynamics. Proc. R. Soc. A 471 (2015), 20140963.CrossRefGoogle ScholarPubMed
38Holm, D. D., Marsden, J. E. and Ratiu, T. S.. Euler–Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 349 (1998), 4173–4177.CrossRefGoogle Scholar
39Holm, D. D., Marsden, J. E. and Ratiu, T. S.. The Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137 (1998), 1–81.CrossRefGoogle Scholar
40Holm, D. D., Jeffery, C., Kurien, S., Livescu, D., Taylor, M. A. and Wingate, B. A.. The LANS-α model for computing turbulence. Los Alamos Sci. 29 (2005), 152–171.Google Scholar
41Holm, D. D., Schmah, T. and Stoica, C.. Geometric mechanics and symmetry: from finite to infinite dimensions.vol. 12 (Oxford University Press, 2009).Google Scholar
42Holm, D. D., Schmah, T. and Stoica, C.. Geometric Mechanics and Symmetry: from Finite to Infinite Dimensions.vol. 12 (Oxford University Press, 2009).Google Scholar
43Inoue, A. and Funaki, T.. On a new derivation of the Navier–Stokes equation. Commun. Math. Phys. 65 (1979), 83–90.CrossRefGoogle Scholar
44Iyer, G.. A stochastic perturbation of inviscid flows. Commun. Math. Phys 266 (2006), 631–645.CrossRefGoogle Scholar
45Kazantsev, A. P.. Enhancement of a magnetic field by a conducting fluid. Sov. Phys. JETP 26 (1968), 1031–1034.Google Scholar
46Kelvin, T.. On vortex motion. Trans. Roy. Soc. Edinb. 25 (1869), 217–260.Google Scholar
47Kraichnan, R. H.. Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11 (1968), 945–953.CrossRefGoogle Scholar
48Kraichnan, R. H. and Nagarajan, S.. Growth of turbulent magnetic fields. Phys. Fluids 10 (1967), 859–870.Google Scholar
49Krylov, N. V. and Rozovskii, B. L.. Stochastic evolution equations. Itogi Nauki i Tekhniki. Seriya “Sovremennye Problemy Matematiki. Noveishie Dostizheniya 14 (1979), 71–146.Google Scholar
50Kunita, H.. Some extensions of Ito's formula. Séminaire de Probabilités XV 1979/80, pp. 118–141 (Berlin, Heidelberg: Springer, 1981).CrossRefGoogle Scholar
51Kunita, H.. Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics,vol. 24 (Cambridge, UK: Cambridge Univ. Press, 1997).Google Scholar
52Le Jan, Y. and Watanabe, S.. Stochastic flows of diffeomorphisms. North-Holland Mathematical Library,vol. 32,pp. 307–332 (Elsevier, 1984).Google Scholar
53Leray, J.. Sur les movements d'un fluide visqueux remplaissant l'espace. Acta Math. 63 (1934), 193.CrossRefGoogle Scholar
54Marsden, J. E. and Hughes, T. J.. Mathematical Foundations of Elasticity. Courier Corporation (1994).Google Scholar
55Mémin, E.. Fluid flow dynamics under location uncertainty. Geophys. Astrophys. Fluid Dyn. 108 (2014), 119–146.CrossRefGoogle Scholar
56Mikulevicius, R. and Rozovskii, B. L.. Global L 2-solutions of stochastic Navier–Stokes equations. Annals of Probab. 33 (2005), 137–176.CrossRefGoogle Scholar
57Protter, P. E.. Stochastic differential equations. In Stochastic integration and differential equations, pp. 249–361 (Berlin, Heidelberg: Springer, 2005.CrossRefGoogle Scholar
58Rapoport, D. L.. Stochastic differential geometry and the random integration of the Navier–Stokes equations and the kinematic dynamo problem on smooth compact manifolds and Euclidean space. Hadronic J. 23 (2000), 637–675.Google Scholar
59Rapoport, D. L.. On the geometry of the random representations for viscous fluids and a remarkable pure noise representation. Rep. Math. Phys. 50 (2002), 211–250.CrossRefGoogle Scholar
60Resseguier, E., Mémin, E. and Chapron, B.. Geophysical flows under location uncertainty, Part I: Random transport and general models. Geophys. Astrophys. Fluid Dyn. 111 (2017a), 149–176.CrossRefGoogle Scholar
61Resseguier, E., Mémin, E. and Chapron, B.. Geophysical flows under location uncertainty, Part II: Quasigeostrophic models and efficient ensemble spreading. Geophys. Astrophys. Fluid Dyn. 111 (2017b), 177–208.CrossRefGoogle Scholar
62Resseguier, E., Mémin, E. and Chapron, B.. Geophysical flows under location uncertainty, Part III: SQG and frontal dynamics under strong turbulence. Geophys. Astrophys. Fluid Dyn. 111 (2017c), 209–227.Google Scholar
63Rezakhanlou, F.. Regular flows for diffusions with rough drifts. arXiv preprint arXiv:1405.5856 (2014).Google Scholar
64Rezakhanlou, F.. Stochastically symplectic maps and their applications to the Navier–Stokes equation. Annales de l'Institut Henri Poincare (C) Non Linear Analysis.vol. 33, No.1 (Masson: Elsevier, 2016).Google Scholar
65Serre, D.. Helicity and other conservation laws in perfect fluid motion. Comptes Rendus Mécanique 346 (2018), 175–183.CrossRefGoogle Scholar
66Shizan, F. and Luo, D.. Constantin and Iyer's Representation Formula for the Navier–Stokes Equations on Manifolds. Potential Anal. 48 (2018), 181–206.Google Scholar