It is shown that the helicity of three-dimensional viscous incompressible flow can be identified with the overall linking of the fluid’s initial vorticity to the expectation of a stochastic mean field limit. The relevant mean field limit is obtained by following the Lagrangian paths in the stochastic Hamiltonian interacting particle system of Hochgerner [S. Hochgerner, Proc. R. Soc. A 474 (2018) 20180178].
Accepté le :
Publié le :
Simon Hochgerner 1
CC-BY 4.0
@article{CRMECA_2022__350_G2_283_0,
author = {Simon Hochgerner},
title = {Probabilistic representation of helicity in viscous fluids},
journal = {Comptes Rendus. M\'ecanique},
pages = {283--295},
publisher = {Acad\'emie des sciences, Paris},
volume = {350},
year = {2022},
doi = {10.5802/crmeca.116},
language = {en},
}
Simon Hochgerner. Probabilistic representation of helicity in viscous fluids. Comptes Rendus. Mécanique, Volume 350 (2022), pp. 283-295. doi : 10.5802/crmeca.116. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.116/
[1] The degree of knottedness of tangled vortex lines, J. Fluid Mech., Volume 35 (1969), pp. 117-129 | DOI | Zbl
[2] Helicity and the Călugăreanu invariant, Proc. R. Soc. Lond. A, Volume 439 (1992), pp. 411-429 | Zbl
[3] Helicity in laminar and turbulent flow, Annu. Rev. Fluid Mech., Volume 24 (1992), pp. 281-312 | DOI | Zbl
[4] Sur la géométrie différentielle de groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier, Volume 16 (1966) no. 1, pp. 319-361 | DOI | Zbl
[5] Constantes d’un îlot tourbillonnaire en fluide parfait barotrope, C. R. Hebd. Séances Acad. Sci., Volume 252 (1961), pp. 2810-2812 | Zbl
[6] Helicity conservation by flow across scales in reconnecting vortex links and knots, Proc. Natl. Acad. Sci. USA, Volume 111 (2014) no. 43, pp. 15350-15355 | DOI
[7] Conservation of writhe helicity under anti-parallel reconnection, Sci. Rep., Volume 5 (2015), 9224 | DOI
[8] Simulated Navier–Stokes trefoil reconnection (Preprint) | arXiv
[9] Trefoil knot structure during reconnection (Preprint) | arXiv
[10] Stochastic mean field approach to fluid dynamics, J. Nonlinear Sci., Volume 28 (2018) no. 2, pp. 725-737 | DOI | MR | Zbl
[11] A Hamiltonian mean field system for the Navier–Stokes equation, Proc. R. Soc. A, Volume 474 (2018) no. 2218, 20180178 | DOI | MR | Zbl
[12] A Hamiltonian interacting particle system for compressible flow, Water, Volume 12 (2020) no. 8, 2109 | DOI | MR
[13] A martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Prob., Volume 12 (1984) no. 2, pp. 458-479 | DOI | MR | Zbl
[14] Stochastic McKean–Vlasov equations, NoDEA, Volume 2 (1995), pp. 199-229 | DOI | MR | Zbl
[15] Mean field limit for stochastic particle systems, Active Particles (N. Bellomo; P. Degond; E. Tadmor, eds.) (Modeling and Simulation in Science, Engineering and Technology), Volume 1, Birkhäuser, Basel, pp. 379-402
[16] A stochastic Lagrangian representation of the three-dimensional incompressible Navier–Stokes equations, Comm. Pure Appl. Math., Volume 61 (2008) no. 3, pp. 330-345 | DOI | MR | Zbl
[17] Variational principles for stochastic fluid dynamics, Proc. R. Soc. A, Volume 471 (2015) no. 2176, 20140963 | MR | Zbl
[18] Topological Methods in Hydrodynamics, Springer, New York, 1998 | DOI
[19] Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., Volume 92 (1970) no. 1, pp. 1037-1041 | DOI | MR | Zbl
[20] Diffeomorphism groups, hydrodynamics and relativity, Proceedings of the 13th Biennial Seminar of Canadian Mathematical Congress (1972), pp. 135-279 | Zbl
[21] Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the hamiltonian approach, Phase Space Analysis of Partial Differential Equations (A. Bove; F. Colombini; D. D. Santo, eds.) (Progress in Non Linear Differential Equations and their Applications), Volume 69, Birkhauser Verlag, Basel, 2006, pp. 133-215 | DOI | MR | Zbl
[22] Stochastic integration and differential equations, Stochastic Modelling and Applied Probability, Springer, New York, 2005 | MR
[23] A semilinear Mckean–Vlasov stochastic evolution equation in Hilbert space, Stoch. Proc. Appl., Volume 60 (1995) no. 1, pp. 65-85 | DOI | MR | Zbl
[24] Remarks on a paper by Gavrilov: Grad–Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications, Geom. Funct. Anal., Volume 29 (2019), pp. 1773-1793 | DOI | MR | Zbl
Cité par Sources :
Commentaires - Politique