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**Equilibrium statistical theory for nearly parallel vortex filaments.**
*(English)*
Zbl 1041.76038

In this outstanding paper the first mathematically rigorous equilibrium statistical theory for three-dimensional vortex filaments is developed in the context of the simplified asymptotic equations for nearly parallel vortex filaments, which have been derived recently by Klein, Majda, and Damodaran. These simplified equations arise from a systematic asymptotic expansion of the Navier-Stokes equation and involve the motion of families of curves, representing the vortex filaments, under linearized self-induction and mutual potential vortex interaction. The authors consider the equilibrium statistical mechanics of arbitrarily large numbers of nearly parallel filaments with equal circulations. First, the equilibrium Gibbs ensemble is written down exactly through function space integrals; then a suitably scaled mean field statistical theory is developed in the limit of infinitely many interacting filaments. The mean field equations involve a novel Hartree-like problem with a two-body logarithmic interaction potential and an inverse temperature given by the normalized length of the filaments. They analyze the mean field problem and show various equivalent variational formulations of it. The mean field statistical theory for nearly parallel vortex filaments is compared and contrasted with the well-known mean field statistical theory for two-dimensional point vortices.

The main ideas are first introduced through heuristic reasoning and then are confirmed by a mathematically rigorous analysis. A potential application of this statistical theory to rapidly rotating convection in geophysical flows is also discussed briefly.

The main ideas are first introduced through heuristic reasoning and then are confirmed by a mathematically rigorous analysis. A potential application of this statistical theory to rapidly rotating convection in geophysical flows is also discussed briefly.

Reviewer: Olaf Ninnemann (Berlin)

### MSC:

76B47 | Vortex flows for incompressible inviscid fluids |

76D17 | Viscous vortex flows |

82D15 | Statistical mechanics of liquids |

76F02 | Fundamentals of turbulence |

35Q30 | Navier-Stokes equations |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76F55 | Statistical turbulence modeling |

76M35 | Stochastic analysis applied to problems in fluid mechanics |

82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |

86A05 | Hydrology, hydrography, oceanography |

### Keywords:

equilibrium Gibbs ensemble; two-body logarithmic interaction potential; rapidly rotating convection### Citations:

Zbl 0846.76015
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\textit{P.-L. Lions} and \textit{A. Majda}, Commun. Pure Appl. Math. 53, No. 1, 76--142 (2000; Zbl 1041.76038)

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### References:

[1] | Angelescu, J Statist Phys 74 pp 147– (1994) |

[2] | Caglioti, Comm Math Phys 143 pp 501– (1992) |

[3] | Caglioti, Comm Math Phys 174 pp 229– (1995) |

[4] | Chorin, Commun Math Phys 83 pp 517– (1982) |

[5] | Chorin, Phys Rev Lett 60 pp 1947– (1988) |

[6] | Vorticity and turbulence. Applied Mathematical Sciences, 103. Springer, New York, 1994. · Zbl 0795.76002 |

[7] | Chorin, Phys D 52 pp 403– (1991) |

[8] | ; A mathematical introduction to fluid mechanics. Second edition. Texts in Applied Mathematics, 4. Springer, New York, 1990. · Zbl 0712.76008 |

[9] | Some applications of functional integration in statistical mechanics. Statistical mechanics and quantum field theory, 327-427. and , eds. Gordon and Breach, New York, 1971. |

[10] | Joyce, J Plasma Phys 10 pp 107– (1973) |

[11] | Julien, J Fluid Mech 332 pp 243– (1996) |

[12] | Kiessling, Comm Pure Appl Math 46 pp 27– (1993) |

[13] | Klein, J Fluid Mech 228 pp 201– (1995) |

[14] | Hydrodynamics. Reprint of the 1932 sixth edition. With a foreword by R. A. Caflisch [Russel E. Caflisch]. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1993. |

[15] | Lebowitz, J Statist Phys 50 pp 657– (1988) |

[16] | Hartree-Fock and related equations. Nonlinear partial differential equations and their applications. Collèege de France Seminar, Vol. IX (Paris, 1985-1986), 304-333. Pitman Res Notes Math Ser, 181. Longman Sci. Tech., Harlow, 1988. |

[17] | On Euler equations and statistical physics. Cattedra Galileiana [Galileo Chair], 1. Scuola Normale Superiore, Classe di Scienze, Pisa, 1998. |

[18] | Simplified asymptotic equations for slender vortex filaments. Recent advances in partial differential equations, Venice 1996, 237-280. Proc Sympos Appl Math, 54. Amer. Math. Soc., Providence, R.I., 1998. |

[19] | ; Vorticity and the mathematical theory of incompressible flow. Cambridge University Press, Cambridge, 2000, to appear. |

[20] | Montgomery, Phys Fluids 17 pp 1139– (1974) |

[21] | Onsager, Nuovo Cimento (9) 6 pp 279– (1949) |

[22] | Statistical mechanics rigorous results. W. A. Benjamin, New York-Amsterdam, 1969. · Zbl 0177.57301 |

[23] | Functional integration and quantum physics. Pure and Applied Mathematics, 86. Academic Press [Harcourt Brace Jovanovich, Publishers] New York-London, 1979. · Zbl 0434.28013 |

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