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Collisions of Vortex Filament Pairs

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Abstract

We consider the problem of collisions of vortex filaments for a model introduced by Klein et al. (J Fluid Mech 288:201–248, 1995) and Zakharov (Sov Phys Usp 31(7):672–674, 1988, Lect. Notes Phys 536:369–385, 1999) to describe the interaction of almost parallel vortex filaments in three-dimensional fluids. Since the results of Crow (AIAA J 8:2172–2179, 1970) examples of collisions are searched as perturbations of antiparallel translating pairs of filaments, with initial perturbations related to the unstable mode of the linearized problem; most results are numerical calculations. In this article, we first consider a related model for the evolution of pairs of filaments, and we display another type of initial perturbation leading to collision in finite time. Moreover, we give numerical evidence that it also leads to collision through the initial model. We finally study the self-similar solutions of the model.

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Notes

  1. In the sense that \(\Psi _1(t)=-\overline{\Psi }_2(t)\).

References

  • Bambusi, D., Faou, E., Grébert, B.: Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation. Numer. Math. 123, 461–492 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  • Banica, V., Miot, E.: Global existence and collisions for symmetric configurations of nearly parallel vortex filaments. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, 813–832 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Banica, V., Miot, E.: Evolution, interaction and collisions of vortex filaments. Differ. Integral Equ. 26, 355–388 (2013)

    MATH  MathSciNet  Google Scholar 

  • Bona, J.L., Ponce, G., Saut, J.-C., Sparber, C.: Dispersive blow up for nonlinear Schrödinger equations revisited. J. Math. Pure. Appl. (to appear, 2014)

  • Craig, W., Garcìa-Azpeitia, C.: https://www.math.uzh.ch/nhpde12/fileadmin/nhpde12/pdf/Craig_Monday1030Ascona2012.pdf

  • Crow, S.C.: Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179 (1970)

    Article  Google Scholar 

  • Faou, E.: Geometric numerical integration and Schrödinger equations. In: Zurich Lectures in Advanced Mathematics, xiii\(+\)138p. European Mathematical Society (EMS), Zürich (2012)

  • Kenig, C., Ponce, G., Vega, L.: On the interaction of nearly parallel vortex filaments. Commun. Math. Phys. 243, 471–483 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Kerr, R.M.: Evidence for a singularity of the three-dimensional, incompressible Euler equations. Phys. Fluids A 5, 1725–1746 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Klein, R., Majda, A.J., Damodaran, K.: Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  • Lions, P.-L., Majda, A.J.: Equilibrium statistical theory for nearly parallel vortex filaments. Commun. Pure Appl. Math. 53, 76–142 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  • Merle, F., Zaag, H.: Reconnection of vortex with the boundary and finite time quenching. Nonlinearity 10, 1497–1550 (1997a)

  • Merle, F., Zaag, H.: Stability of the blow-up profile for equations of the type \(u_t=\Delta u+\vert u\vert ^{p-1}u\). Duke Math. J. 86, 143–195 (1997b)

  • Zakharov, V.E.: Wave collapse. Sov. Phys. Usp. 31(7), 672–674 (1988)

    Article  Google Scholar 

  • Zakharov, V.E.: Quasi-two-dimensional hydrodynamics and interaction of vortex tubes. Lect. Notes Phys. 536, 369–385 (1999)

    Article  Google Scholar 

Download references

Acknowledgments

Valeria Banica and Evelyne Miot are partially supported by the French ANR Project SchEq ANR-12-JS-0005-01. Erwan Faou is supported by the ERC starting Grant GEOPARDI No. 279389. Evelyne Miot is partially supported by the French ANR Project GEODISP ANR-12-BS01-0015-01.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et de Modélisation d’Évry (UMR 8071), Université d’Évry, 23 Bd. de France, 91037, Evry, France

    Valeria Banica

  2. Département de mathématiques et applications (UMR 8553), ENS, 45 rue d’Ulm, 75005, Paris, France

    Erwan Faou

  3. Centre de Mathématiques Laurent Schwartz (UMR 7640), École Polytechnique, 91128, Palaiseau, France

    Evelyne Miot

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  1. Valeria Banica
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  2. Erwan Faou
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Correspondence to Valeria Banica.

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Communicated by Paul Newton.

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Banica, V., Faou, E. & Miot, E. Collisions of Vortex Filament Pairs. J Nonlinear Sci 24, 1263–1284 (2014). https://doi.org/10.1007/s00332-014-9218-5

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  • DOI: https://doi.org/10.1007/s00332-014-9218-5

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