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Dynamics of nearly parallel vortex filaments for the Gross–Pitaevskii equation

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Abstract

Klein et al. (J Fluid Mech 288:201–248, 1995) have formally derived a simplified asymptotic motion law for the evolution of nearly parallel vortex filaments in the context of the three dimensional Euler equation for incompressible fluids. In the present work, we rigorously derive the corresponding asymptotic motion law in the context of the Gross–Pitaevskii equation.

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Notes

  1. Described further down, otherwise they wouldn’t be anything close to parallel!

  2. Since a rescaling will eventually be made in the description that sends the lateral boundary to infinity, the exact shape of \(\omega \) is of limited impact on the analysis, and the limit flow for the filaments does not depend at all on \(\omega .\) Still, some of our later assumptions for establishing convergence do depend on \(\omega \), see e.g. (9).

  3. The other two components of the 3D Jacobian also have interpretations, see e.g. Proposition 2 below, but they do not enter in the statement of our main theorem.

  4. After this work was completed, Dávila, del Pino, Musso and Wei have announced the construction of solutions to the Euler equation exhibiting the leapfrogging phenomenon.

  5. We will not need the exact definition of \(\kappa (n,\varepsilon ,\omega )\) or \(\gamma \) in this paper, but these constants will appear in various formulas.

  6. Note that the sets \({\mathcal {G}}^\varepsilon _2\) and \({\mathcal {B}}^\varepsilon _2\) from [7] coincide exactly with our sets \({\mathcal {G}}(u_\varepsilon )\) and \({\mathcal {B}}(u_\varepsilon )\); compare our definitions (45) with [7], equation (3.16).

  7. These results assume that (54), (55) hold for every \(z\in I\), but the proofs extend to our situation with essentially no change.

  8. after extracting a uniformly convergent subsequence of \(\{ \tilde{f}^\varepsilon \}\)

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Acknowledgements

This work was partially supported by the ANR project ODA (ANR-18-CE40-0020-01) of the Agence Nationale de la Recherche, and by the Natural Sciences and Engineering Research Council of Canada under Operating Grant 261955. The meticulous reading of the manuscript and pertinent suggestions made by an anonymous referee were highly appreciated.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    R. L. Jerrard

  2. Laboratoire Jacques-Louis Lions, Sorbonne Université, 4 place Jussieu BC 187, 75252, Paris Cedex 05, France

    D. Smets

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  1. R. L. Jerrard
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  2. D. Smets
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Correspondence to R. L. Jerrard.

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Communicated by A. Malchiodi.

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Jerrard, R.L., Smets, D. Dynamics of nearly parallel vortex filaments for the Gross–Pitaevskii equation. Calc. Var. 60, 127 (2021). https://doi.org/10.1007/s00526-021-01984-w

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