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Stability of the Self-similar Dynamics of a Vortex Filament

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Abstract

In this paper we continue our investigation of self-similar solutions of the vortex filament equation, also known as the binormal flow or the localized induction equation. Our main result is the stability of the self-similar dynamics of small perturbations of a given self-similar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic one-dimensional Schrödinger equation, connected to the binormal flow by Hasimoto’s transform.

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Authors and Affiliations

  1. Laboratoire Analyse et probabilités (EA 2172), Déptartement de Mathématiques, Université d’Evry, 23 Bd. de France, 91037, Evry, France

    Valeria Banica

  2. Departamento de Matematicas, Universidad del Pais Vasco, Aptdo. 644, 48080, Bilbao, Spain

    Luis Vega

Authors
  1. Valeria Banica
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  2. Luis Vega
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Correspondence to Valeria Banica.

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Communicated by L. Saint-Raymond

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Banica, V., Vega, L. Stability of the Self-similar Dynamics of a Vortex Filament. Arch Rational Mech Anal 210, 673–712 (2013). https://doi.org/10.1007/s00205-013-0660-6

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  • DOI: https://doi.org/10.1007/s00205-013-0660-6

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