[Évolution par le flot binormal de courbes à un coin]
Dans cet article on considère le flot binormal avec données initiales des courbes régulières partout sauf en un point où elles ont un coin. On montre sous des conditions appropriées sur la donnée initiale qu'il existe une unique solution régulière pour des temps strictement positifs et négatifs. De plus, cette solution satisfait le flot binormal en un sens faible et peut être vue comme une perturbation d'une solution auto-similaire bien choisie. Réciproquement, on montre aussi que si à temps on prend comme donnée initiale une petite perturbation régulière d'une solution auto-similaire, alors il existe une unique solution, qui à temps est régulière partout sauf en un point où elle a un coin de même angle que celui formé par la solution auto-similaire. Cette solution peut être prolongée aux temps négatifs. La preuve s'appuie sur les résultats des articles précédents [9], [2], [3] et [4] sur l'étude des petites perturbations des solutions auto-similaires. Un argument de compacité est utilisé pour éviter les conditions à poids imposées dans [4], ainsi qu'une analyse plus raffinée des asymptotiques en temps et en espace des vecteurs tangents et normaux.
In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique regular solution exists for strictly positive and strictly negative times. Moreover, this solution satisfies a weak version of the equation for all times and can be seen as a perturbation of a suitably chosen self-similar solution. Conversely, we also prove that if at time a small regular perturbation of a self-similar solution is taken as initial condition then there exists a unique solution that at time is regular except at a point where it has a corner with the same angle as the one of the self-similar solution. This solution can be extended for negative times. The proof uses the full strength of the previous papers [9], [2], [3] and [4] on the study of small perturbations of self-similar solutions. A compactness argument is used to avoid the weighted conditions we needed in [4], as well as a more refined analysis of the asymptotic in time and in space of the tangent and normal vectors.
DOI : 10.24033/asens.2273
Keywords: Vortex filaments, binormal flow, singular data, nonlinear Schrödinger equations, selfsimilar solutions, scattering.
Mot clés : Tourbillons filamentaires, flot binormal, données singulières, équations de Schrödinger non-linéaires, solutions autosimilaires, diffusion.
@article{ASENS_2015__48_6_1423_0,
author = {Banica, Valeria and Vega, Luis},
title = {The initial value problem for the {Binormal} {Flow} with rough data},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {1423--1455},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 48},
number = {6},
year = {2015},
doi = {10.24033/asens.2273},
mrnumber = {3429472},
zbl = {1336.53073},
language = {en},
url = {http://www.numdam.org/articles/10.24033/asens.2273/}
}
TY - JOUR AU - Banica, Valeria AU - Vega, Luis TI - The initial value problem for the Binormal Flow with rough data JO - Annales scientifiques de l'École Normale Supérieure PY - 2015 SP - 1423 EP - 1455 VL - 48 IS - 6 PB - Société Mathématique de France. Tous droits réservés UR - http://www.numdam.org/articles/10.24033/asens.2273/ DO - 10.24033/asens.2273 LA - en ID - ASENS_2015__48_6_1423_0 ER -
%0 Journal Article %A Banica, Valeria %A Vega, Luis %T The initial value problem for the Binormal Flow with rough data %J Annales scientifiques de l'École Normale Supérieure %D 2015 %P 1423-1455 %V 48 %N 6 %I Société Mathématique de France. Tous droits réservés %U http://www.numdam.org/articles/10.24033/asens.2273/ %R 10.24033/asens.2273 %G en %F ASENS_2015__48_6_1423_0
Banica, Valeria; Vega, Luis. The initial value problem for the Binormal Flow with rough data. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 6, pp. 1423-1455. doi : 10.24033/asens.2273. http://www.numdam.org/articles/10.24033/asens.2273/
Localized-induction concept on a curved vortex and motion of an elliptic vortex ring, Phys. Fluids, Volume 8 (1965), pp. 553-560 | DOI
A numerical study of superfluid turbulence in the Self Induction Approximation, J. of Compt. Physics, Volume 76 (1988), pp. 301-326 | DOI | Zbl
On the stability of a singular vortex dynamics, Comm. Math. Phys., Volume 286 (2009), pp. 593-627 (ISSN: 0010-3616) | DOI | MR | Zbl
Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc. (JEMS), Volume 14 (2012), pp. 209-253 (ISSN: 1435-9855) | DOI | MR | Zbl
Stability of the self-similar dynamics of a vortex filament, Arch. Ration. Mech. Anal., Volume 210 (2013), pp. 673-712 (ISSN: 0003-9527) | DOI | MR | Zbl
Stability of small-amplitude torus knot solutions of the localized induction approximation, J. Phys. A, Volume 44 (2011), 335204 pages (ISSN: 1751-8113) | DOI | MR | Zbl
On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo, Volume 22 (1906), 117 pages
A numerical study of the self-similar solutions of the Schrödinger map, SIAM J. Appl. Math., Volume 70 (2009), pp. 1047-1077 (ISSN: 0036-1399) | DOI | MR | Zbl
Three-dimensional distortions of a vortex filament with axial velocity, J. Fluid Mech., Volume 222 (1991), pp. 369-416 (ISSN: 0022-1120) | DOI | MR | Zbl
Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Partial Differential Equations, Volume 28 (2003), pp. 927-968 (ISSN: 0360-5302) | DOI | MR | Zbl
A soliton in a vortex filament, J. Fluid Mech., Volume 51 (1972), pp. 477-485 | DOI | MR | Zbl
On Schrödinger maps from to , Ann. Sci. Éc. Norm. Supér., Volume 45 (2012), pp. 637-680 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl
On the motion of a curve by its binormal curvature, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 1487-1515 (ISSN: 1435-9855) | DOI | MR | Zbl
, Nonlinear waves (Sapporo, 1995) (GAKUTO Internat. Ser. Math. Sci. Appl.), Volume 10, Gakkōtosho, Tokyo, 1997, pp. 231-236 | MR | Zbl
Stability of solitons on vortex filaments, Phys. Lett. A, Volume 377 (2013), pp. 766-769 (ISSN: 0375-9601) | DOI | MR | Zbl
On the evolution of higher-dimensional Heisenberg continuum spin systems, Phys. A, Volume 107 (1981), pp. 533-552 (ISSN: 0378-4371) | DOI | MR
On the dynamics of a continuum spin system, Phys. A, Volume 84 (1976), pp. 577-590 (ISSN: 0378-4371) | DOI | MR
Velocity, energy, and helicity of vortex knots and unknots, Phys. Rev. E, Volume 82 (2010), 026309 pages (ISSN: 1539-3755) | DOI | MR
Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets, Am. J. Physiol., Volume 266 (1994), p. H319-H328
Rediscovery of Da Rios equations, Nature, Volume 352 (1991), pp. 561-562 | DOI
The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dynam. Res., Volume 18 (1996), pp. 245-268 (ISSN: 0169-5983) | DOI | MR | Zbl
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