Sur la stabilité d’une dynamique singulière de vortex
Séminaire Équations aux dérivées partielles (Polytechnique) (2007-2008), Talk no. 3, 11 p.

On étudie la stabilité de la dynamique singulière de vortex filamentaire décrite dans [13], qui engendre un coin en temps fini. On montre que sous certaines perturbations petites et régulières, le coin est encore formé. Notre approche utilise le flot binormal et la transformation de Hasimoto. On se ramène aux propriétés de scattering longue portée pour une équation de type Gross-Pitaesvski avec coefficients variables en temps. Ce travail a été obtenu en collaboration avec Luis Vega.

@article{SEDP_2007-2008____A3_0,
     author = {Banica, Valeria},
     title = {Sur la stabilit\'e d'une dynamique singuli\`ere de vortex},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2007-2008},
     note = {talk:3},
     language = {fr},
     url = {http://www.numdam.org/item/SEDP_2007-2008____A3_0}
}
Banica, Valeria. Sur la stabilité d’une dynamique singulière de vortex. Séminaire Équations aux dérivées partielles (Polytechnique) (2007-2008), Talk no. 3, 11 p. http://www.numdam.org/item/SEDP_2007-2008____A3_0/

[1] R.J. Arms, F.R. Hama, Localized-induction concept on a curved vortex and motion of an elliptic vortex ring, Phys. Fluids, (1965), 553.

[2] V. Banica, L. Vega, On the Dirac delta as initial condition for nonlinear Schrödinger equations, à paraître Ann. I. H. Poincaré, An. Non Lin.

[3] G.K. Batchelor, An Introduction to the Fluid Dynamics, Cambridge University Press, Cambridge, 1967. | MR 1744638 | Zbl 0152.44402

[4] R. Betchov, On the curvature and torsion of an isolated filament, J. Fluid Mech. 22 (1965), 471. | MR 178656 | Zbl 0133.43803

[5] T. F. Buttke, A numerical study of superfluid turbulence in the Self-Induction Approximation, J. of Comp. Physics 76, (1988), 301-326. | Zbl 0639.76136

[6] R. Carles, Geometric Optics and Long Range Scattering for One-Dimensional Nonlinear Schrödinger Equations, Comm. Math. Phys. 220 (2001), no. 1, 41-67. | MR 1882399 | Zbl 1029.35211

[7] M. Lakshmanan, M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems, Physics A 107, (1981), 533-552 . | MR 624580

[8] M. Lakshmanan, TH. W. Ruijgrok, C.J. Thompson, On the dynamics of a continuum spin system, Physica A 84, (1976), 577-590. | MR 449262

[9] L.D. Landau, Collected papers of L. D. Landau, Gordon and Breach, New York, (1965). | MR 237287

[10] L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo 22 (1906), 117.

[11] F. de la Hoz, Self-similar solutions for the 1-D Schrödinger map on the Hyperbolic plane, Math. Z. 257 (2007), 61-80. | MR 2318570 | Zbl 1128.35099

[12] S. Gustafson, K. Nakanishi, T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, arXiv :math/0605655. | MR 2360438

[13] S. Gutiérrez, J. Rivas, L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Part. Diff. Eq. 28 (2003), 927-968. | MR 1986056 | Zbl 1044.35089

[14] H. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-485. | Zbl 0237.76010

[15] N. Hayashi, P. Naumkin, Domain and range of the modified wave operator for Schrödinger equations with critical nonlinearity, Comm. Math. Phys. 267 (2006), no. 2, 477-492. | MR 2249776 | Zbl 1113.81121

[16] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys. 139, no.3 (1991), 479-493. | MR 1121130 | Zbl 0742.35043

[17] R.L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dynam. Res. 18, no. 5 (1996), 245–268. | MR 1408546 | Zbl 1006.01505

[18] P.G. Saffman, Vortex dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge U. Press, New York, 1992. | MR 1217252 | Zbl 0777.76004

[19] M. Spivak, A comprehensive introduction to differential geometry. Vol. II.Second edition. Publish or Perish, Inc., Wilmington, Del., 1979. | MR 532831