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On the Stability of a Singular Vortex Dynamics

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Abstract

In this paper we address the question of the singular vortex dynamics exhibited in [15], which generates a corner in finite time. The purpose is to prove that under some appropriate small regular perturbation the corner still remains. Our approach uses the Hasimoto transform and deals with the long range scattering properties of a Gross-Pitaevski equation with time-variable coefficients.

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References

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

  2. Banica V., Vega L.: On the Dirac delta as initial condition for nonlinear Schrödinger equations. Ann. I. H. Poincaré, (c) Non-lin Anal 52(7), 697–711 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  3. Batchelor G.K.: An Introduction to the Fluid Dynamics. Cambridge University Press, Cambridge (1967)

    Google Scholar 

  4. Betchov R.: On the curvature and torsion of an isolated filament. J. Fluid Mech. 22, 471 (1965)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Bourgain J., Wang W.: Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(1–2), 197–215 (1997)

    MATH  MathSciNet  Google Scholar 

  6. Buttke T.F.: A numerical study of superfluid turbulence in the Self-Induction Approximation. J. of Comp. Physics 76, 301–326 (1988)

    Article  MATH  ADS  Google Scholar 

  7. Carles R.: Geometric Optics and Long Range Scattering for One-Dimensional Nonlinear Schrödinger Equations. Commun. Math. Phys. 220(1), 41–67 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Christ, M.: Power series solution of a nonlinear Schrödinger equation. In: Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud. 163, Princeton, NJ: Princeton Univ. Press, 2007, pp. 131–155

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

    Article  MATH  Google Scholar 

  10. De la Hoz F.: Self-similar solutions for the 1-D Schrödinger map on the Hyperbolic plane. Math. Z. 257, 61–80 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fefferman C.: Inequalities for strongly singular convolution operators. Acta Math. 124, 9–36 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ginibre J., Velo G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. I. H. Poincaré, An. Non Lin. 2(4), 309–327 (1985)

    MATH  MathSciNet  Google Scholar 

  13. Grünrock, A.: Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS. Int. Math. Res. Not. 41, 2525–2558 (2005)

    Google Scholar 

  14. Gustafson S., Nakanishi K., Tsai T.-P.: Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions. Ann. Henri Poincaré 8(7), 1303–1331 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gutiérrez S., Rivas J., Vega L.: Formation of singularities and self-similar vortex motion under the localized induction approximation. Comm. Part. Diff. Eq. 28, 927–968 (2003)

    Article  MATH  Google Scholar 

  16. Hasimoto H.: A soliton on a vortex filament. J. Fluid Mech. 51, 477–485 (1972)

    Article  MATH  ADS  Google Scholar 

  17. Hayashi N., Naumkin P.: Domain and range of the modified wave operator for Schrödinger equations with critical nonlinearity. Commun. Math. Phys. 267(2), 477–492 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Koiso, N.: Vortex filament equation and semilinear Schrödinger equation. In: Nonlinear waves (Sapporo, 1995) GAKUTO Internat. Ser. Math. Sci. Appl., 10, Tokyo: Gakkōtosho, 1997, pp. 231–236

  19. Kenig C., Ponce G., Vega L.: On the ill-posedness of some canonical non-linear dispersive equations. Duke Math. J. 106(3), 617–633 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lakshmanan M., Daniel M.: On the evolution of higher dimensional Heisenberg continuum spin systems. Physics A 107, 533–552 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  21. Lakshmanan M., Ruijgrok TH.W., Thompson C.J.: On the dynamics of a continuum spin system. Physica A 84, 577–590 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  22. Landau L.D.: Collected papers of L. D. Landau. New York, Gordon and Breach (1965)

    Google Scholar 

  23. Lipniacki T.: Quasi-static solutions for quantum vortex motion under the localized induction approximation. J. Fluid Mech. 477, 321–337 (2002)

    ADS  MathSciNet  Google Scholar 

  24. Lipniacki T.: Shape-preserving solutions for quantum vortex motion. Phys. Fluids 15, 6 (2003)

    Article  MathSciNet  Google Scholar 

  25. Nahmod, A., Shatah, J., Vega, L., Zeng, C.: Schrödinger Maps and their associated Frame Systems. Int. Math. Res. Not. 2007, article ID mm 088, 29 pages, 2007

  26. Ozawa T.: Long range scattering for nonlinear Schrödinger equations in one space dimension. Commun. Math. Phys. 139(3), 479–493 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Ricca R.L.: Physical interpretation of certain invariants for vortex filament motion under LIA. Phys. Fluids A 4, 938–944 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Ricca R.L.: The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics. Fluid Dynam. Res. 18(5), 245–268 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Saffman, P.G.: Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics, New York: Cambridge U. Press, 1992

  30. Schwarz K.W.: Three-dimensional vortex dynamics in superfluid 4 He: line-line and line-boundary interactions. Phys. Rev. B 31, 5782 (1985)

    Article  ADS  Google Scholar 

  31. Strichartz R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equation. Duke Math. J. 44(3), 705–714 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tsutsumi Y., Yajima K.: The asymptotic behavior of nonlinear Schrödinger equations. Bull. Amer. Math. Soc. (N.S.) 11(1), 186–188 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  33. Vargas A., Vega L.: Global wellposedness of 1D cubic nonlinear Schrödinger equation for data with infinity L 2 norm. J. Math. Pures Appl. 80(10), 1029–1044 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Département de Mathématiques, Université d’Evry, Evry Cedex, France

    V. Banica

  2. Departamento de Matemticas, Universidad del Pais Vasco, Bilbao, Spain

    L. Vega

Authors
  1. V. Banica
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  2. L. Vega
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Correspondence to L. Vega.

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Communicated by P. Constantin

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Banica, V., Vega, L. On the Stability of a Singular Vortex Dynamics. Commun. Math. Phys. 286, 593–627 (2009). https://doi.org/10.1007/s00220-008-0682-3

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  • DOI: https://doi.org/10.1007/s00220-008-0682-3

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