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Global existence for the defocusing nonlinear Schrödinger equations with limit periodic initial data
The dynamics of vortex filaments with corners
| 1. | Departamento de Matemáticas, UPV/EHU, Apdo 644, 48080 Bilbao, Spain |
References:
| [1] |
V. Banica and L. Vega, On the stability of a singular vortex dynamics, Comm. Math. Phys., 286 (2009), 593-627.
doi: 10.1007/s00220-008-0682-3. |
| [2] |
V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc.l, 14 (2012), 209-253.
doi: 10.4171/JEMS/300. |
| [3] |
V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament, Arch. Ration. Mech. Anal., 210 (2013), 673-712.
doi: 10.1007/s00205-013-0660-6. |
| [4] |
V. Banica and L. Vega, The initial value problem for the binormal flow with rough data, preprint, arXiv:1304.0996. |
| [5] |
T. F. Buttke, A numerical study of superfluid turbulence in the Self Induction Approximation, J. of Compt. Physics, 76 (1988), 301-326. |
| [6] |
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. |
| [7] |
M. B. Erdogan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus, preprint, arXiv:1303.3604.
doi: 10.4310/MRL.2013.v20.n6.a7. |
| [8] |
S. Jaffard, The spectrum of singularities of Riemanns function, Rev. Mat. Iberoamericana, 12 (1996), 44-460.
doi: 10.4171/RMI/203. |
| [9] |
U. Frisch and G. Parisi, Fully developed turbulence and intermittency, in Proc. Int. Sch. Phys. Enrico Fermi, North-Holland, Amsterdam, 1985. |
| [10] |
U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. |
| [11] |
L. Kapitanski and I. Rodnianski, Does a Quantum Particle Know the Time? Emerging Applications of Number Theory, IMA Vol. Math. Appl., 109 (1999), 355-371.
doi: 10.1007/978-1-4612-1544-8_14. |
| [12] |
S. Gutiérrez and L. Vega, Self-similar solutions of the localized induction approximation: singularity formation, Nonlinearity, 17 (2004), 2091-2136.
doi: 10.1088/0951-7715/17/6/006. |
| [13] |
S. Gutiérrez and L. Vega, On the stability of self-similar solutions of 1D cubic Schrödinger equations, Math. Ann., 356 (2013), 259-300.
doi: 10.1007/s00208-012-0847-4. |
| [14] |
S. Gutiérrez, J. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Part. Diff. Eq., 28 (2003), 927-968.
doi: 10.1081/PDE-120021181. |
| [15] |
H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech., 51 (1972), 477-485. |
| [16] |
J. C. Hardin, The velocity field induced by a helical vortex filament, Phys. Fluids, 25 (1982), 1949-1952. |
| [17] |
F. de la Hoz, Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane, Math. Z., 257 (2007), 61-80.
doi: 10.1007/s00209-007-0115-6. |
| [18] |
F. de la Hoz and L. Vega, Vortex Filament Equation for a Regular Polygon, prepint, arXiv:1304.5521. |
| [19] |
F. de la Hoz, C. García-Cervera and L. Vega, A numerical study of the self-similar solutions of the Schrödinger Map, SIAM J. Appl. Math., 70 (2009), 1047-1077.
doi: 10.1137/080741720. |
| [20] |
C. E. Kenig, G. Ponce, and L.Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633.
doi: 10.1215/S0012-7094-01-10638-8. |
| [21] |
M. Lakshmanan and M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems, Physica A, 107 (1981), 533-552.
doi: 10.1016/0378-4371(81)90186-2. |
| [22] |
M. Lakshmanan, T. W. Ruijgrok and C. J. Thompson, On the the dynamics of a continuum spin system, Physica A, 84 (1976), 577-590. |
| [23] |
T, Lipniacki, Quasi-static solutions for quantum vortex motion under the localized induction approximation, J. Fluid Mech., 477 (2003), 321-337.
doi: 10.1017/S0022112002003282. |
| [24] |
K. I. Oskolkov, A class of I. M. Vinogradov's series and its applications in harmonic analysis, in Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math. 19, Springer, New York, 1992, 353-402.
doi: 10.1007/978-1-4612-2966-7_16. |
| [25] |
C. S. Peskin and D. M. McQueen, Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets, Am. J. Physiol. 266 (Heart Circ. Physiol. 35), (1994), H319-H328. |
| [26] |
R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dynam. Res., 18 (1996), 245-268.
doi: 10.1016/0169-5983(96)82495-6. |
| [27] |
R. L. Ricca, Rediscovery of Da Rios equations, Nature, 352 (1991), 561-562. |
| [28] |
P. G. Saffman, Vortex dynamics, in Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. |
show all references
References:
| [1] |
V. Banica and L. Vega, On the stability of a singular vortex dynamics, Comm. Math. Phys., 286 (2009), 593-627.
doi: 10.1007/s00220-008-0682-3. |
| [2] |
V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc.l, 14 (2012), 209-253.
doi: 10.4171/JEMS/300. |
| [3] |
V. Banica and L. Vega, Stability of the self-similar dynamics of a vortex filament, Arch. Ration. Mech. Anal., 210 (2013), 673-712.
doi: 10.1007/s00205-013-0660-6. |
| [4] |
V. Banica and L. Vega, The initial value problem for the binormal flow with rough data, preprint, arXiv:1304.0996. |
| [5] |
T. F. Buttke, A numerical study of superfluid turbulence in the Self Induction Approximation, J. of Compt. Physics, 76 (1988), 301-326. |
| [6] |
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. |
| [7] |
M. B. Erdogan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus, preprint, arXiv:1303.3604.
doi: 10.4310/MRL.2013.v20.n6.a7. |
| [8] |
S. Jaffard, The spectrum of singularities of Riemanns function, Rev. Mat. Iberoamericana, 12 (1996), 44-460.
doi: 10.4171/RMI/203. |
| [9] |
U. Frisch and G. Parisi, Fully developed turbulence and intermittency, in Proc. Int. Sch. Phys. Enrico Fermi, North-Holland, Amsterdam, 1985. |
| [10] |
U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, 1995. |
| [11] |
L. Kapitanski and I. Rodnianski, Does a Quantum Particle Know the Time? Emerging Applications of Number Theory, IMA Vol. Math. Appl., 109 (1999), 355-371.
doi: 10.1007/978-1-4612-1544-8_14. |
| [12] |
S. Gutiérrez and L. Vega, Self-similar solutions of the localized induction approximation: singularity formation, Nonlinearity, 17 (2004), 2091-2136.
doi: 10.1088/0951-7715/17/6/006. |
| [13] |
S. Gutiérrez and L. Vega, On the stability of self-similar solutions of 1D cubic Schrödinger equations, Math. Ann., 356 (2013), 259-300.
doi: 10.1007/s00208-012-0847-4. |
| [14] |
S. Gutiérrez, J. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Comm. Part. Diff. Eq., 28 (2003), 927-968.
doi: 10.1081/PDE-120021181. |
| [15] |
H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech., 51 (1972), 477-485. |
| [16] |
J. C. Hardin, The velocity field induced by a helical vortex filament, Phys. Fluids, 25 (1982), 1949-1952. |
| [17] |
F. de la Hoz, Self-similar solutions for the 1-D Schrödinger map on the hyperbolic plane, Math. Z., 257 (2007), 61-80.
doi: 10.1007/s00209-007-0115-6. |
| [18] |
F. de la Hoz and L. Vega, Vortex Filament Equation for a Regular Polygon, prepint, arXiv:1304.5521. |
| [19] |
F. de la Hoz, C. García-Cervera and L. Vega, A numerical study of the self-similar solutions of the Schrödinger Map, SIAM J. Appl. Math., 70 (2009), 1047-1077.
doi: 10.1137/080741720. |
| [20] |
C. E. Kenig, G. Ponce, and L.Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633.
doi: 10.1215/S0012-7094-01-10638-8. |
| [21] |
M. Lakshmanan and M. Daniel, On the evolution of higher dimensional Heisenberg continuum spin systems, Physica A, 107 (1981), 533-552.
doi: 10.1016/0378-4371(81)90186-2. |
| [22] |
M. Lakshmanan, T. W. Ruijgrok and C. J. Thompson, On the the dynamics of a continuum spin system, Physica A, 84 (1976), 577-590. |
| [23] |
T, Lipniacki, Quasi-static solutions for quantum vortex motion under the localized induction approximation, J. Fluid Mech., 477 (2003), 321-337.
doi: 10.1017/S0022112002003282. |
| [24] |
K. I. Oskolkov, A class of I. M. Vinogradov's series and its applications in harmonic analysis, in Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math. 19, Springer, New York, 1992, 353-402.
doi: 10.1007/978-1-4612-2966-7_16. |
| [25] |
C. S. Peskin and D. M. McQueen, Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets, Am. J. Physiol. 266 (Heart Circ. Physiol. 35), (1994), H319-H328. |
| [26] |
R. L. Ricca, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dynam. Res., 18 (1996), 245-268.
doi: 10.1016/0169-5983(96)82495-6. |
| [27] |
R. L. Ricca, Rediscovery of Da Rios equations, Nature, 352 (1991), 561-562. |
| [28] |
P. G. Saffman, Vortex dynamics, in Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992. |
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