July 2018, 17(4): 1317-1329. doi: 10.3934/cpaa.2018064

Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$

1. 

Sorbonne Université, Laboratoire Jacques-Louis Lions (UMR 7598), BC 187, 4 place Jussieu, 75005 Paris, France

2. 

Departamento de Matemáticas, Universidad del Pais Vasco, BCAM Alameda Mazarredo 14, 48009 Bilbao, Spain

* Corresponding author: Luis Vega

Received  February 2017 Revised  June 2017 Published  April 2018

Fund Project: The first author is partially supported by ANR project "SchEq" ANR-12-JS01-0005-01. The second author is partially supported by MINECO projects MTM2014-53850-P and SEV-2013-0323. and by the Basque Government projects IT-641-13 and BERC 2014-2017. Both authors are partially supported by ERC Advanced Grant 2014 669689 - HADE

In this note we consider the 1-D cubic Schrödinger equation with data given as small perturbations of a Dirac-δ function and some other related equations. We first recall that although the problem for this type of data is ill-posed one can use the geometric framework of the Schrödinger map to define the solution beyond the singularity time. Then, we find some natural and well defined geometric quantities that are not regular at time zero. Finally, we make a link between these results and some known phenomena in fluid mechanics that inspired this note.

Citation: Valeria Banica, Luis Vega. Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1317-1329. doi: 10.3934/cpaa.2018064
References:
[1]

V. Banica and L. Vega, On the stability of a singular vortex dynamics, Comm. Math. Phys., 286 (2009), 593-627.

[2]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc., 14 (2012), 209-253.

[3]

V. Banica and L. Vega, The initial value problem for the binormal flow with rough data, Ann. Sci. Éc. Norm. Supér., 48 (2015), 1421-1453.

[4]

T. Cazenave and F.B. Weissler, The Cauchy problem for the critical nonlinear Schringer equation, Non. Anal. TMA, 14 (1990), 807-836.

[5]

M. Christ, Power series solution of a nonlinear Schringer equation. Mathematical aspects of nonlinear dispersive equations, 131–155, Ann. of Math. Stud., 163 (2007), Princeton Univ. Press, Princeton, NJ.

[6]

M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, arXiv: 0311048.

[7]

F. de la Hoz and L. Vega, Vortex filament equation for a regular polygon, Nonlinearity, 27 (2014), 3031-3057.

[8]

F. de la Hoz and L. Vega, On the relationship between the one-corner problem and the M-corner problem for the vortex filament equation, in preparation.

[9]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo, 22 (1906), 117-135.

[10]

M. B. Erdogan and N. Tzirakis, Talbot effect for the cubic non-linear Schringer equation on the torus, Math. Res. Lett., 20 (2013), 1081-1090.

[11]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅱ Scattering theory, general case, J. Funct. Anal., 32 (1979), 33-71.

[12]

F. F Grinstein and E. J. Gutmark, Flow control with noncircular jets, Annu. Rev. Fluid Mech., 31 (1999), 239-272.

[13]

F. F GrinsteinE. J. Gutmark and T. Parr, Near field dynamics of subsonic free square jets. A computational and experimental study, Physics of Fluids, 7 (1995), 1483-1497.

[14]

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

[15]

S. GutiérrezJ. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Commun. PDE, 28 (2003), 927-968.

[16]

R. Jerrard, C. Seis, On the vortex filament conjecture for Euler flows, arXiv: 1603.00227.

[17]

R. Jerrard and D. Smets, On the motion of a curve by its binormal curvature, J. Eur. Math. Soc. (JEMS), 17 (2015), 148-1515.

[18]

T. Kappeler and J. C. Molnar, On the wellposedness of the defocusing mKdV equation below L2, arXiv: 1606.07052.

[19]

C. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical non-linear dispersive equations, Duke Math. J., 106 (2001), 716-633.

[20]

H. Koch and D. Tataru, Conserved energies for the cubic NLS in 1-d, arXiv: 1607.02534.

[21]

R. Killip, M. Visan Monica and X. Zhang, Talk Bonn March 2016 and private communication, 2016.

[22]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flows, Cambridge Texts in Applied Mathematics, Cambridge U. Press, 2002.

[23]

P. J. Olver, Dispersive quantization, Am. Math. Monthly, 117 (2010), 599-610.

[24]

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

[25]

A. Vargas and L. Vega, Global well-posedness for 1d non-linear Schrödinger equation for data with an infinite L2 norm,, J. Math. Pures Appl., 80 (2001), 1029-1044.

[26]

L. Vega, The dynamics of vortex filaments with corners, Commun. Pure Appl. Anal., 14 (2015), 1581-1601.

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.

[2]

V. Banica and L. Vega, Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc., 14 (2012), 209-253.

[3]

V. Banica and L. Vega, The initial value problem for the binormal flow with rough data, Ann. Sci. Éc. Norm. Supér., 48 (2015), 1421-1453.

[4]

T. Cazenave and F.B. Weissler, The Cauchy problem for the critical nonlinear Schringer equation, Non. Anal. TMA, 14 (1990), 807-836.

[5]

M. Christ, Power series solution of a nonlinear Schringer equation. Mathematical aspects of nonlinear dispersive equations, 131–155, Ann. of Math. Stud., 163 (2007), Princeton Univ. Press, Princeton, NJ.

[6]

M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, arXiv: 0311048.

[7]

F. de la Hoz and L. Vega, Vortex filament equation for a regular polygon, Nonlinearity, 27 (2014), 3031-3057.

[8]

F. de la Hoz and L. Vega, On the relationship between the one-corner problem and the M-corner problem for the vortex filament equation, in preparation.

[9]

L. S. Da Rios, On the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo, 22 (1906), 117-135.

[10]

M. B. Erdogan and N. Tzirakis, Talbot effect for the cubic non-linear Schringer equation on the torus, Math. Res. Lett., 20 (2013), 1081-1090.

[11]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅱ Scattering theory, general case, J. Funct. Anal., 32 (1979), 33-71.

[12]

F. F Grinstein and E. J. Gutmark, Flow control with noncircular jets, Annu. Rev. Fluid Mech., 31 (1999), 239-272.

[13]

F. F GrinsteinE. J. Gutmark and T. Parr, Near field dynamics of subsonic free square jets. A computational and experimental study, Physics of Fluids, 7 (1995), 1483-1497.

[14]

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

[15]

S. GutiérrezJ. Rivas and L. Vega, Formation of singularities and self-similar vortex motion under the localized induction approximation, Commun. PDE, 28 (2003), 927-968.

[16]

R. Jerrard, C. Seis, On the vortex filament conjecture for Euler flows, arXiv: 1603.00227.

[17]

R. Jerrard and D. Smets, On the motion of a curve by its binormal curvature, J. Eur. Math. Soc. (JEMS), 17 (2015), 148-1515.

[18]

T. Kappeler and J. C. Molnar, On the wellposedness of the defocusing mKdV equation below L2, arXiv: 1606.07052.

[19]

C. KenigG. Ponce and L. Vega, On the ill-posedness of some canonical non-linear dispersive equations, Duke Math. J., 106 (2001), 716-633.

[20]

H. Koch and D. Tataru, Conserved energies for the cubic NLS in 1-d, arXiv: 1607.02534.

[21]

R. Killip, M. Visan Monica and X. Zhang, Talk Bonn March 2016 and private communication, 2016.

[22]

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flows, Cambridge Texts in Applied Mathematics, Cambridge U. Press, 2002.

[23]

P. J. Olver, Dispersive quantization, Am. Math. Monthly, 117 (2010), 599-610.

[24]

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

[25]

A. Vargas and L. Vega, Global well-posedness for 1d non-linear Schrödinger equation for data with an infinite L2 norm,, J. Math. Pures Appl., 80 (2001), 1029-1044.

[26]

L. Vega, The dynamics of vortex filaments with corners, Commun. Pure Appl. Anal., 14 (2015), 1581-1601.

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