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On the one dimensional cubic NLS in a critical space
BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country - Spain |
In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.
References:
| [1] |
V. Banica and L. Vega,
Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc. (JEMS), 14 (2012), 209-253.
doi: 10.4171/JEMS/300. |
| [2] |
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. |
| [3] |
V. Banica and L. Vega, Evolution of polygonal lines by the binormal flow, Annals of PDE, 6 (2020).
doi: 10.1007/s40818-020-0078-z. |
| [4] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
| [5] |
R. Carles and T. Kappeler,
Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces, Bull. Soc. Math. France, 145 (2017), 623-642.
|
| [6] |
M. Christ, J. Colliander and T. Tao,
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.
doi: 10.1353/ajm.2003.0040. |
| [7] |
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.
|
| [8] |
F. De la Hoz and L. Vega,
Vortex filament equation for a regular polygon, Nonlinearity, 27 (2014), 3031-3057.
doi: 10.1088/0951-7715/27/12/3031. |
| [9] |
A. Grünrock,
Bi-and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not., 2005 (2005), 2525-2558.
doi: 10.1155/IMRN.2005.2525. |
| [10] |
A. Grünrock and S. Herr,
Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.
doi: 10.1137/070689139. |
| [11] |
B. Harrop-Griffiths, R. Killip and M. Visan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, arXiv: 2003.05011. |
| [12] |
H. Hasimoto,
A solution on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.
doi: 10.1017/S0022112072002307. |
| [13] |
R. L. Jerrard and D. Smets,
On the motion of a curve by its binormal curvature, J. Eur. Math. Soc., 17 (2015), 1487-1515.
doi: 10.4171/JEMS/536. |
| [14] |
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. |
| [15] |
N. Kishimoto,
Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.
|
| [16] |
N. Kita,
Mode generating property of solutions to the nonlinear Schrödinger equations in one space dimension, GAKUTO Internat. Ser., Math. Sci. Appl., 26 (2006), 111-128.
|
| [17] |
T. Oh,
A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.
doi: 10.1619/fesi.60.259. |
| [18] |
T. Oh and Y. Wang,
Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.
doi: 10.1016/j.jde.2019.12.017. |
| [19] |
Y. Tsutsumi,
$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.
|
| [20] |
A. Vargas and L. Vega,
Global well-posedness for 1D non-linear Schrödinger equation for data with an infinite $L^2$ norm, J. Math. Pures Appl, 80 (2001), 1029-1044.
doi: 10.1016/S0021-7824(01)01224-7. |
show all references
References:
| [1] |
V. Banica and L. Vega,
Scattering for 1D cubic NLS and singular vortex dynamics, J. Eur. Math. Soc. (JEMS), 14 (2012), 209-253.
doi: 10.4171/JEMS/300. |
| [2] |
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. |
| [3] |
V. Banica and L. Vega, Evolution of polygonal lines by the binormal flow, Annals of PDE, 6 (2020).
doi: 10.1007/s40818-020-0078-z. |
| [4] |
J. Bourgain,
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.
doi: 10.1007/BF01895688. |
| [5] |
R. Carles and T. Kappeler,
Norm-inflation with infinite loss of regularity for periodic NLS equations in negative Sobolev spaces, Bull. Soc. Math. France, 145 (2017), 623-642.
|
| [6] |
M. Christ, J. Colliander and T. Tao,
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.
doi: 10.1353/ajm.2003.0040. |
| [7] |
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.
|
| [8] |
F. De la Hoz and L. Vega,
Vortex filament equation for a regular polygon, Nonlinearity, 27 (2014), 3031-3057.
doi: 10.1088/0951-7715/27/12/3031. |
| [9] |
A. Grünrock,
Bi-and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not., 2005 (2005), 2525-2558.
doi: 10.1155/IMRN.2005.2525. |
| [10] |
A. Grünrock and S. Herr,
Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.
doi: 10.1137/070689139. |
| [11] |
B. Harrop-Griffiths, R. Killip and M. Visan, Sharp well-posedness for the cubic NLS and mKdV in $H^s(\mathbb{R})$, arXiv: 2003.05011. |
| [12] |
H. Hasimoto,
A solution on a vortex filament, J. Fluid Mech., 51 (1972), 477-485.
doi: 10.1017/S0022112072002307. |
| [13] |
R. L. Jerrard and D. Smets,
On the motion of a curve by its binormal curvature, J. Eur. Math. Soc., 17 (2015), 1487-1515.
doi: 10.4171/JEMS/536. |
| [14] |
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. |
| [15] |
N. Kishimoto,
Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464.
|
| [16] |
N. Kita,
Mode generating property of solutions to the nonlinear Schrödinger equations in one space dimension, GAKUTO Internat. Ser., Math. Sci. Appl., 26 (2006), 111-128.
|
| [17] |
T. Oh,
A remark on norm inflation with general initial data for the cubic nonlinear Schrödinger equations in negative Sobolev spaces, Funkcial. Ekvac., 60 (2017), 259-277.
doi: 10.1619/fesi.60.259. |
| [18] |
T. Oh and Y. Wang,
Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, J. Differential Equations, 269 (2020), 612-640.
doi: 10.1016/j.jde.2019.12.017. |
| [19] |
Y. Tsutsumi,
$L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.
|
| [20] |
A. Vargas and L. Vega,
Global well-posedness for 1D non-linear Schrödinger equation for data with an infinite $L^2$ norm, J. Math. Pures Appl, 80 (2001), 1029-1044.
doi: 10.1016/S0021-7824(01)01224-7. |
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