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June  2022, 42(6): 2563-2584. doi: 10.3934/dcds.2021203

On the one dimensional cubic NLS in a critical space

BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country - Spain

*Corresponding author: Marco Bravin

Received  July 2021 Published  June 2022 Early access  January 2022

Fund Project: Marco Bravin is supported by ERC-2014-ADG project HADE Id. 669689 (European Research Council). Luis Vega is supported by ERC-2014-ADG project HADE Id. 669689 (European Research Council), MINECO grant BERC 2018-2021, PGC2018-094522-B-I00 and SEV-2017-0718 (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.

Citation: Marco Bravin, Luis Vega. On the one dimensional cubic NLS in a critical space. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2563-2584. doi: 10.3934/dcds.2021203
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. ChristJ. 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. KenigG. 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. ChristJ. 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. KenigG. 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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