May 2019, 18(3): 1375-1402. doi: 10.3934/cpaa.2019067

A remark on norm inflation for nonlinear Schrödinger equations

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received  July 2018 Revised  July 2018 Published  November 2018

We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on $\mathbb{R}^d$ or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.

Citation: Nobu Kishimoto. A remark on norm inflation for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1375-1402. doi: 10.3934/cpaa.2019067
References:
[1]

T. Alazard and R. Carles, Loss of regularity for supercritical nonlinear Schrödinger equations, Math. Ann., 343 (2009), 397-420. doi: 10.1007/s00208-008-0276-6.

[2]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259. doi: 10.1016/j.jfa.2005.08.004.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Ⅰ, Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

N. BurqP. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett., 9 (2002), 323-335. doi: 10.4310/MRL.2002.v9.n3.a8.

[5]

R. Carles, Geometric optics and instability for semi-classical Schrödinger equations, Arch. Ration. Mech. Anal., 183 (2007), 525-553. doi: 10.1007/s00205-006-0017-5.

[6]

R. CarlesE. Dumas and C. Sparber, Geometric optics and instability for NLS and Davey-Stewartson models, J. Eur. Math. Soc., 14 (2012), 1885-1921. doi: 10.4171/JEMS/350.

[7]

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. doi: 10.24033/bsmf.2749.

[8]

A. Choffrut and O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not. IMRN, 2018, 699-738. doi: 10.1093/imrn/rnw246.

[9]

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.

[10]

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

[11]

M. Christ, J. Colliander and T. Tao, Instability of the periodic nonlinear Schrödinger equation, preprint, arXiv: math/0311227.

[12]

M. ChristJ. Colliander and T. Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal., 254 (2008), 368-395. doi: 10.1016/j.jfa.2007.09.005.

[13]

F. FalkE. W. Laedke and K. H. Spatschek, Stability of solitary-wave pulses in shape-memory alloys, Phys. Rev. B, 36 (1987), 3031-3041. doi: 10.1103/PhysRevB.36.3031.

[14]

H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983.

[15]

A. Grünrock, Some local wellposedness results for nonlinear Schrödinger equations below $L^2$, preprint, arXiv: math/0011157.

[16]

S. Guo, On the 1D cubic nonlinear Schrödinger equation in an almost critical space, J. Fourier Anal. Appl., 23 (2017), 91-124. doi: 10.1007/s00041-016-9464-z.

[17]

Z. Guo and T. Oh, Non-existence of solutions for the periodic cubic NLS below $L^2$, Int. Math. Res. Not. IMRN, (2018), 1656-1729. doi: 10.1093/imrn/rnw271.

[18]

S. GustafsonK. Nakanishi and T. P. Tsai, Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math., 11 (2009), 657-707. doi: 10.1142/S0219199709003491.

[19]

H. HuhS. Machihara and M. Okamoto, Well-posedness and ill-posedness of the Cauchy problem for the generalized Thirring model, Differential Integral Equations, 29 (2016), 401-420.

[20]

T. Iwabuchi and T. Ogawa, Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc., 367 (2015), 2613-2630. doi: 10.1090/S0002-9947-2014-06000-5.

[21]

T. Iwabuchi and K. Uriya, Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$, Commun. Pure Appl. Anal., 14 (2015), 1395-1405. doi: 10.3934/cpaa.2015.14.1395.

[22]

C. E. KenigG. Ponce and L. Vega, Quadratic forms for the $1$-D semilinear Schrödinger equation, Trans. Amer. Math. Soc., 348 (1996), 3323-3353. doi: 10.1090/S0002-9947-96-01645-5.

[23]

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.

[24]

R. KillipM. Vişan and X. Zhang, Low regularity conservation laws for integrable PDE, Geom. Funct. Anal., 28 (2018), 1062-1090. doi: 10.1007/s00039-018-0444-0.

[25]

N. Kishimoto, Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation, J. Differential Equations, 247 (2009), 1397-1439. doi: 10.1016/j.jde.2009.06.009.

[26]

N. Kishimoto and K. Tsugawa, Local well-posedness for quadratic nonlinear Schrödinger equations and the "good" Boussinesq equation, Differential Integral Equations, 23 (2010), 463-493.

[27]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, 2007, no. 16, Art.ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.

[28]

H. Koch and D. Tataru, Energy and local energy bounds for the 1-d cubic NLS equation in $H^{-1/4}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988. doi: 10.1016/j.anihpc.2012.05.006.

[29]

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

[30]

S. Machihara and M. Okamoto, Ill-posedness of the Cauchy problem for the Chern-Simons-Dirac system in one dimension, J. Differential Equations, 258 (2015), 1356-1394. doi: 10.1016/j.jde.2014.10.020.

[31]

S. Machihara and M. Okamoto, Sharp well-posedness and ill-posedness for the Chern-SimonsDirac system in one dimension, Int. Math. Res. Not. IMRN, 2016, 1640-1694. doi: 10.1093/imrn/rnv160.

[32]

L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett., 16 (2009), 111-120. doi: 10.4310/MRL.2009.v16.n1.a11.

[33]

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.

[34]

T. Oh and C. Sulem, On the one-dimensional cubic nonlinear Schrödinger equation below $L^2$, Kyoto J. Math., 52 (2012), 99-115. doi: 10.1215/21562261-1503772.

[35]

T. Oh and Y. Wang, On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle, to appear in An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), available from arXiv: 1508.00827.

[36]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, preprint, arXiv: 1806.08761.

[37]

M. Okamoto, Norm inflation for the generalized Boussinesq and Kawahara equations, Nonlinear Anal., 157 (2017), 44-61. doi: 10.1016/j.na.2017.03.011.

[38]

M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math., 301, Birkhäuser/Springer Basel AG, (2012), 267–283. doi: 10.1007/978-3-0348-0454-7_14.

[39]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

show all references

References:
[1]

T. Alazard and R. Carles, Loss of regularity for supercritical nonlinear Schrödinger equations, Math. Ann., 343 (2009), 397-420. doi: 10.1007/s00208-008-0276-6.

[2]

I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal., 233 (2006), 228-259. doi: 10.1016/j.jfa.2005.08.004.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Ⅰ, Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

N. BurqP. Gérard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett., 9 (2002), 323-335. doi: 10.4310/MRL.2002.v9.n3.a8.

[5]

R. Carles, Geometric optics and instability for semi-classical Schrödinger equations, Arch. Ration. Mech. Anal., 183 (2007), 525-553. doi: 10.1007/s00205-006-0017-5.

[6]

R. CarlesE. Dumas and C. Sparber, Geometric optics and instability for NLS and Davey-Stewartson models, J. Eur. Math. Soc., 14 (2012), 1885-1921. doi: 10.4171/JEMS/350.

[7]

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. doi: 10.24033/bsmf.2749.

[8]

A. Choffrut and O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Not. IMRN, 2018, 699-738. doi: 10.1093/imrn/rnw246.

[9]

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.

[10]

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

[11]

M. Christ, J. Colliander and T. Tao, Instability of the periodic nonlinear Schrödinger equation, preprint, arXiv: math/0311227.

[12]

M. ChristJ. Colliander and T. Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal., 254 (2008), 368-395. doi: 10.1016/j.jfa.2007.09.005.

[13]

F. FalkE. W. Laedke and K. H. Spatschek, Stability of solitary-wave pulses in shape-memory alloys, Phys. Rev. B, 36 (1987), 3031-3041. doi: 10.1103/PhysRevB.36.3031.

[14]

H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983.

[15]

A. Grünrock, Some local wellposedness results for nonlinear Schrödinger equations below $L^2$, preprint, arXiv: math/0011157.

[16]

S. Guo, On the 1D cubic nonlinear Schrödinger equation in an almost critical space, J. Fourier Anal. Appl., 23 (2017), 91-124. doi: 10.1007/s00041-016-9464-z.

[17]

Z. Guo and T. Oh, Non-existence of solutions for the periodic cubic NLS below $L^2$, Int. Math. Res. Not. IMRN, (2018), 1656-1729. doi: 10.1093/imrn/rnw271.

[18]

S. GustafsonK. Nakanishi and T. P. Tsai, Scattering theory for the Gross-Pitaevskii equation in three dimensions, Commun. Contemp. Math., 11 (2009), 657-707. doi: 10.1142/S0219199709003491.

[19]

H. HuhS. Machihara and M. Okamoto, Well-posedness and ill-posedness of the Cauchy problem for the generalized Thirring model, Differential Integral Equations, 29 (2016), 401-420.

[20]

T. Iwabuchi and T. Ogawa, Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc., 367 (2015), 2613-2630. doi: 10.1090/S0002-9947-2014-06000-5.

[21]

T. Iwabuchi and K. Uriya, Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$, Commun. Pure Appl. Anal., 14 (2015), 1395-1405. doi: 10.3934/cpaa.2015.14.1395.

[22]

C. E. KenigG. Ponce and L. Vega, Quadratic forms for the $1$-D semilinear Schrödinger equation, Trans. Amer. Math. Soc., 348 (1996), 3323-3353. doi: 10.1090/S0002-9947-96-01645-5.

[23]

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.

[24]

R. KillipM. Vişan and X. Zhang, Low regularity conservation laws for integrable PDE, Geom. Funct. Anal., 28 (2018), 1062-1090. doi: 10.1007/s00039-018-0444-0.

[25]

N. Kishimoto, Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation, J. Differential Equations, 247 (2009), 1397-1439. doi: 10.1016/j.jde.2009.06.009.

[26]

N. Kishimoto and K. Tsugawa, Local well-posedness for quadratic nonlinear Schrödinger equations and the "good" Boussinesq equation, Differential Integral Equations, 23 (2010), 463-493.

[27]

H. Koch and D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN, 2007, no. 16, Art.ID rnm053, 36 pp. doi: 10.1093/imrn/rnm053.

[28]

H. Koch and D. Tataru, Energy and local energy bounds for the 1-d cubic NLS equation in $H^{-1/4}$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 955-988. doi: 10.1016/j.anihpc.2012.05.006.

[29]

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

[30]

S. Machihara and M. Okamoto, Ill-posedness of the Cauchy problem for the Chern-Simons-Dirac system in one dimension, J. Differential Equations, 258 (2015), 1356-1394. doi: 10.1016/j.jde.2014.10.020.

[31]

S. Machihara and M. Okamoto, Sharp well-posedness and ill-posedness for the Chern-SimonsDirac system in one dimension, Int. Math. Res. Not. IMRN, 2016, 1640-1694. doi: 10.1093/imrn/rnv160.

[32]

L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett., 16 (2009), 111-120. doi: 10.4310/MRL.2009.v16.n1.a11.

[33]

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.

[34]

T. Oh and C. Sulem, On the one-dimensional cubic nonlinear Schrödinger equation below $L^2$, Kyoto J. Math., 52 (2012), 99-115. doi: 10.1215/21562261-1503772.

[35]

T. Oh and Y. Wang, On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle, to appear in An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), available from arXiv: 1508.00827.

[36]

T. Oh and Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces, preprint, arXiv: 1806.08761.

[37]

M. Okamoto, Norm inflation for the generalized Boussinesq and Kawahara equations, Nonlinear Anal., 157 (2017), 44-61. doi: 10.1016/j.na.2017.03.011.

[38]

M. Ruzhansky, M. Sugimoto and B. Wang, Modulation spaces and nonlinear evolution equations, in Evolution Equations of Hyperbolic and Schrödinger Type, Progr. Math., 301, Birkhäuser/Springer Basel AG, (2012), 267–283. doi: 10.1007/978-3-0348-0454-7_14.

[39]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

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