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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

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