July  2015, 14(4): 1395-1405. doi: 10.3934/cpaa.2015.14.1395

Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$

1. 

Department of Mathematics, Chuo University, Kasuga, Bunkyoku, Tokyo, 112-8551, Japan

2. 

Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

Received  June 2014 Revised  September 2014 Published  April 2014

We are concerned with the ill-posedness issue for the nonlinear Schrödinger equation with the quadratic nonlinearity $|u|^2$ and prove the norm inflation in the dimensions $1 \le n \le 3$. This is the extension of the ill-posed result by Kishimoto-Tsugawa [12] in one dimension and also the remaining case of Iwabuchi-Ogawa [7].
Citation: Tsukasa Iwabuchi, Kota Uriya. Ill-posedness for the quadratic nonlinear Schrödinger equation with nonlinearity $|u|^2$. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1395-1405. doi: 10.3934/cpaa.2015.14.1395
References:
[1]

I. Bejenaru and D. De Silva, Low regularity solutions for 2D quadratic nonlinear Schrödinger equation,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 5805. doi: 10.1090/S0002-9947-08-04415-2. Google Scholar

[2]

J. Bourgain and N. Pavlović, Ill-posedness of the incompressible Navier-Stokes equations in the critical space in 3D,, \emph{J. Funct. Anal.}, 255 (2008), 2233. doi: 10.1016/j.jfa.2008.07.008. Google Scholar

[3]

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

[4]

J. E. Colliander, J. -M. Delrot, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS,, \emph{Trans. Amer. Math. Soc.}, 353 (2001), 3307. doi: 10.1090/S0002-9947-01-02760-X. Google Scholar

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, \emph{Nonlinear Anal.}, 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

[6]

H. G. Feichtinger, Modulation spaces on locally compact Abelian groups,, Technical Report, (1983), 1. Google Scholar

[7]

T. Iwabuchi and T. Ogawa, Ill-posedness for nonlinear Schrödinger equation with quadratic non-linearity in low dimensions,, \emph{Trans. Amer. Math. Soc.}, (). doi: 10.1090/S0002-9947-2014-06000-5. Google Scholar

[8]

T. Kato, On nonlinear Schrödinger equations,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 46 (1987), 113. Google Scholar

[9]

C. E. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation,, \emph{Trans. Amer. Math. Soc.}, 348 (1996), 3323. doi: 10.1090/S0002-9947-96-01645-5. Google Scholar

[10]

C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations,, \emph{Duke Math. J.}, 106 (2000), 617. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar

[11]

N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\overlineu^2$,, \emph{Commun. Pure. Appl. Anal.}, 7 (2008), 1123. doi: 10.3934/cpaa.2008.7.1123. Google Scholar

[12]

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

[13]

T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations,, \emph{Amer. J. Math.}, 123 (2001), 839. Google Scholar

[14]

J. Toft, Continuity properties for modulation spaces, with application to pseudo-differential calculus, I,, \emph{J. Funct. Anal.}, 207 (2004), 399. doi: 10.1016/j.jfa.2003.10.003. Google Scholar

[15]

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

[16]

B. Wang, L. Zhao and B. Guo, Isometric decomposition operators, function spaces $E_{p,q}^\lambda$ and applications to nonlinear evolution equations,, \emph{J. Funct. Anal.}, 233 (2006), 1. doi: 10.1016/j.jfa.2005.06.018. Google Scholar

show all references

References:
[1]

I. Bejenaru and D. De Silva, Low regularity solutions for 2D quadratic nonlinear Schrödinger equation,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 5805. doi: 10.1090/S0002-9947-08-04415-2. Google Scholar

[2]

J. Bourgain and N. Pavlović, Ill-posedness of the incompressible Navier-Stokes equations in the critical space in 3D,, \emph{J. Funct. Anal.}, 255 (2008), 2233. doi: 10.1016/j.jfa.2008.07.008. Google Scholar

[3]

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

[4]

J. E. Colliander, J. -M. Delrot, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS,, \emph{Trans. Amer. Math. Soc.}, 353 (2001), 3307. doi: 10.1090/S0002-9947-01-02760-X. Google Scholar

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, \emph{Nonlinear Anal.}, 14 (1990), 807. doi: 10.1016/0362-546X(90)90023-A. Google Scholar

[6]

H. G. Feichtinger, Modulation spaces on locally compact Abelian groups,, Technical Report, (1983), 1. Google Scholar

[7]

T. Iwabuchi and T. Ogawa, Ill-posedness for nonlinear Schrödinger equation with quadratic non-linearity in low dimensions,, \emph{Trans. Amer. Math. Soc.}, (). doi: 10.1090/S0002-9947-2014-06000-5. Google Scholar

[8]

T. Kato, On nonlinear Schrödinger equations,, \emph{Ann. Inst. H. Poincar\'e Phys. Th\'eor.}, 46 (1987), 113. Google Scholar

[9]

C. E. Kenig, G. Ponce and L. Vega, Quadratic forms for the 1-D semilinear Schrödinger equation,, \emph{Trans. Amer. Math. Soc.}, 348 (1996), 3323. doi: 10.1090/S0002-9947-96-01645-5. Google Scholar

[10]

C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations,, \emph{Duke Math. J.}, 106 (2000), 617. doi: 10.1215/S0012-7094-01-10638-8. Google Scholar

[11]

N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\overlineu^2$,, \emph{Commun. Pure. Appl. Anal.}, 7 (2008), 1123. doi: 10.3934/cpaa.2008.7.1123. Google Scholar

[12]

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

[13]

T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations,, \emph{Amer. J. Math.}, 123 (2001), 839. Google Scholar

[14]

J. Toft, Continuity properties for modulation spaces, with application to pseudo-differential calculus, I,, \emph{J. Funct. Anal.}, 207 (2004), 399. doi: 10.1016/j.jfa.2003.10.003. Google Scholar

[15]

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

[16]

B. Wang, L. Zhao and B. Guo, Isometric decomposition operators, function spaces $E_{p,q}^\lambda$ and applications to nonlinear evolution equations,, \emph{J. Funct. Anal.}, 233 (2006), 1. doi: 10.1016/j.jfa.2005.06.018. Google Scholar

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