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 and 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, Trans. Amer. Math. Soc., 360 (2008), 5805-5830. doi: 10.1090/S0002-9947-08-04415-2.

[2]

J. Bourgain and N. Pavlović, Ill-posedness of the incompressible Navier-Stokes equations in the critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247. doi: 10.1016/j.jfa.2008.07.008.

[3]

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.

[4]

J. E. Colliander, J. -M. Delrot, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325. doi: 10.1090/S0002-9947-01-02760-X.

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[6]

H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983, in Proc. Internat. Conf. on Wavelets and Applications (R. Radha, M. Krishna and S. Yhangavelu eds.), New Delhi Allied Publishers, 2003, 1-56.

[7]

T. Iwabuchi and T. Ogawa, Ill-posedness for nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc., to appear. doi: 10.1090/S0002-9947-2014-06000-5.

[8]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.

[9]

C. E. Kenig, G. 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.

[10]

C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2000), 617-633. doi: 10.1215/S0012-7094-01-10638-8.

[11]

N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\overlineu^2$, Commun. Pure. Appl. Anal., 7 (2008), 1123-1143. doi: 10.3934/cpaa.2008.7.1123.

[12]

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.

[13]

T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.

[14]

J. Toft, Continuity properties for modulation spaces, with application to pseudo-differential calculus, I, J. Funct. Anal., 207 (2004), 399-429. doi: 10.1016/j.jfa.2003.10.003.

[15]

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

[16]

B. Wang, L. Zhao and B. Guo, Isometric decomposition operators, function spaces $E_{p,q}^\lambda$ and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018.

show all references

References:
[1]

I. Bejenaru and D. De Silva, Low regularity solutions for 2D quadratic nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 360 (2008), 5805-5830. doi: 10.1090/S0002-9947-08-04415-2.

[2]

J. Bourgain and N. Pavlović, Ill-posedness of the incompressible Navier-Stokes equations in the critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247. doi: 10.1016/j.jfa.2008.07.008.

[3]

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.

[4]

J. E. Colliander, J. -M. Delrot, C. E. Kenig and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325. doi: 10.1090/S0002-9947-01-02760-X.

[5]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.

[6]

H. G. Feichtinger, Modulation spaces on locally compact Abelian groups, Technical Report, University of Vienna, 1983, in Proc. Internat. Conf. on Wavelets and Applications (R. Radha, M. Krishna and S. Yhangavelu eds.), New Delhi Allied Publishers, 2003, 1-56.

[7]

T. Iwabuchi and T. Ogawa, Ill-posedness for nonlinear Schrödinger equation with quadratic non-linearity in low dimensions, Trans. Amer. Math. Soc., to appear. doi: 10.1090/S0002-9947-2014-06000-5.

[8]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.

[9]

C. E. Kenig, G. 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.

[10]

C. E. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2000), 617-633. doi: 10.1215/S0012-7094-01-10638-8.

[11]

N. Kishimoto, Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\overlineu^2$, Commun. Pure. Appl. Anal., 7 (2008), 1123-1143. doi: 10.3934/cpaa.2008.7.1123.

[12]

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.

[13]

T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.

[14]

J. Toft, Continuity properties for modulation spaces, with application to pseudo-differential calculus, I, J. Funct. Anal., 207 (2004), 399-429. doi: 10.1016/j.jfa.2003.10.003.

[15]

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

[16]

B. Wang, L. Zhao and B. Guo, Isometric decomposition operators, function spaces $E_{p,q}^\lambda$ and applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018.

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