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, Trans. Amer. Math. Soc., 360 (2008), 5805-5830. 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, J. Funct. Anal., 255 (2008), 2233-2247. 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, J. Funct. Anal., 233 (2006), 228-259. 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, Trans. Amer. Math. Soc., 353 (2001), 3307-3325. 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 $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[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. 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, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.  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, J. Funct. Anal., 233 (2006), 1-39. 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, Trans. Amer. Math. Soc., 360 (2008), 5805-5830. 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, J. Funct. Anal., 255 (2008), 2233-2247. 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, J. Funct. Anal., 233 (2006), 228-259. 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, Trans. Amer. Math. Soc., 353 (2001), 3307-3325. 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 $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[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. 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, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[15]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.  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, J. Funct. Anal., 233 (2006), 1-39. doi: 10.1016/j.jfa.2005.06.018.  Google Scholar

[1]

In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012

[2]

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

[3]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863

[4]

Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253

[5]

Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565

[6]

Piero D'Ancona, Mamoru Okamoto. Blowup and ill-posedness results for a Dirac equation without gauge invariance. Evolution Equations & Control Theory, 2016, 5 (2) : 225-234. doi: 10.3934/eect.2016002

[7]

Adán J. Corcho. Ill-Posedness for the Benney system. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 965-972. doi: 10.3934/dcds.2006.15.965

[8]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327

[9]

Yannis Angelopoulos. Well-posedness and ill-posedness results for the Novikov-Veselov equation. Communications on Pure & Applied Analysis, 2016, 15 (3) : 727-760. doi: 10.3934/cpaa.2016.15.727

[10]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082

[11]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm. Communications on Pure & Applied Analysis, 2003, 2 (1) : 33-50. doi: 10.3934/cpaa.2003.2.33

[12]

Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019

[13]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[14]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261

[15]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37

[16]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010

[17]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023

[18]

Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093

[19]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205

[20]

Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021122

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (93)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]