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Communications on Pure and Applied Analysis (CPAA)
 

Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data
Pages: 1563 - 1591, Issue 4, July 2014

doi:10.3934/cpaa.2014.13.1563      Abstract        References        Full text (564.0K)           Related Articles

Hiroyuki Hirayama - Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan (email)

1 I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part I, Int. Math. Res. Pap., 2006 (2006), 84pp.       
2 I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part II, Trans. Amer. Math. Soc., 360 (2008), 5925-5957.       
3 H. Chihara, Local existence for semilinear Schrödinger equations, Math. Japon., 42 (1995), 35-51.       
4 H. Chihara, Gain of regularity for semilinear Schrödinger equations, Math. Ann., 315 (1999), 529-567.       
5 M. Christ, Illposedness of a Schrödinger equation with derivative nonlinearity, preprint, Available from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.1363.
6 M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations., 17 (2004), 297-330.       
7 M. Colin, T. Colin and M. Ohta, Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction, Ann. Inst. H. Poincaré Anal. Nonlin\éaire., 26 (2009), 2211-2226.       
8 M. Colin, T. Colin and M. Ohta, Instability of standing waves for a system of nonlinear Schrödinger equations with three-wave interaction, Funkcialaj Ekvacioj., 52 (2009), 371-380.       
9 M. Colin and M. Ohta, Bifurcation from semitrivial standing waves and ground states for a system of nonlinear Schrödinger equations, SIAM J. Math. Anal., 44 (2012), 206-223.       
10 J. Colliander, J. Delort, C. Kenig, and G. Staffilani, Bilinear estimates and applications to 2D NLS, Trans. Amer. Math. Soc., 353 (2001), 3307-3325.       
11 J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436.       
12 A. Grünrock, On the Cauchy - and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations, preprint, arXiv:math/0006195v1.
13 M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non linéaie., 26 (2009), 917-941.       
14 M. Hadac, S. Herr and H. Koch, Errantum to "Well-posedness and scattering for the KP-II equation in a critical space'' [Ann. I. H. Poincaré-AN26 (3) (2009) 917-941], Ann. Inst. H. Poincaré Anal. Non linéaie., 27 (2010), 971-972.       
15 N. Hayashi, C. Li and P. Naumkin, On a system of nonlinear Schrödinger equations in 2D, Differential Integral Equations., 24 (2011), 417-434.       
16 N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations, Differ. Equ. Appl., 3 (2011), 415-426.       
17 S. Herr, D. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^{1}(T^{3})$, Duke. Math. J., 159 (2011), 329-349.       
18 A. Ionescu and C. Kenig, Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J. Amer. Math. Soc., 20 (2007), 753-798.       
19 M. Ikeda, S. Katayama and H. Sunagawa, Null structure in a system of quadratic derivative nonlinear Schrödinger equations, preprint, arXiv:1305.3662v1.
20 C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.       
21 C. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math., 134 (1998), 489-545.       
22 H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not., 2005 (2005), 1833-1847.       
23 S. Mizohata, On the Cauchy Problem, Notes and Reports in Mathematics in Science and Engineering, Science Press & Academic Press., 3 (1985), 177pp.       
24 L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.       
25 T. Ozawa and H. Sunagawa, Small data blow-up for a system of nonlinear Schrodinger equations, J. Math. Anal. Appl., 399 (2013), 147-155.       
26 T. Schottdorf, Global existence without decay for quadratic Klein-Gordon equations, preprint, arXiv:1209.1518v2.
27 A. Stefanov, On quadratic derivative Schrödinger equations in one space dimension, Trans. Amer. Math. Soc., 359 (2007), 3589-3607.       
28 T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^{1}(\R )$, J. Hyperbolic Differ. Equ., 1 (2004), 27-49.       

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