# American Institute of Mathematical Sciences

July  2014, 13(4): 1563-1591. doi: 10.3934/cpaa.2014.13.1563

## Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data

 1 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

Received  September 2013 Revised  November 2013 Published  February 2014

In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of the solution of the system in the Sobolev space $H^s$ for $s > d/2+3$ is proved by M. Colin and T. Colin. We prove the well-posedness of the system with low regularity initial data. For some cases, we also prove the well-posedness and the scattering at the scaling critical regularity by using $U^2$ space and $V^2$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).
Citation: Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
##### References:
 [1] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part I, Int. Math. Res. Pap., 2006 (2006), 84pp.  Google Scholar [2] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part II, Trans. Amer. Math. Soc., 360 (2008), 5925-5957. doi: 10.1090/S0002-9947-08-04471-1.  Google Scholar [3] H. Chihara, Local existence for semilinear Schrödinger equations, Math. Japon., 42 (1995), 35-51.  Google Scholar [4] H. Chihara, Gain of regularity for semilinear Schrödinger equations, Math. Ann., 315 (1999), 529-567. doi: 10.1007/s002080050328.  Google Scholar [5] M. Christ, Illposedness of a Schrödinger equation with derivative nonlinearity,, preprint, ().   Google Scholar [6] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations., 17 (2004), 297-330.  Google Scholar [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. doi: 10.1016/j.anihpc.2009.01.011.  Google Scholar [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. doi: 10.1619/fesi.52.371.  Google Scholar [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. doi: 10.1137/110823808.  Google Scholar [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. doi: 10.1090/S0002-9947-01-02760-X.  Google Scholar [11] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.  Google Scholar [12] A. Grünrock, On the Cauchy - and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations,, preprint, ().   Google Scholar [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. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar [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. doi: 10.1016/j.anihpc.2010.01.006.  Google Scholar [15] N. Hayashi, C. Li and P. Naumkin, On a system of nonlinear Schrödinger equations in 2D, Differential Integral Equations., 24 (2011), 417-434.  Google Scholar [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. doi: 10.7153/dea-03-26.  Google Scholar [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. doi: 10.1215/00127094-1415889.  Google Scholar [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. doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar [19] M. Ikeda, S. Katayama and H. Sunagawa, Null structure in a system of quadratic derivative nonlinear Schrödinger equations,, preprint, ().   Google Scholar [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. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar [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. doi: 10.1007/s002220050272.  Google Scholar [22] H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not., 2005 (2005), 1833-1847. doi: 10.1155/IMRN.2005.1833.  Google Scholar [23] S. Mizohata, On the Cauchy Problem, Notes and Reports in Mathematics in Science and Engineering, Science Press & Academic Press., 3 (1985), 177pp.  Google Scholar [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. doi: 10.1137/S0036141001385307.  Google Scholar [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. doi: 10.1016/j.jmaa.2012.10.003.  Google Scholar [26] T. Schottdorf, Global existence without decay for quadratic Klein-Gordon equations,, preprint, ().   Google Scholar [27] A. Stefanov, On quadratic derivative Schrödinger equations in one space dimension, Trans. Amer. Math. Soc., 359 (2007), 3589-3607. doi: 10.1090/S0002-9947-07-04207-9.  Google Scholar [28] T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^{1}(\R )$, J. Hyperbolic Differ. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.  Google Scholar

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##### References:
 [1] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part I, Int. Math. Res. Pap., 2006 (2006), 84pp.  Google Scholar [2] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part II, Trans. Amer. Math. Soc., 360 (2008), 5925-5957. doi: 10.1090/S0002-9947-08-04471-1.  Google Scholar [3] H. Chihara, Local existence for semilinear Schrödinger equations, Math. Japon., 42 (1995), 35-51.  Google Scholar [4] H. Chihara, Gain of regularity for semilinear Schrödinger equations, Math. Ann., 315 (1999), 529-567. doi: 10.1007/s002080050328.  Google Scholar [5] M. Christ, Illposedness of a Schrödinger equation with derivative nonlinearity,, preprint, ().   Google Scholar [6] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential Integral Equations., 17 (2004), 297-330.  Google Scholar [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. doi: 10.1016/j.anihpc.2009.01.011.  Google Scholar [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. doi: 10.1619/fesi.52.371.  Google Scholar [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. doi: 10.1137/110823808.  Google Scholar [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. doi: 10.1090/S0002-9947-01-02760-X.  Google Scholar [11] J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384-436. doi: 10.1006/jfan.1997.3148.  Google Scholar [12] A. Grünrock, On the Cauchy - and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations,, preprint, ().   Google Scholar [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. doi: 10.1016/j.anihpc.2008.04.002.  Google Scholar [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. doi: 10.1016/j.anihpc.2010.01.006.  Google Scholar [15] N. Hayashi, C. Li and P. Naumkin, On a system of nonlinear Schrödinger equations in 2D, Differential Integral Equations., 24 (2011), 417-434.  Google Scholar [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. doi: 10.7153/dea-03-26.  Google Scholar [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. doi: 10.1215/00127094-1415889.  Google Scholar [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. doi: 10.1090/S0894-0347-06-00551-0.  Google Scholar [19] M. Ikeda, S. Katayama and H. Sunagawa, Null structure in a system of quadratic derivative nonlinear Schrödinger equations,, preprint, ().   Google Scholar [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. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar [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. doi: 10.1007/s002220050272.  Google Scholar [22] H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not., 2005 (2005), 1833-1847. doi: 10.1155/IMRN.2005.1833.  Google Scholar [23] S. Mizohata, On the Cauchy Problem, Notes and Reports in Mathematics in Science and Engineering, Science Press & Academic Press., 3 (1985), 177pp.  Google Scholar [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. doi: 10.1137/S0036141001385307.  Google Scholar [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. doi: 10.1016/j.jmaa.2012.10.003.  Google Scholar [26] T. Schottdorf, Global existence without decay for quadratic Klein-Gordon equations,, preprint, ().   Google Scholar [27] A. Stefanov, On quadratic derivative Schrödinger equations in one space dimension, Trans. Amer. Math. Soc., 359 (2007), 3589-3607. doi: 10.1090/S0002-9947-07-04207-9.  Google Scholar [28] T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^{1}(\R )$, J. Hyperbolic Differ. Equ., 1 (2004), 27-49. doi: 10.1142/S0219891604000032.  Google Scholar
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