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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

Hiroyuki Hirayama 1,

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).
Keywords: Schr\"odinger equation, scaling critical, well-posedness, Bilinear estimate, Cauchy problem, bounded $p$-variation..
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B6.
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,, \emph{Int. Math. Res. Pap.}, 2006 (2006).   Google Scholar

[2]

I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part II,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 5925.  doi: 10.1090/S0002-9947-08-04471-1.  Google Scholar

[3]

H. Chihara, Local existence for semilinear Schrödinger equations,, \emph{Math. Japon.}, 42 (1995), 35.   Google Scholar

[4]

H. Chihara, Gain of regularity for semilinear Schrödinger equations,, \emph{Math. Ann.}, 315 (1999), 529.  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,, \emph{Differential Integral Equations.}, 17 (2004), 297.   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,, \emph{Ann. Inst. H. Poincar\'e Anal. Nonlin\'eaire.}, 26 (2009), 2211.  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,, \emph{Funkcialaj Ekvacioj.}, 52 (2009), 371.  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,, \emph{SIAM J. Math. Anal.}, 44 (2012), 206.  doi: 10.1137/110823808.  Google Scholar

[10]

J. Colliander, J. Delort, C. 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

[11]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, \emph{J. Funct. Anal.}, 151 (1997), 384.  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,, \emph{Ann. Inst. H. Poincar\'e Anal. Non lin\'eaie.}, 26 (2009), 917.  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],, \emph{Ann. Inst. H. Poincar\'e Anal. Non lin\'eaie.}, 27 (2010), 971.  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,, \emph{Differential Integral Equations.}, 24 (2011), 417.   Google Scholar

[16]

N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations,, \emph{Differ. Equ. Appl.}, 3 (2011), 415.  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 $H^{1}(T^{3})$,, \emph{Duke. Math. J.}, 159 (2011), 329.  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,, \emph{J. Amer. Math. Soc.}, 20 (2007), 753.  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,, \emph{J. Amer. Math. Soc.}, 9 (1996), 573.  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,, \emph{Invent. Math.}, 134 (1998), 489.  doi: 10.1007/s002220050272.  Google Scholar

[22]

H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation,, \emph{Int. Math. Res. Not.}, 2005 (2005), 1833.  doi: 10.1155/IMRN.2005.1833.  Google Scholar

[23]

S. Mizohata, On the Cauchy Problem,, Notes and Reports in Mathematics in Science and Engineering, 3 (1985).   Google Scholar

[24]

L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982.  doi: 10.1137/S0036141001385307.  Google Scholar

[25]

T. Ozawa and H. Sunagawa, Small data blow-up for a system of nonlinear Schrodinger equations,, \emph{J. Math. Anal. Appl.}, 399 (2013), 147.  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,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 3589.  doi: 10.1090/S0002-9947-07-04207-9.  Google Scholar

[28]

T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^{1}(\R )$,, \emph{J. Hyperbolic Differ. Equ.}, 1 (2004), 27.  doi: 10.1142/S0219891604000032.  Google Scholar

show all references

References:
[1]

I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part I,, \emph{Int. Math. Res. Pap.}, 2006 (2006).   Google Scholar

[2]

I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Part II,, \emph{Trans. Amer. Math. Soc.}, 360 (2008), 5925.  doi: 10.1090/S0002-9947-08-04471-1.  Google Scholar

[3]

H. Chihara, Local existence for semilinear Schrödinger equations,, \emph{Math. Japon.}, 42 (1995), 35.   Google Scholar

[4]

H. Chihara, Gain of regularity for semilinear Schrödinger equations,, \emph{Math. Ann.}, 315 (1999), 529.  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,, \emph{Differential Integral Equations.}, 17 (2004), 297.   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,, \emph{Ann. Inst. H. Poincar\'e Anal. Nonlin\'eaire.}, 26 (2009), 2211.  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,, \emph{Funkcialaj Ekvacioj.}, 52 (2009), 371.  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,, \emph{SIAM J. Math. Anal.}, 44 (2012), 206.  doi: 10.1137/110823808.  Google Scholar

[10]

J. Colliander, J. Delort, C. 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

[11]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system,, \emph{J. Funct. Anal.}, 151 (1997), 384.  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,, \emph{Ann. Inst. H. Poincar\'e Anal. Non lin\'eaie.}, 26 (2009), 917.  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],, \emph{Ann. Inst. H. Poincar\'e Anal. Non lin\'eaie.}, 27 (2010), 971.  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,, \emph{Differential Integral Equations.}, 24 (2011), 417.   Google Scholar

[16]

N. Hayashi, C. Li and T. Ozawa, Small data scattering for a system of nonlinear Schrödinger equations,, \emph{Differ. Equ. Appl.}, 3 (2011), 415.  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 $H^{1}(T^{3})$,, \emph{Duke. Math. J.}, 159 (2011), 329.  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,, \emph{J. Amer. Math. Soc.}, 20 (2007), 753.  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,, \emph{J. Amer. Math. Soc.}, 9 (1996), 573.  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,, \emph{Invent. Math.}, 134 (1998), 489.  doi: 10.1007/s002220050272.  Google Scholar

[22]

H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation,, \emph{Int. Math. Res. Not.}, 2005 (2005), 1833.  doi: 10.1155/IMRN.2005.1833.  Google Scholar

[23]

S. Mizohata, On the Cauchy Problem,, Notes and Reports in Mathematics in Science and Engineering, 3 (1985).   Google Scholar

[24]

L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982.  doi: 10.1137/S0036141001385307.  Google Scholar

[25]

T. Ozawa and H. Sunagawa, Small data blow-up for a system of nonlinear Schrodinger equations,, \emph{J. Math. Anal. Appl.}, 399 (2013), 147.  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,, \emph{Trans. Amer. Math. Soc.}, 359 (2007), 3589.  doi: 10.1090/S0002-9947-07-04207-9.  Google Scholar

[28]

T. Tao, Global well-posedness of the Benjamin-Ono equation in $H^{1}(\R )$,, \emph{J. Hyperbolic Differ. Equ.}, 1 (2004), 27.  doi: 10.1142/S0219891604000032.  Google Scholar

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