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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, Volume 13, Issue 4, July 2014      doi:10.3934/cpaa.2014.13.1563

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

Abstract: 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, well-posedness, Cauchy problem, scaling critical, Bilinear estimate, bounded $p$-variation.
Mathematics Subject Classification:  Primary: 35Q55; Secondary: 35B65.

Received: September 2013;      Revised: November 2013;      Available Online: February 2014.

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