A remark on the well-posedness for a system of quadratic derivative nonlinear Schrödinger equations

Hiroyuki Hirayama; Shinya Kinoshita; Mamoru Okamoto

doi:10.3934/cpaa.2022101

Article Contents

2022, Volume 21, Issue 10: 3309-3334. Doi: 10.3934/cpaa.2022101 open access

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A remark on the well-posedness for a system of quadratic derivative nonlinear Schrödinger equations

1.
Faculty of Education, University of Miyazaki, 1-1, Gakuenkibanadai-nishi, Miyazaki, 889-2192 Japan
2.
Department of Mathematics, Graduate School of Science and Engineering, Saitama University, 255 Shimo-Okubo, Saitama 338-8570, Japan
3.
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

^*Corresponding author
^*Corresponding author

Received: December 31, 2021

Revised: April 30, 2022

Early access: June 2022

Published: October 2022

This work was supported by JSPS KAKENHI Grant Numbers JP17K14220, JP20K14342, and JP21J00514

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Abstract

We consider the Cauchy problem for the system of quadratic derivative nonlinear Schrödinger equations, which was introduced by Colin and Colin (2004). In the previous paper, the authors (2021) determined the almost optimal Sobolev regularity to be well-posed in $ H^s ( \mathbb{R}^d) $ as long as we use the iteration argument. In this paper, we consider the well-posedness under the conditions where the flow map fails to be twice differentiable. To prove the well-posedness, we construct a modified energy and apply the energy method.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B30.

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Table 1. Regularities to be well-posed when $ \kappa \neq 0 $

$ \kappa \neq 0 $	$ d=1 $	$ d=2 $	$ d=3 $	$ d \ge 4 $

$ \mu>0 $	$ s \ge 0 $		$ s \ge s_c $
$ \mu=0 $		$ s \ge 1 $
$ \mu<0 $	$ s \ge \frac 12 $		$ s>s_c $

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Table 2. Regularities to be well-posed when $ \alpha= \beta $ and $ ( \alpha- \gamma)( \beta+ \gamma) \neq 0 $

$ d=1 $	$ d=2 $	$ d=3 $	$ d \ge 4 $
$ s \ge \frac 12 $		$ s>s_c $

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References

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