# American Institute of Mathematical Sciences

October  2022, 21(10): 3309-3334. doi: 10.3934/cpaa.2022101

## 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

Received  January 2022 Revised  May 2022 Published  October 2022 Early access  June 2022

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

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.

Citation: Hiroyuki Hirayama, Shinya Kinoshita, Mamoru Okamoto. A remark on the well-posedness for a system of quadratic derivative nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2022, 21 (10) : 3309-3334. doi: 10.3934/cpaa.2022101
##### References:
 [1] H. Bahouri and G. Perelman, Global well-posedness for the derivative nonlinear Schrödinger equation, Invent. math., (2022), 50 pp doi: 10.1007/s00222-022-01113-0. [2] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equation. I, IMRP Int. Math. Res. Pap., 2006 (2006), 84 pp. doi: 10.1155/IMRP/2006/70630. [3] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Ⅱ, Trans. Amer. Math. Soc., 360 (2008), 5925-5957.  doi: 10.1090/S0002-9947-08-04471-1. [4] I. Bejenaru and D. Tataru, Large data local solutions for the derivative NLS equation, J. Eur. Math. Soc., 10 (2008), 957-985.  doi: 10.4171/JEMS/136. [5] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [6] M. Christ, Illposedness of a Schrödinger equation with derivative nonlinearity, preprint (http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.1363). [7] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differ. Integral Equ., 17 (2004), 297-330. [8] M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562.  doi: 10.1016/j.cam.2005.05.031. [9] 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. Nonlinéaire., 6 (2009), 2211-2226.  doi: 10.1016/j.anihpc.2009.01.011. [10] A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations, preprint, arXiv: math/0006195. [11] M. Hayashi and T. Ozawa, Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differ. Equ., 261 (2016), 5424-5445.  doi: 10.1016/j.jde.2016.08.018. [12] H. Hirayama, Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data, Commun. Pure Appl. Anal., 13 (2014), 1563-1591.  doi: 10.3934/cpaa.2014.13.1563. [13] H. Hirayama and S. Kinoshita, Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations, Nonlinear Anal., 178 (2019), 205-226.  doi: 10.1016/j.na.2018.07.013. [14] H. Hirayama, S. Kinoshita and and M. Okamoto, Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with radial initial data, Ann. H. Poincaré, 21 (2020), 2611-2636.  doi: 10.1007/s00023-020-00931-3. [15] H. Hirayama, S. Kinoshita and M. Okamoto, Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces, J. Math. Anal. Appl., 499 (2021), 29 pp. doi: 10.1016/j. jmaa. 2021.125028. [16] R. J. Iorio and V. M. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics, 70. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511623745. [17] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704. [18] 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. [19] R. Killip, M. Ntekoume and M. Visan, On the well-posedness problem for the derivative nonlinear Schrödinger equation, preprint, arXiv: 2101.12274 [20] T. Ozawa, Finite energy solutions for the Schrödinger equations with quadratic nonlinearity in one space dimension, Funkcial. Ekvac., 41 (1998), 451-468. [21] D. Pornnopparath, Small data well-posedness for derivative nonlinear Schrödinger equations, J. Differ. Equ., 265 (2018), 3792-3840.  doi: 10.1016/j.jde.2018.05.016.

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##### References:
 [1] H. Bahouri and G. Perelman, Global well-posedness for the derivative nonlinear Schrödinger equation, Invent. math., (2022), 50 pp doi: 10.1007/s00222-022-01113-0. [2] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equation. I, IMRP Int. Math. Res. Pap., 2006 (2006), 84 pp. doi: 10.1155/IMRP/2006/70630. [3] I. Bejenaru, Quadratic nonlinear derivative Schrödinger equations. Ⅱ, Trans. Amer. Math. Soc., 360 (2008), 5925-5957.  doi: 10.1090/S0002-9947-08-04471-1. [4] I. Bejenaru and D. Tataru, Large data local solutions for the derivative NLS equation, J. Eur. Math. Soc., 10 (2008), 957-985.  doi: 10.4171/JEMS/136. [5] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035. [6] M. Christ, Illposedness of a Schrödinger equation with derivative nonlinearity, preprint (http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.1363). [7] M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differ. Integral Equ., 17 (2004), 297-330. [8] M. Colin and T. Colin, A numerical model for the Raman amplification for laser-plasma interaction, J. Comput. Appl. Math., 193 (2006), 535-562.  doi: 10.1016/j.cam.2005.05.031. [9] 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. Nonlinéaire., 6 (2009), 2211-2226.  doi: 10.1016/j.anihpc.2009.01.011. [10] A. Grünrock, On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schrödinger equations, preprint, arXiv: math/0006195. [11] M. Hayashi and T. Ozawa, Well-posedness for a generalized derivative nonlinear Schrödinger equation, J. Differ. Equ., 261 (2016), 5424-5445.  doi: 10.1016/j.jde.2016.08.018. [12] H. Hirayama, Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data, Commun. Pure Appl. Anal., 13 (2014), 1563-1591.  doi: 10.3934/cpaa.2014.13.1563. [13] H. Hirayama and S. Kinoshita, Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations, Nonlinear Anal., 178 (2019), 205-226.  doi: 10.1016/j.na.2018.07.013. [14] H. Hirayama, S. Kinoshita and and M. Okamoto, Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations with radial initial data, Ann. H. Poincaré, 21 (2020), 2611-2636.  doi: 10.1007/s00023-020-00931-3. [15] H. Hirayama, S. Kinoshita and M. Okamoto, Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces, J. Math. Anal. Appl., 499 (2021), 29 pp. doi: 10.1016/j. jmaa. 2021.125028. [16] R. J. Iorio and V. M. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge Studies in Advanced Mathematics, 70. Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511623745. [17] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704. [18] 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. [19] R. Killip, M. Ntekoume and M. Visan, On the well-posedness problem for the derivative nonlinear Schrödinger equation, preprint, arXiv: 2101.12274 [20] T. Ozawa, Finite energy solutions for the Schrödinger equations with quadratic nonlinearity in one space dimension, Funkcial. Ekvac., 41 (1998), 451-468. [21] D. Pornnopparath, Small data well-posedness for derivative nonlinear Schrödinger equations, J. Differ. Equ., 265 (2018), 3792-3840.  doi: 10.1016/j.jde.2018.05.016.
Regularities to be well-posed when $\kappa \neq 0$
 $\kappa \neq 0$ $d=1$ $d=2$ $d=3$ $d \ge 4$ $\mu>0$ $s \ge 0$ $s \ge s_c$ $\mu=0$ $s \ge 1$ $\mu<0$ $s \ge \frac 12$ $s>s_c$
 $\kappa \neq 0$ $d=1$ $d=2$ $d=3$ $d \ge 4$ $\mu>0$ $s \ge 0$ $s \ge s_c$ $\mu=0$ $s \ge 1$ $\mu<0$ $s \ge \frac 12$ $s>s_c$
Regularities to be well-posed when $\alpha= \beta$ and $( \alpha- \gamma)( \beta+ \gamma) \neq 0$
 $d=1$ $d=2$ $d=3$ $d \ge 4$ $s \ge \frac 12$ $s>s_c$
 $d=1$ $d=2$ $d=3$ $d \ge 4$ $s \ge \frac 12$ $s>s_c$
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