# American Institute of Mathematical Sciences

July  2015, 35(7): 2863-2880. doi: 10.3934/dcds.2015.35.2863

## Well-posedness and ill-posedness for the cubic fractional Schrödinger equations

 1 Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756 2 Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, 689-798, South Korea 3 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon, Korea 305-701 4 Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea

Received  May 2014 Revised  December 2014 Published  January 2015

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < \alpha < 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-\alpha}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow map is not locally uniformly continuous.
Citation: Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
##### References:
 [1] N. Burq, P. Gerard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett, 9 (2002), 323-335. doi: 10.4310/MRL.2002.v9.n3.a8. [2] W. Chen and Z. Guo, Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation, J. Anal. Math., 114 (2011), 121-156. doi: 10.1007/s11854-011-0014-y. [3] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math, 125 (2003), 1235-1293. doi: 10.1353/ajm.2003.0040. [4] _______, Instability of the periodic nonlinear Schrödinger equation,, preprint, (). [5] Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcialaj Ekvacioj, 56 (2013), 193-224. doi: 10.1619/fesi.56.193. [6] _______, On the orbital stability of fractional Schrödinger equations, Comm. Pure Appl. Anal, 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267. [7] Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions and blowup phenomena of mass critical fractional Schrödinger equations, Nolinear Analysis, 86 (2013), 12-29. doi: 10.1016/j.na.2013.03.002. [8] _______, Profile decompositions of fractional Schrodinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037. doi: 10.1016/j.jde.2014.01.030. [9] Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970. [10] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. [11] S. Demirbas, M. B. Erdoǧan and N. Tzirakis, Existence and uniqueness theory for the fractional Schrödinger equation on the torus,, preprint, (). [12] B. Guo and Z. Huo, Global Well-Posedness for the Fractional Nonlinear Schrödinger Equation, Comm. Partial Differential Equations, 36 (2011), 247-255. doi: 10.1080/03605302.2010.503769. [13] A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Func. Anal., 266 (2014), 139-176. doi: 10.1016/j.jfa.2013.08.027. [14] S. Kwon, Well-posedness and ill-posedness of the fifth-order modified KdV equations, Elec. J. Diff. Eqns, (2008), 1-15. [15] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. [16] L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett, 16 (2009), 111-120. doi: 10.4310/MRL.2009.v16.n1.a11. [17] T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.

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##### References:
 [1] N. Burq, P. Gerard and N. Tzvetkov, An instability property of the nonlinear Schrödinger equation on $S^d$, Math. Res. Lett, 9 (2002), 323-335. doi: 10.4310/MRL.2002.v9.n3.a8. [2] W. Chen and Z. Guo, Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation, J. Anal. Math., 114 (2011), 121-156. doi: 10.1007/s11854-011-0014-y. [3] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math, 125 (2003), 1235-1293. doi: 10.1353/ajm.2003.0040. [4] _______, Instability of the periodic nonlinear Schrödinger equation,, preprint, (). [5] Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcialaj Ekvacioj, 56 (2013), 193-224. doi: 10.1619/fesi.56.193. [6] _______, On the orbital stability of fractional Schrödinger equations, Comm. Pure Appl. Anal, 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267. [7] Y. Cho, G. Hwang, S. Kwon and S. Lee, Profile decompositions and blowup phenomena of mass critical fractional Schrödinger equations, Nolinear Analysis, 86 (2013), 12-29. doi: 10.1016/j.na.2013.03.002. [8] _______, Profile decompositions of fractional Schrodinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037. doi: 10.1016/j.jde.2014.01.030. [9] Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020. doi: 10.1512/iumj.2013.62.4970. [10] Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. [11] S. Demirbas, M. B. Erdoǧan and N. Tzirakis, Existence and uniqueness theory for the fractional Schrödinger equation on the torus,, preprint, (). [12] B. Guo and Z. Huo, Global Well-Posedness for the Fractional Nonlinear Schrödinger Equation, Comm. Partial Differential Equations, 36 (2011), 247-255. doi: 10.1080/03605302.2010.503769. [13] A. D. Ionescu and F. Pusateri, Nonlinear fractional Schrödinger equations in one dimension, J. Func. Anal., 266 (2014), 139-176. doi: 10.1016/j.jfa.2013.08.027. [14] S. Kwon, Well-posedness and ill-posedness of the fifth-order modified KdV equations, Elec. J. Diff. Eqns, (2008), 1-15. [15] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2. [16] L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schrödinger equation, Math. Res. Lett, 16 (2009), 111-120. doi: 10.4310/MRL.2009.v16.n1.a11. [17] T. Tao, Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. doi: 10.1353/ajm.2001.0035.
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