# American Institute of Mathematical Sciences

November  2015, 14(6): 2265-2282. doi: 10.3934/cpaa.2015.14.2265

## On Fractional Schrödinger Equations in sobolev spaces

 1 University of Texas at Austin, United States 2 Université Aix-Marseille, I2M, France

Received  December 2014 Revised  June 2015 Published  September 2015

Let $\sigma \in (0,1)$ with $\sigma \neq \frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: \begin{eqnarray} i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0, u(0)=u_0\in H^s, \end{eqnarray} where $(-\Delta)^\sigma$ is the Fourier multiplier of symbol $|\xi|^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics [23]. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.
Citation: Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
##### References:
 [1] Jean Bertoin, Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. [2] Thierry Cazenave, Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. [3] M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear schrödinger and wave equations, arXiv:math/0311048, 2003. [4] Michael Christ, James Colliander and Terence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. [5] Y. Cho, M. Fall, H. Hajaiej, P. Markowich and S. Trabelsi, Orbital stability of standing waves of a class of fractional schrödinger equations with a general hartree-type integrand, Preprint, 2013. [6] Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang and Tohru Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193. [7] Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang and Tohru Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267. [8] Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional schrödinger equations, arxiv.org/abs/1311.0082, 2014. doi: 10.3934/dcds.2015.35.2863. [9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $R^3$, Ann. of Math. (2), 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. [10] Yonggeun Cho, Tohru Ozawa and Suxia Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. [11] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. [12] Rupert L. Frank and Enno Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $R$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. [13] R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian,, \emph{Preprint}., (). [14] Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional schrödinger equation in the radial case, Preprint, 2013. [15] Z. Guo and Y. Wang, Improved strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear schrodinger and wave equation, to appear J. Anal. Math., 2014. doi: 10.1007/s11854-014-0025-6. [16] Taoufik Hmidi and Sahbi Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 46 (2005), 2815-2828. doi: 10.1155/IMRN.2005.2815. [17] Joachim Krieger, Enno Lenzmann and Pierre Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129. doi: 10.1007/s00205-013-0620-1. [18] Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. [19] Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6. [20] Carlos E. Kenig, Gustavo Ponce and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. doi: 10.1215/S0012-7094-01-10638-8. [21] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. [22] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972, [23] Nick Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7. doi: 10.1103/PhysRevE.66.056108. [24] Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.

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##### References:
 [1] Jean Bertoin, Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. [2] Thierry Cazenave, Semilinear Schrödinger equations, volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. [3] M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear schrödinger and wave equations, arXiv:math/0311048, 2003. [4] Michael Christ, James Colliander and Terence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. [5] Y. Cho, M. Fall, H. Hajaiej, P. Markowich and S. Trabelsi, Orbital stability of standing waves of a class of fractional schrödinger equations with a general hartree-type integrand, Preprint, 2013. [6] Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang and Tohru Ozawa, On the Cauchy problem of fractional Schrödinger equation with Hartree type nonlinearity, Funkcial. Ekvac., 56 (2013), 193-224. doi: 10.1619/fesi.56.193. [7] Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang and Tohru Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267. [8] Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional schrödinger equations, arxiv.org/abs/1311.0082, 2014. doi: 10.3934/dcds.2015.35.2863. [9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $R^3$, Ann. of Math. (2), 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. [10] Yonggeun Cho, Tohru Ozawa and Suxia Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128. doi: 10.3934/cpaa.2011.10.1121. [11] F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. [12] Rupert L. Frank and Enno Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $R$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. [13] R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian,, \emph{Preprint}., (). [14] Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional schrödinger equation in the radial case, Preprint, 2013. [15] Z. Guo and Y. Wang, Improved strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear schrodinger and wave equation, to appear J. Anal. Math., 2014. doi: 10.1007/s11854-014-0025-6. [16] Taoufik Hmidi and Sahbi Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 46 (2005), 2815-2828. doi: 10.1155/IMRN.2005.2815. [17] Joachim Krieger, Enno Lenzmann and Pierre Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129. doi: 10.1007/s00205-013-0620-1. [18] Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. [19] Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6. [20] Carlos E. Kenig, Gustavo Ponce and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633. doi: 10.1215/S0012-7094-01-10638-8. [21] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. [22] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972, [23] Nick Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7. doi: 10.1103/PhysRevE.66.056108. [24] Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.
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