American Institute of Mathematical Sciences

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

On Fractional Schrödinger Equations in sobolev spaces

Younghun Hong 1, and  Yannick Sire 2,

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.
Keywords: Sobolev spaces, ill-posedness., Fractional Schrödinger, Local and global well-posedness.
Mathematics Subject Classification: 35A01, 35B35, 35B45, 35B6.
Citation: Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[3]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear schrödinger and wave equations, arXiv:math/0311048, 2003.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian,, \emph{Preprint}., ().   Google Scholar

[14]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional schrödinger equation in the radial case, Preprint, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  Google Scholar

[22]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972,  Google Scholar

[23]

Nick Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[24]

Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.   Google Scholar

show all references

References:
[1]

Jean Bertoin, Lévy processes, volume 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.  Google Scholar

[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.  Google Scholar

[3]

M. Christ, J. Colliander and T. Tao, Ill-posedness for nonlinear schrödinger and wave equations, arXiv:math/0311048, 2003.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional laplacian,, \emph{Preprint}., ().   Google Scholar

[14]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional schrödinger equation in the radial case, Preprint, 2013. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  Google Scholar

[22]

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972,  Google Scholar

[23]

Nick Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[24]

Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, \emph{Comm. Math. Phys.}, 87 (): 567.   Google Scholar

[1]

Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[2]

G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[3]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072
[4]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082
[5]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[6]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[7]

Fabio S. Bemfica, Marcelo M. Disconzi, P. Jameson Graber. Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, 2021, 20 (9) : 2885-2914. doi: 10.3934/cpaa.2021068
[8]

Michael S. Jolly, Anuj Kumar, Vincent R. Martinez. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021169
[9]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053
[10]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197
[11]

Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021147
[12]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[13]

Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066
[14]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123
[15]

Xuan Liu, Ting Zhang. Local well-posedness and finite time blowup for fourth-order Schrödinger equation with complex coefficient. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021156
[16]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[17]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261
[18]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[19]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37
[20]

Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (237)
  • HTML views (0)
  • Cited by (45)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences