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On Dirac equation with a potential and critical Sobolev exponent
On Fractional Schrödinger Equations in sobolev spaces
1. | University of Texas at Austin, United States |
2. | Université Aix-Marseille, I2M, France |
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 $\mathbb{R}^{N}$, 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, 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, Comm. Math. Phys., 87 (1982/83), 567-576. |
show all references
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 $\mathbb{R}^{N}$, 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, 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, Comm. Math. Phys., 87 (1982/83), 567-576. |
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