Communications on Pure and Applied Analysis (CPAA)

On Fractional Schrödinger Equations in sobolev spaces
Pages: 2265 - 2282, Issue 6, November 2015

doi:10.3934/cpaa.2015.14.2265      Abstract        References        Full text (445.8K)           Related Articles

Younghun Hong - University of Texas at Austin, United States (email)
Yannick Sire - Université Aix-Marseille, I2M, France (email)

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.       
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.       
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.       
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.       
10 Yonggeun Cho, Tohru Ozawa and Suxia Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128.       
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.       
12 Rupert L. Frank and Enno Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $R$, Acta Math., 210 (2013), 261-318.       
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.       
16 Taoufik Hmidi and Sahbi Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 46 (2005), 2815-2828.       
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.       
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.       
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.       
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.       
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.       
24 Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.       

Go to top