Global Well-Posedness and Scattering for the Defocusing H 1 2 -Subcritical Hartree Equation in R d
Miao, Changxing ; Xu, Guixiang ; Zhao, Lifeng
Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009), p. 1831-1852 / Harvested from Numdam
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@article{AIHPC_2009__26_5_1831_0,
     author = {Miao, Changxing and Xu, Guixiang and Zhao, Lifeng},
     title = {Global Well-Posedness and Scattering for the Defocusing ${H}^{\frac{1}{2}}$-Subcritical Hartree Equation in ${R}^{d}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {26},
     year = {2009},
     pages = {1831-1852},
     doi = {10.1016/j.anihpc.2009.01.003},
     mrnumber = {2566712},
     zbl = {1176.35140},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2009__26_5_1831_0}
}

Miao, Changxing; Xu, Guixiang; Zhao, Lifeng. Global Well-Posedness and Scattering for the Defocusing ${H}^{\frac{1}{2}}$-Subcritical Hartree Equation in ${R}^{d}$. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) pp. 1831-1852. doi : 10.1016/j.anihpc.2009.01.003. http://gdmltest.u-ga.fr/item/AIHPC_2009__26_5_1831_0/

[1] Cazenave T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University Courant Institute of Mathematical Sciences, New York, 2003. | MR 2002047 | Zbl 1055.35003

[2] J. Colliander, M. Grillakis, N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on R 2 , IMRN 23 (2007), Art. ID rnm090, 30 pp. | MR 2377216 | Zbl 1142.35085

[3] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Global Existence and Scattering for Rough Solutions to a Nonlinear Schrödinger Equations on R 3 , Comm. Pure Appl. Math. 57 (8) (2004) 987-1014. | MR 2053757 | Zbl 1060.35131

[4] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Resonant Decompositions and the I-Method for Cubic Nonlinear Schrödinger on R 2 , Discrete Contin. Dynam. Systems 21 (3) (2008) 665-686. | MR 2399431 | Zbl 1147.35095

[5] Chae M., Hong S., Kim J., Yang C. W., Scattering Theory Below Energy for a Class of Hartree Type Equations, Comm. Partial Differential Equations 33 (2008) 321-348. | MR 2398232 | Zbl pre05270170

[6] De Silva D., Pavlovic N., Staffilani G., Tzirakis N., Global Well-Posedness and Polynomial Bounds for the Defocusing L 2 -Critical Nonlinear Schrödinger Equation in R, Comm. Partial Differential Equations 33 (2008) 1395-1429. | MR 2450163 | Zbl 1155.35310

[7] Ginibre J., Ozawa T., Long Range Scattering for Nonlinear Schrödinger and Hartree Equations in Space Dimension n≥2, Comm. Math. Phys. 151 (1993) 619-645. | MR 1207269 | Zbl 0776.35070

[8] Ginibre J., Velo G., Scattering Theory in the Energy Space for a Class of Hartree Equations, in: Nonlinear Wave Equations, Providence, RI, 1998, Contemp. Math., vol. 263, Amer. Math. Soc., Providence, RI, 2000, pp. 29-60. | MR 1777634 | Zbl 0966.35095

[9] Ginibre J., Velo G., Long Range Scattering and Modified Wave Operators for Some Hartree Type Equations, Rev. Math. Phys. 12 (3) (2000) 361-429. | MR 1755906 | Zbl 1044.35041

[10] Ginibre J., Velo G., Long Range Scattering and Modified Wave Operators for Some Hartree Type Equations II, Ann. Inst. H. Poincaré 1 (4) (2000) 753-800. | MR 1785187 | Zbl 1024.35084

[11] Ginibre J., Velo G., Long Range Scattering and Modified Wave Operators for Some Hartree Type Equations. III: Gevrey Spaces and Low Dimensions, J. Differential Equations 175 (2) (2001) 415-501. | MR 1855975 | Zbl 0991.35057

[12] A. Grünrock, New applications of the Fourier restriction norm method to wellposedness problems for nonlinear evolution equations, Dissertation Univ. Wuppertal, 2002.

[13] Hayashi N., Tsutsumi Y., Scattering Theory for the Hartree Equations, Ann. Inst. H. Poincaré Phys. Théor. 61 (1987) 187-213. | Numdam | MR 887147 | Zbl 0634.35059

[14] Keel M., Tao T., Endpoint Strichartz Estimates, Amer. J. Math. 120 (5) (1998) 955-980. | MR 1646048 | Zbl 0922.35028

[15] Li D., Miao C., Zhang X., The Focusing Energy-Critical Hartree Equation, J. Differential Equations 246 (3) (2009) 1139-1163. | MR 2474589 | Zbl 1155.35421

[16] Lin J. E., Strauss W. A., Decay and Scattering of Solutions of a Nonlinear Schrödinger Equation, J. Funct. Anal. 30 (2) (1978) 245-263. | MR 515228 | Zbl 0395.35070

[17] Miao C., H m -Modified Wave Operator for Nonlinear Hartree Equation in the Space Dimensions n≥2, Acta Math. Sinica 13 (2) (1997) 247-268. | MR 1481805 | Zbl 0877.35089

[18] Miao C., Xu G., Zhao L., The Cauchy Problem of the Hartree Equation. Dedicated to Professor Li Daqian on the Occasion of Seventieth Birthday, J. Partial Differential Equations 21 (2008) 22-44. | MR 2394047 | Zbl 1174.35099

[19] Miao C., Xu G., Zhao L., Global Well-Posedness and Scattering for the Energy-Critical, Defocusing Hartree Equation for Radial Data, J. Funct. Anal. 253 (2007) 605-627. | MR 2370092 | Zbl 1136.35092

[20] Miao C., Xu G., Zhao L., Global Well-Posedness and Scattering for the Energy-Critical, Defocusing Hartree Equation in R 1+n , arXiv:0707.3254.

[21] Miao C., Xu G., Zhao L., Global Well-Posedness and Scattering for the Mass-Critical Hartree Equation With Radial Data, J. Math. Pures Appl. 91 (2009) 49-79. | MR 2487900 | Zbl 1154.35078

[22] Nakanishi K., Energy Scattering for Hartree Equations, Math. Res. Lett. 6 (1999) 107-118. | MR 1682697 | Zbl 0949.35104

[23] Nawa H., Ozawa T., Nonlinear Scattering With Nonlocal Interactions, Comm. Math. Phys. 146 (1992) 259-275. | MR 1165183 | Zbl 0748.35046

[24] Tao T., Multilinear Weighted Convolution of L 2 Functions, and Applications to Non-Linear Dispersive Equations, Amer. J. Math. 123 (2001) 839-908. | MR 1854113 | Zbl 0998.42005

[25] Tao T., Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conf. Ser. in Math., vol. 106, Amer. Math. Soc., 2006. | MR 2233925 | Zbl 1106.35001

[26] http://tosio.math.toronto.edu/wiki/index.php/Main_Page.