Global Well-Posedness and Scattering for the Defocusing ${H}^{\frac{1}{2}}$-Subcritical Hartree Equation in ${R}^{d}$

Miao, Changxing; Xu, Guixiang; Zhao, Lifeng

doi:10.1016/j.anihpc.2009.01.003

Global Well-Posedness and Scattering for the Defocusing

H^{\frac{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) no. 5, pp. 1831-1852

MR Zbl

DOI : 10.1016/j.anihpc.2009.01.003

@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},
     pages = {1831--1852},
     year = {2009},
     publisher = {Elsevier},
     volume = {26},
     number = {5},
     doi = {10.1016/j.anihpc.2009.01.003},
     mrnumber = {2566712},
     zbl = {1176.35140},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2009.01.003/}
}

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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) no. 5, pp. 1831-1852. doi: 10.1016/j.anihpc.2009.01.003

Bibliographie
Cité par

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

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR | Numdam

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

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

[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. | Zbl | MR

[17] Miao C., $H^{m}$ -Modified Wave Operator for Nonlinear Hartree Equation in the Space Dimensions $n \geq 2$ , Acta Math. Sinica 13 (2) (1997) 247-268. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

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

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

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