Global Well-Posedness and Scattering for the Derivative Nonlinear Schrödinger Equation With Small Rough Data

Wang, Baoxiang; Han, Lijia; Huang, Chunyan

doi:10.1016/j.anihpc.2009.03.004

Wang, Baoxiang ; Han, Lijia ; Huang, Chunyan

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 6, pp. 2253-2281.

MR Zbl

DOI : 10.1016/j.anihpc.2009.03.004

@article{AIHPC_2009__26_6_2253_0,
     author = {Wang, Baoxiang and Han, Lijia and Huang, Chunyan},
     title = {Global {Well-Posedness} and {Scattering} for the {Derivative} {Nonlinear} {Schr\"odinger} {Equation} {With} {Small} {Rough} {Data}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {2253--2281},
     publisher = {Elsevier},
     volume = {26},
     number = {6},
     year = {2009},
     doi = {10.1016/j.anihpc.2009.03.004},
     mrnumber = {2569894},
     zbl = {1180.35492},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2009.03.004/}
}

TY  - JOUR
AU  - Wang, Baoxiang
AU  - Han, Lijia
AU  - Huang, Chunyan
TI  - Global Well-Posedness and Scattering for the Derivative Nonlinear Schrödinger Equation With Small Rough Data
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2009
SP  - 2253
EP  - 2281
VL  - 26
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2009.03.004/
DO  - 10.1016/j.anihpc.2009.03.004
LA  - en
ID  - AIHPC_2009__26_6_2253_0
ER  -

%0 Journal Article
%A Wang, Baoxiang
%A Han, Lijia
%A Huang, Chunyan
%T Global Well-Posedness and Scattering for the Derivative Nonlinear Schrödinger Equation With Small Rough Data
%J Annales de l'I.H.P. Analyse non linéaire
%D 2009
%P 2253-2281
%V 26
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2009.03.004/
%R 10.1016/j.anihpc.2009.03.004
%G en
%F AIHPC_2009__26_6_2253_0

Wang, Baoxiang; Han, Lijia; Huang, Chunyan. Global Well-Posedness and Scattering for the Derivative Nonlinear Schrödinger Equation With Small Rough Data. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 6, pp. 2253-2281. doi : 10.1016/j.anihpc.2009.03.004. http://www.numdam.org/articles/10.1016/j.anihpc.2009.03.004/

Bibliographie
Cité par

[1] Bergh J., Löfström J., Interpolation Spaces, Springer-Verlag, 1976. | Zbl

[2] Bejenaru I., Tataru D., Large Data Local Solutions for the Derivative NLS Equation, arXiv:math.AP/0610092v1. | MR

[3] Chihara H., Global Existence of Small Solutions to Semilinear Schrödinger Equations With Gauge Invariance, Publ. Res. Inst. Math. Sci. 31 (1995) 731-753. | MR | Zbl

[4] Chihara H., The Initial Value Problem for Cubic Semilinear Schrödinger Equations With Gauge Invariance, Publ. Res. Inst. Math. Sci. 32 (1996) 445-471. | MR | Zbl

[5] Christ M., Illposedness of a Schrödinger Equation With Derivative Regularity, preprint, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.1363&rep=rep1&type=pdf.

[6] Christ M., Kiselev A., Maximal Functions Associated to Filtrations, J. Funct. Anal. 179 (2001) 406-425. | MR | Zbl

[7] Constantin P., Saut J. C., Local Smoothing Properties of Dispersive Equations, J. Amer. Math. Soc. 1 (1988) 413-446. | MR | Zbl

[8] Clarkson P. A., Tuszynski J. A., Exact Solutions of the Multidimensional Derivative Nonlinear Schrödinger Equation for Many-Body Systems Near Criticality, J. Phys. A: Math. Gen. 23 (1990) 4269-4288. | MR | Zbl

[9] Ding Wei-Yue, Wang You-De, Schrödinger Flow of Maps Into Symplectic Manifolds, Sci. China Ser. A 41 (1998) 746-755. | MR | Zbl

[10] Dixon J. M., Tuszynski J. A., Coherent Structures in Strongly Interacting Many-Body Systems: II. Classical Solutions and Quantum Fluctuations, J. Phys. A: Math. Gen. 22 (1989) 4895-4920. | MR | Zbl

[11] H.G. Feichtinger, Modulation spaces on locally compact Abelian group, Technical Report, University of Vienna, 1983, in: Proc. Internat. Conf. on Wavelet and Applications, New Delhi Allied Publishers, India, 2003, pp. 99-140, http://www.unive.ac.at/nuhag-php/bibtex/open_files/fe03-1_modspa03.pdf.

[12] Ionescu A., Kenig C. E., Low-Regularity Schrödinger Maps, II: Global Well Posedness in Dimensions $d \geq 3$ , Comm. Math. Phys. 271 (2007) 523-559, arXiv:math/0605209v1. | MR | Zbl

[13] Kenig C. E., Ponce G., Vega L., Oscillatory Integrals and Regularity of Dispersive Equations, Indiana Univ. Math. J. 40 (1991) 253-288. | MR | Zbl

[14] Kenig C. E., Ponce G., Vega L., Small Solutions to Nonlinear Schrodinger Equation, Ann. Inst. H. Poincaré Sect. C 10 (1993) 255-288. | Numdam | MR | Zbl

[15] Kenig C. E., Ponce G., Vega L., Smoothing Effects and Local Existence Theory for the Generalized Nonlinear Schrödinger Equations, Invent. Math. 134 (1998) 489-545. | MR | Zbl

[16] Kenig C. E., Ponce G., Vega L., The Cauchy Problem for Quasi-Linear Schrodinger Equations, Invent. Math. 158 (2004) 343-388. | MR | Zbl

[17] Kenig C. E., Ponce G., Rolvent C., Vega L., The Genereal Quasilinear Untrahyperbolic Schrodinger Equation, Adv. Math. 206 (2006) 402-433. | MR | Zbl

[18] Klainerman S., Long-Time Behavior of Solutions to Nonlinear Evolution Equations, Arch. Ration. Mech. Anal. 78 (1982) 73-98. | MR | Zbl

[19] Klainerman S., Ponce G., Global Small Amplitude Solutions to Nonlinear Evolution Equations, Comm. Pure Appl. Math. 36 (1983) 133-141. | MR | Zbl

[20] Linares F., Ponce G., On the Davey-Stewartson Systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993) 523-548. | Numdam | MR | Zbl

[21] Molinet L., Ribaud F., Well-Posedness Results for the Generalized Benjamin-Ono Equation With Small Initial Data, J. Math. Pures Appl. 83 (2004) 277-311. | MR | Zbl

[22] Ozawa T., Zhai J., Global Existence of Small Classical Solutions to Nonlinear Schrödinger Equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 303-311. | Numdam | MR | Zbl

[23] Shatah J., Global Existence of Small Classical Solutions to Nonlinear Evolution Equations, J. Differential Equations 46 (1982) 409-423. | MR | Zbl

[24] Sjölin P., Regularity of Solutions to the Schrödinger Equations, Duke Math. J. 55 (1987) 699-715. | MR | Zbl

[25] Smith H. F., Sogge C. D., Global Strichartz Estimates for Nontrapping Perturbations of Laplacian, Comm. Partial Differential Equations 25 (2000) 2171-2183. | MR | Zbl

[26] Sugimoto M., Tomita N., The Dilation Property of Modulation Spaces and Their Inclusion Relation With Besov Spaces, J. Funct. Anal. 248 (2007) 79-106. | MR | Zbl

[27] Tuszynski J. A., Dixon J. M., Coherent Structures in Strongly Interacting Many-Body Systems: I. Derivation of Dynamics, J. Phys. A: Math. Gen. 22 (1989) 4877-4894. | MR | Zbl

[28] Toft J., Continuity Properties for Modulation Spaces, With Applications to Pseudo-Differential Calculus, I, J. Funct. Anal. 207 (2004) 399-429. | MR | Zbl

[29] Triebel H., Theory of Function Spaces, Birkhäuser-Verlag, 1983. | MR | Zbl

[30] Wang B. X., Zhao L. F., Guo B. L., Isometric Decomposition Operators, Function Spaces $E_{p, q}^{λ}$ and Applications to Nonlinear Evolution Equations, J. Funct. Anal. 233 (2006) 1-39. | MR | Zbl

[31] Wang B. X., Hudzik H., The Global Cauchy Problem for the NLS and NLKG With Small Rough Data, J. Differential Equations 231 (2007) 36-73. | MR | Zbl

[32] Wang B. X., Huang C. Y., Frequency-Uniform Decomposition Method for the Generalized BO, KdV and NLS Equations, J. Differential Equations 239 (2007) 213-250. | MR

[33] Wang B. X., Wang Y. Z., Global Well Posedness and Scattering for the Elliptic and Non-Elliptic Derivative Nonlinear Schrödinger Equations With Small Data, preprint, arXiv:0803.2634.

[34] Vega L., The Schrödinger Equation: Pointwise Convergence to the Initial Data, Proc. Amer. Math. Soc. 102 (1988) 874-878. | MR | Zbl

Cité par Sources :