Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data. (English) Zbl 1180.35492
Summary: We study the Cauchy problem for the generalized elliptic and non-elliptic derivative nonlinear Schrödinger equations (DNLS) and get the global well posedness of solutions with small data in modulation spaces \(M_{2,1}^s(\mathbb R^n)\). Noticing that \(B_{2,1}^{s+n/2}\subset M_{2,1}^s\subset B_{2,1}^s\) are optimal inclusions, we have shown the global well posedness of DNLS with a class of rough data. As a by-product, the existence of the scattering operators in modulation spaces with small data is also obtained.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B45 | A priori estimates in context of PDEs |
| 81U99 | Quantum scattering theory |
Keywords:
derivative nonlinear Schrödinger equation; scattering; global well posedness; small rough data; modulation spacesReferences:
| [1] | Bergh, J.; Löfström, J., Interpolation Spaces (1976), Springer-Verlag · Zbl 0344.46071 |
| [2] | Bejenaru, I.; Tataru, D., Large data local solutions for the derivative NLS equation · Zbl 1250.35160 |
| [3] | Chihara, H., Global existence of small solutions to semilinear Schrödinger equations with gauge invariance, Publ. Res. Inst. Math. Sci., 31, 731-753 (1995) · Zbl 0847.35126 |
| [4] | Chihara, H., The initial value problem for cubic semilinear Schrödinger equations with gauge invariance, Publ. Res. Inst. Math. Sci., 32, 445-471 (1996) · Zbl 0870.35095 |
| [5] | Christ, M., Illposedness of a Schrödinger equation with derivative regularity (2003), preprint |
| [6] | Christ, M.; Kiselev, A., Maximal functions associated to filtrations, J. Funct. Anal., 179, 406-425 (2001) · Zbl 0974.47025 |
| [7] | Constantin, P.; Saut, J. C., Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1, 413-446 (1988) · Zbl 0667.35061 |
| [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, 4269-4288 (1990) · Zbl 0738.35090 |
| [9] | Ding, Wei-Yue; Wang, You-De, Schrödinger flow of maps into symplectic manifolds, Sci. China Ser. A, 41, 746-755 (1998) · Zbl 0918.53017 |
| [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, 4895-4920 (1989) · Zbl 0714.70021 |
| [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; 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 \(d \geqslant 3\), Comm. Math. Phys., 271, 523-559 (2007) · Zbl 1137.35068 |
| [13] | Kenig, C. E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40, 253-288 (1991) |
| [14] | Kenig, C. E.; Ponce, G.; Vega, L., Small solutions to nonlinear Schrodinger equation, Ann. Inst. H. Poincaré Sect. C, 10, 255-288 (1993) · Zbl 0786.35121 |
| [15] | Kenig, C. E.; Ponce, G.; Vega, L., Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math., 134, 489-545 (1998) · Zbl 0928.35158 |
| [16] | Kenig, C. E.; Ponce, G.; Vega, L., The Cauchy problem for quasi-linear Schrodinger equations, Invent. Math., 158, 343-388 (2004) · Zbl 1177.35221 |
| [17] | Kenig, C. E.; Ponce, G.; Rolvent, C.; Vega, L., The genereal quasilinear untrahyperbolic Schrodinger equation, Adv. Math., 206, 402-433 (2006) · Zbl 1122.35137 |
| [18] | Klainerman, S., Long-time behavior of solutions to nonlinear evolution equations, Arch. Ration. Mech. Anal., 78, 73-98 (1982) · Zbl 0502.35015 |
| [19] | Klainerman, S.; Ponce, G., Global small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math., 36, 133-141 (1983) · Zbl 0509.35009 |
| [20] | Linares, F.; Ponce, G., On the Davey-Stewartson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10, 523-548 (1993) · Zbl 0807.35136 |
| [21] | Molinet, L.; Ribaud, F., Well-posedness results for the generalized Benjamin-Ono equation with small initial data, J. Math. Pures Appl., 83, 277-311 (2004) · Zbl 1084.35094 |
| [22] | Ozawa, T.; Zhai, J., Global existence of small classical solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 303-311 (2008) · Zbl 1143.35370 |
| [23] | Shatah, J., Global existence of small classical solutions to nonlinear evolution equations, J. Differential Equations, 46, 409-423 (1982) · Zbl 0518.35046 |
| [24] | Sjölin, P., Regularity of solutions to the Schrödinger equations, Duke Math. J., 55, 699-715 (1987) · Zbl 0631.42010 |
| [25] | Smith, H. F.; Sogge, C. D., Global Strichartz estimates for nontrapping perturbations of Laplacian, Comm. Partial Differential Equations, 25, 2171-2183 (2000) · Zbl 0972.35014 |
| [26] | Sugimoto, M.; Tomita, N., The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248, 79-106 (2007) · Zbl 1124.42018 |
| [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, 4877-4894 (1989) · Zbl 0714.70020 |
| [28] | Toft, J., Continuity properties for modulation spaces, with applications to pseudo-differential calculus, I, J. Funct. Anal., 207, 399-429 (2004) · Zbl 1083.35148 |
| [29] | Triebel, H., Theory of Function Spaces (1983), Birkhäuser-Verlag · Zbl 0546.46028 |
| [30] | Wang, B. X.; Zhao, L. F.; Guo, B. L., Isometric decomposition operators, function spaces \(E_{p, q}^\lambda\) and applications to nonlinear evolution equations, J. Funct. Anal., 233, 1-39 (2006) · Zbl 1099.46023 |
| [31] | Wang, B. X.; Hudzik, H., The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 231, 36-73 (2007) · Zbl 1121.35132 |
| [32] | Wang, B. X.; Huang, C. Y., Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239, 213-250 (2007) · Zbl 1219.35289 |
| [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 |
| [34] | Vega, L., The Schrödinger equation: Pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102, 874-878 (1988) · Zbl 0654.42014 |
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