Small solutions to nonlinear Schrödinger equations
Annales de l'I.H.P. Analyse non linéaire, Volume 10 (1993) no. 3, pp. 255-288.
@article{AIHPC_1993__10_3_255_0,
     author = {Kenig, Carlos E. and Ponce, Gustavo and Vega, Luis},
     title = {Small solutions to nonlinear {Schr\"odinger} equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {255--288},
     publisher = {Gauthier-Villars},
     volume = {10},
     number = {3},
     year = {1993},
     zbl = {0786.35121},
     mrnumber = {1230709},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1993__10_3_255_0/}
}
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Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis. Small solutions to nonlinear Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Volume 10 (1993) no. 3, pp. 255-288. http://www.numdam.org/item/AIHPC_1993__10_3_255_0/

[1] A. Carbery, Radial Fourier multipliers and associated maximal function, North Holland Math. Studies, III, 1985, pp. 49-55. | MR | Zbl

[2] T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos, Vol. 22, Universidade Federal do Rio de Janeiro.

[3] T. Cazenave and F.B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Lecture in Math, Vol. 1392, Springer-Verlag, Berlin, New York, 1989, pp. 18-29. | MR | Zbl

[4] F.M. Christ and M. Weinstein, Dispersive small amplitude solution to the generalized Korteweg-de Vries equation, J. Funct. Anal., Vol. 100, 1991, pp. 87-109. | MR | Zbl

[5] R.R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiel, Astérisque, Vol. 57, 1973. | Zbl

[6] P. Constantin, and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math., Soc., Vol. 1, 1989, pp. 413-446. | MR | Zbl

[7] B. Dahlberg and C.E. Kenig, A note an almost every where behavior of solutions to the schrödinger equations, Lecture Notes in Math., Vol. 908, Springer-Verlag, Berlin, New York, 1982, pp. 205-208. | MR | Zbl

[8] J. Ginibre, and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equation, J. Math. pures et appl., Vol. 64, 1985, pp. 363-401. | MR | Zbl

[9] J. Ginibre and G. Velo, On a class of Schrödinger equations, J. Funct. Anal., Vol. 32, 1979, pp. 1-71. | MR | Zbl

[10] J. Ginibre and Y. Tsutsumi, Uniqueness for the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., Vol. 20, 1989, pp. 1388-1425. | MR | Zbl

[11] J.M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewarson systems, Nonlinearity, Vol. 3, 1990, pp. 475-506. | MR | Zbl

[12] R.T. Glassey, On the blowing up solutions to the Cauchy problem for nonlinar Schrödinger equations, J. Math. Phys., Vol. 18, 1979, pp. 1794-1797. | MR | Zbl

[13] N. Hayashi, Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J, Vol. 62, 1991, pp. 575-592. | MR

[14] N. Hayashi, K. Nakamitsu and M. Tsutsumi, On solutions to the initial value problem for the nonlinear Schrödinger equations in one dimensions, Math. Z., Vol. 192, 1986, pp. 637-650. | MR | Zbl

[15] N. Hayashi, K. Nakamitsu and M. Tsutsumi, On solutions to the initial value problem for the nonlinear Schrödinger equations, J. Funct. Anal., Vol. 71, 1987, pp. 218-245. | MR | Zbl

[16] N. Hayashi and S. Saitoh, Analyticity and global existence of small solutions to some nonlinear Shrödinger equations, Comm. Math. Phys., Vol. 129, 1990, pp. 27-41. | MR | Zbl

[17] T. Kato, Quasilinear evolution equation, with applications to partial differential equations, Lecture Notes in Math., Vol. 448, pringer-Verlag, pp. 27-50.

[18] T. Kato, Nonlinear Schrödinger equation, Schrodinger operators, H. Holden and A. Jensen Eds, Lecture Notes in Physics, Vol. 345, Springer-Verlag, Berlin, New York, 1989, pp. 218-263. | Zbl

[19] T. Kato, On the Cauchy problem for the (generalized) Kortewed-de Vries equation, Advances in Math. Supp. Studies, Studies in Applied Math., Vol. 8, 1983, pp. 93-128. | MR | Zbl

[20] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., Vol. 41, 1988, pp. 891-907. | MR | Zbl

[21] D.J. Kaup and A.C. Newell, An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., Vol. 19, 1978, pp. 798-801. | MR | Zbl

[22] C.E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., Vol. 4, 1991, pp. 323-347. | MR | Zbl

[23] C.E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana University Math. J., Vol. 40, 1991, pp. 33-69. | MR | Zbl

[24] C.E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc., (to appear). | MR | Zbl

[25] C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for generalized Korteweg-de Vries via contraction principle, Comm. Pure Appl. Math., Vol. 46, 1993, p. 527-620. | MR | Zbl

[26] C.E. Kenig, and A. Ruiz, A strong type (2, 2) estimate for the maximal function associated to the Schrödinger equation, Trans. Amer. Math. Soc., Vol. 280, 1983, pp. 239-246. | MR | Zbl

[27] S. Klainerman, Long time behavior of solutions to nonlinear evolutions equations, Arch. Ration. Mech. and Analysis, 78, 1981, pp. 73-98. | MR | Zbl

[28] S. Klainerman and G. Ponce, Global small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math., Vol. 36, 1983, pp. 133-141. | MR | Zbl

[29] J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Diff. Eqs., Vol. 46, 1982, pp. 409-423. | MR | Zbl

[30] J. Simon and E. Taflin, Wave operators and analytic solutions for systems of systems of nonlinear Klein-Gordon equations and of non-linear Schrödinger equations, Comm. Math. Phys., 99, 1985, pp. 541-562. | MR | Zbl

[31] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math., 55, 1987, pp. 699-715. | MR | Zbl

[32] E.M. Stein, Oscillaroty integrals in Fourier Analysis, Beijing Lectures in Harmonic Analysis, Princeton University Press, 1986, pp. 307-355. | MR | Zbl

[33] E.M. Stein and G. Weiss, Introduction to Fourier Analysis in Eucliden Spaces, Princeton University Press, 1971. | MR | Zbl

[34] W.A. Srauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41, 1981, pp. 110-133. | Zbl

[35] R.S. Strichartz, Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Vol. 44, 1977, pp. 705-714. | MR | Zbl

[36] Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equation, Nonlinear Anal., 11, 1987, pp. 1143-1154. | MR | Zbl

[37] Y. Tsutsumi, L2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, Vol. 31, 1987, pp. 115-125. | MR | Zbl

[38] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schrödinger equation. Existence and Uniqueness Theorem, Funkcialaj Ekvacioj, 23, 1980, pp. 259-277. | MR | Zbl

[39] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schröndinger equation. II, Funkcialaj Ekvacioj, 24, 1981, pp. 85-94. | MR | Zbl

[40] L. Vega, Doctoral Thesis, Universidad Autonoma de Madrid, Spain, 1987.

[41] L. Vega, The Schrödinger eqution: pointwise convergence to the initial date, Proc. Amer. Math. Soc., Vol. 102, 1988, pp. 874-878. | MR | Zbl