@article{AIHPC_1993__10_3_255_0, author = {Kenig, Carlos 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}, volume = {10}, year = {1993}, pages = {255-288}, mrnumber = {1230709}, zbl = {0786.35121}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_1993__10_3_255_0} }
Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis. Small solutions to nonlinear Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Tome 10 (1993) pp. 255-288. http://gdmltest.u-ga.fr/item/AIHPC_1993__10_3_255_0/
[1] Radial Fourier multipliers and associated maximal function, North Holland Math. Studies, III, 1985, pp. 49-55. | MR 848141 | Zbl 0632.42012
,[2] An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos, Vol. 22, Universidade Federal do Rio de Janeiro.
,[3] 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 1021011 | Zbl 0694.35170
and ,[4] Dispersive small amplitude solution to the generalized Korteweg-de Vries equation, J. Funct. Anal., Vol. 100, 1991, pp. 87-109. | MR 1124294 | Zbl 0743.35067
and ,[5] Au-delà des opérateurs pseudo-différentiel, Astérisque, Vol. 57, 1973. | Zbl 0483.35082
and ,[6] Local smoothing properties of dispersive equations, J. Amer. Math., Soc., Vol. 1, 1989, pp. 413-446. | MR 928265 | Zbl 0667.35061
, and ,[7] 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 654188 | Zbl 0519.35022
and ,[8] 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 839728 | Zbl 0535.35069
, and ,[9] On a class of Schrödinger equations, J. Funct. Anal., Vol. 32, 1979, pp. 1-71. | MR 533219 | Zbl 0396.35029 | Zbl 0396.35028
and ,[10] Uniqueness for the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., Vol. 20, 1989, pp. 1388-1425. | MR 1019307 | Zbl 0702.35224
and ,[11] On the initial value problem for the Davey-Stewarson systems, Nonlinearity, Vol. 3, 1990, pp. 475-506. | MR 1054584 | Zbl 0727.35111
and ,[12] On the blowing up solutions to the Cauchy problem for nonlinar Schrödinger equations, J. Math. Phys., Vol. 18, 1979, pp. 1794-1797. | MR 460850 | Zbl 0372.35009
,[13] Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J, Vol. 62, 1991, pp. 575-592. | MR 1054532
,[14] On solutions to the initial value problem for the nonlinear Schrödinger equations in one dimensions, Math. Z., Vol. 192, 1986, pp. 637-650. | MR 847012 | Zbl 0617.35025
, and ,[15] On solutions to the initial value problem for the nonlinear Schrödinger equations, J. Funct. Anal., Vol. 71, 1987, pp. 218-245. | MR 880978 | Zbl 0657.35033
, and ,[16] Analyticity and global existence of small solutions to some nonlinear Shrödinger equations, Comm. Math. Phys., Vol. 129, 1990, pp. 27-41. | MR 1046275 | Zbl 0705.35132
and ,[17] Quasilinear evolution equation, with applications to partial differential equations, Lecture Notes in Math., Vol. 448, pringer-Verlag, pp. 27-50.
,[18] 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 0698.35131
,[19] 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 759907 | Zbl 0549.34001
,[20] Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., Vol. 41, 1988, pp. 891-907. | MR 951744 | Zbl 0671.35066
and ,[21] An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys., Vol. 19, 1978, pp. 798-801. | MR 464963 | Zbl 0383.35015
and ,[22] Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., Vol. 4, 1991, pp. 323-347. | MR 1086966 | Zbl 0737.35102
, and ,[23] Oscillatory integrals and regularity of dispersive equations, Indiana University Math. J., Vol. 40, 1991, pp. 33-69. | MR 1101221 | Zbl 0738.35022
, and ,[24] On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc., (to appear). | MR 1153015 | Zbl 0804.35105
, and ,[25] Well-posedness and scattering results for generalized Korteweg-de Vries via contraction principle, Comm. Pure Appl. Math., Vol. 46, 1993, p. 527-620. | MR 1211741 | Zbl 0808.35128
, and ,[26] 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 712258 | Zbl 0525.42011
, and ,[27] Long time behavior of solutions to nonlinear evolutions equations, Arch. Ration. Mech. and Analysis, 78, 1981, pp. 73-98. | MR 654553 | Zbl 0502.35015
,[28] Global small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math., Vol. 36, 1983, pp. 133-141. | MR 680085 | Zbl 0509.35009
and ,[29] Global existence of small solutions to nonlinear evolution equations, J. Diff. Eqs., Vol. 46, 1982, pp. 409-423. | MR 681231 | Zbl 0518.35046
,[30] 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 796012 | Zbl 0615.47034
and ,[31] Regularity of solutions to the Schrödinger equations, Duke Math., 55, 1987, pp. 699-715. | MR 904948 | Zbl 0631.42010
,[32] Oscillaroty integrals in Fourier Analysis, Beijing Lectures in Harmonic Analysis, Princeton University Press, 1986, pp. 307-355. | MR 864375 | Zbl 0618.42006
,[33] Introduction to Fourier Analysis in Eucliden Spaces, Princeton University Press, 1971. | MR 304972 | Zbl 0232.42007
and ,[34] Nonlinear scattering theory at low energy, J. Funct. Anal., 41, 1981, pp. 110-133. | Zbl 0466.47006
,[35] Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Vol. 44, 1977, pp. 705-714. | MR 512086 | Zbl 0372.35001
,[36] Global strong solutions for nonlinear Schrödinger equation, Nonlinear Anal., 11, 1987, pp. 1143-1154. | MR 913674 | Zbl 0657.35032
,[37] L2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, Vol. 31, 1987, pp. 115-125. | MR 915266 | Zbl 0638.35021
,[38] On solutions of the derivative nonlinear Schrödinger equation. Existence and Uniqueness Theorem, Funkcialaj Ekvacioj, 23, 1980, pp. 259-277. | MR 621533 | Zbl 0478.35032
and ,[39] On solutions of the derivative nonlinear Schröndinger equation. II, Funkcialaj Ekvacioj, 24, 1981, pp. 85-94. | MR 634894 | Zbl 0491.35016
and ,[40]
, Doctoral Thesis, Universidad Autonoma de Madrid, Spain, 1987.[41] The Schrödinger eqution: pointwise convergence to the initial date, Proc. Amer. Math. Soc., Vol. 102, 1988, pp. 874-878. | MR 934859 | Zbl 0654.42014
,