Small solutions to nonlinear Schrödinger equations

Carlos E. Kenig; Gustavo Ponce; Luis Vega

Annales de l'I.H.P. Analyse non linéaire (1993)

  • Volume: 10, Issue: 3, page 255-288
  • ISSN: 0294-1449

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Kenig, Carlos E., Ponce, Gustavo, and Vega, Luis. "Small solutions to nonlinear Schrödinger equations." Annales de l'I.H.P. Analyse non linéaire 10.3 (1993): 255-288. <http://eudml.org/doc/78303>.

@article{Kenig1993,
author = {Kenig, Carlos E., Ponce, Gustavo, Vega, Luis},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {local existence; Cauchy problem; nonlinear Schrödinger equation; boundedness of the initial data; smoothing effect of Kato type},
language = {eng},
number = {3},
pages = {255-288},
publisher = {Gauthier-Villars},
title = {Small solutions to nonlinear Schrödinger equations},
url = {http://eudml.org/doc/78303},
volume = {10},
year = {1993},
}

TY - JOUR
AU - Kenig, Carlos E.
AU - Ponce, Gustavo
AU - Vega, Luis
TI - Small solutions to nonlinear Schrödinger equations
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1993
PB - Gauthier-Villars
VL - 10
IS - 3
SP - 255
EP - 288
LA - eng
KW - local existence; Cauchy problem; nonlinear Schrödinger equation; boundedness of the initial data; smoothing effect of Kato type
UR - http://eudml.org/doc/78303
ER -

References

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  1. [1] A. Carbery, Radial Fourier multipliers and associated maximal function, North Holland Math. Studies, III, 1985, pp. 49-55. Zbl0632.42012MR848141
  2. [2] T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos, Vol. 22, Universidade Federal do Rio de Janeiro. Zbl0584.35022
  3. [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. Zbl0694.35170MR1021011
  4. [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. Zbl0743.35067MR1124294
  5. [5] R.R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiel, Astérisque, Vol. 57, 1973. Zbl0483.35082
  6. [6] P. Constantin, and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math., Soc., Vol. 1, 1989, pp. 413-446. Zbl0667.35061MR928265
  7. [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. Zbl0519.35022MR654188
  8. [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. Zbl0535.35069MR839728
  9. [9] J. Ginibre and G. Velo, On a class of Schrödinger equations, J. Funct. Anal., Vol. 32, 1979, pp. 1-71. Zbl0396.35028MR533219
  10. [10] J. Ginibre and Y. Tsutsumi, Uniqueness for the generalized Korteweg-de Vries equations, SIAM J. Math. Anal., Vol. 20, 1989, pp. 1388-1425. Zbl0702.35224MR1019307
  11. [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. Zbl0727.35111MR1054584
  12. [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. Zbl0372.35009MR460850
  13. [13] N. Hayashi, Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J, Vol. 62, 1991, pp. 575-592. MR1054532
  14. [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. Zbl0617.35025MR847012
  15. [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. Zbl0657.35033MR880978
  16. [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. Zbl0705.35132MR1046275
  17. [17] T. Kato, Quasilinear evolution equation, with applications to partial differential equations, Lecture Notes in Math., Vol. 448, pringer-Verlag, pp. 27-50. 
  18. [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. Zbl0698.35131
  19. [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. Zbl0549.34001MR759907
  20. [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. Zbl0671.35066MR951744
  21. [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. Zbl0383.35015MR464963
  22. [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. Zbl0737.35102MR1086966
  23. [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. Zbl0738.35022MR1101221
  24. [24] C.E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc., (to appear). Zbl0804.35105MR1153015
  25. [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. Zbl0808.35128MR1211741
  26. [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. Zbl0525.42011MR712258
  27. [27] S. Klainerman, Long time behavior of solutions to nonlinear evolutions equations, Arch. Ration. Mech. and Analysis, 78, 1981, pp. 73-98. Zbl0502.35015MR654553
  28. [28] S. Klainerman and G. Ponce, Global small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math., Vol. 36, 1983, pp. 133-141. Zbl0509.35009MR680085
  29. [29] J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Diff. Eqs., Vol. 46, 1982, pp. 409-423. Zbl0518.35046MR681231
  30. [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. Zbl0615.47034MR796012
  31. [31] P. Sjölin, Regularity of solutions to the Schrödinger equations, Duke Math., 55, 1987, pp. 699-715. Zbl0631.42010MR904948
  32. [32] E.M. Stein, Oscillaroty integrals in Fourier Analysis, Beijing Lectures in Harmonic Analysis, Princeton University Press, 1986, pp. 307-355. Zbl0618.42006MR864375
  33. [33] E.M. Stein and G. Weiss, Introduction to Fourier Analysis in Eucliden Spaces, Princeton University Press, 1971. Zbl0232.42007MR304972
  34. [34] W.A. Srauss, Nonlinear scattering theory at low energy, J. Funct. Anal., 41, 1981, pp. 110-133. Zbl0466.47006
  35. [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. Zbl0372.35001MR512086
  36. [36] Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equation, Nonlinear Anal., 11, 1987, pp. 1143-1154. Zbl0657.35032MR913674
  37. [37] Y. Tsutsumi, L2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj, Vol. 31, 1987, pp. 115-125. Zbl0638.35021MR915266
  38. [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. Zbl0478.35032MR621533
  39. [39] M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schröndinger equation. II, Funkcialaj Ekvacioj, 24, 1981, pp. 85-94. Zbl0491.35016MR634894
  40. [40] L. Vega, Doctoral Thesis, Universidad Autonoma de Madrid, Spain, 1987. 
  41. [41] L. Vega, The Schrödinger eqution: pointwise convergence to the initial date, Proc. Amer. Math. Soc., Vol. 102, 1988, pp. 874-878. Zbl0654.42014MR934859

Citations in EuDML Documents

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  1. XS. Feng, F. Wei, On global solutions to a nonlinear Alfvén wave equation
  2. Gustavo Ponce, Local existence theory for the generalized Schrödinger equation
  3. Nakao Hayashi, Local existence in time of small solutions to the Davey-Stewartson systems
  4. Felipe Linares, Gustavo Ponce, On the Davey-Stewartson systems
  5. Jean-Marc Delort, Solutions globales pour l’équation de Schrödinger à nonlinéarités quadratiques et à données petites
  6. Alberto Ruiz, Regularizing estimates for Schrödinger and wave equations
  7. Baoxiang Wang, Lijia Han, Chunyan Huang, Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data
  8. Tohru Ozawa, Jian Zhai, Global existence of small classical solutions to nonlinear Schrödinger equations
  9. Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli, Strichartz Estimates for the Schrödinger Equation with small Magnetic Potential
  10. Jean-David Benamou, François Castella, Thodoros Katsaounis, Benoît Perthame, High Frequency limit of the Helmholtz Equations

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