Small solutions to nonlinear Schrödinger equations

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 10, Issue 3, 1993, Pages 255-288

  Kenig, Carlos E ^a   Ponce, Gustavo ^b   Vega, Luis ^c  

^a UNIVERSITY OF CHICAGO   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

Author keywords

35 B 45; 35 Q 20; 44 B 99; Nonlinear Schr dinger equations; smoothing effects

Indexed keywords

INITIAL VALUE PROBLEMS;

35 B 45; 35 Q 20; 44 B 99; DINGER EQUATION; SMOOTHING EFFECTS;

NONLINEAR EQUATIONS;

EID: 84894536209     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0294-1449(16)30213-X     Document Type: Article

References (41)

1. Carbery, A., Radial Fourier multipliers and associated maximal function. North Holland Math. Studies III (1985), 49–55.
2. T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos, Vol. 22, Universidade Federal do Rio de Janeiro.
3. Cazenave, T., Weissler, F.B., Some remarks on the nonlinear Schrödinger equation in the critical case. Lecture in Math, Vol. 1392, 1989, Springer-Verlag, Berlin, New York, 18–29.
4. Christ, F.M., Weinstein, M., Dispersive small amplitude solution to the generalized Korteweg-de Vries equation. J. Funct. Anal. Vol. 100 (1991), 87–109.
5. Coifman, R.R., Meyer, Y., Au-delà des opérateurs pseudo-différentiel. Astérisque, Vol. 57, 1973.
6. Constantin, P., Saut, J.-C., Local smoothing properties of dispersive equations. J. Amer. Math., Soc. Vol. 1 (1989), 413–446.
7. Dahlberg, B., Kenig, C.E., A note an almost every where behavior of solutions to the schrödinger equations. Lecture Notes in Math., Vol. 908, 1982, Springer-Verlag, Berlin, New York, 205–208.
8. Ginibre, J., Velo, G., Scattering theory in the energy space for a class of nonlinear Schrödinger equation. J. Math, pures et appl. Vol. 64 (1985), 363–401.
9. Ginibre, J., Velo, G., On a class of Schrödinger equations. J. Funct. Anal. Vol. 32 (1979), 1–71.
10. Ginibre, J., Tsutsumi, Y., Uniqueness for the generalized Korteweg-de Vries equations. SIAM J. Math. Anal. Vol. 20 (1989), 1388–1425.
11. Ghidaglia, J.M., Saut, J.-C., On the initial value problem for the Davey-Stewarson systems. Nonlinearity Vol. 3 (1990), 475–506.
12. Glassey, R.T., On the blowing up solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. Vol. 18 (1979), 1794–1797.
13. Hayashi, N., Global existence of small analytic solutions to nonlinear Schrödinger equations. Duke Math. J Vol. 62 (1991), 575–592.
14. Hayashi, N., Nakamitsu, K., Tsutsumi, M., On solutions to the initial value problem for the nonlinear Schrödinger equations in one dimensions. Math. Z. Vol. 192 (1986), 637–650.
15. Hayashi, N., Nakamitsu, K., Tsutsumi, M., On solutions to the initial value problem for the nonlinear Schrödinger equations. J. Funct. Anal. Vol. 71 (1987), 218–245.
16. Hayashi, N., Saitoh, S., Analyticity and global existence of small solutions to some nonlinear Shrödinger equations. Comm. Math. Phys. Vol. 129 (1990), 27–41.
17. T. Kato, Quasilinear evolution equation, with applications to partial differential equations, Lecture Notes in Math., Vol. 448, Springer-Verlag, pp. 27-50.
18. Kato, T., Nonlinear Schrödinger equation, Schrodinger operators. Holden, H., Jensen, A., (eds.) Lecture Notes in Physics, Vol. 345, 1989, Springer-Verlag, Berlin, New York, 218–263.
19. Kato, T., On the Cauchy problem for the (generalized) Kortewed-de Vries equation, Advances in Math. Supp. Studies. Studies in Applied Math. Vol. 8 (1983), 93–128.
20. Kato, T., Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. Vol. 41 (1988), 891–907.
21. Kaup, D.J., Newell, A.C., An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. Vol. 19 (1978), 798–801.
22. Kenig, C.E., Ponce, G., Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Amer. Math. Soc. Vol. 4 (1991), 323–347.
23. Kenig, C.E., Ponce, G., Vega, L., Oscillatory integrals and regularity of dispersive equations. Indiana University Math. J. Vol. 40 (1991), 33–69.
24. C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc., (to appear).
25. Kenig, C.E., Ponce, G., Vega, L., Well-posedness and scattering results for generalized Korteweg-de Vries via contraction principle. Comm. Pure Appl. Math. Vol. 46 (1993), 527–620.
26. Kenig, C.E., Ruiz, A., A strong type (2, 2) estimate for the maximal function associated to the Schrödinger equation. Trans. Amer. Math. Soc. Vol. 280 (1983), 239–246.
27. Klainerman, S., Long time behavior of solutions to nonlinear evolutions equations. Arch. Ration. Mech. and Analysis 78 (1981), 73–98.
28. Klainerman, S., Ponce, G., Global small amplitude solutions to nonlinear evolution equations. Comm. Pure Appl. Math. Vol. 36 (1983), 133–141.
29. Shatah, J., Global existence of small solutions to nonlinear evolution equations. J. Diff. Eqs. Vol. 46 (1982), 409–423.
30. Simon, J., Taflin, E., 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), 541–562.
31. Sjölin, P., Regularity of solutions to the Schrödinger equations. Duke Math. 55 (1987), 699–715.
32. Stein, E.M., Oscillaroty integrals in Fourier Analysis. Beijing Lectures in Harmonic Analysis, 1986, Princeton University Press, 307–355.
33. Stein, E.M., Weiss, G., Introduction to Fourier Analysis in Eucliden Spaces. 1971, Princeton University Press.
34. Srauss, W.A., Nonlinear scattering theory at low energy. J. Funct. Anal. 41 (1981), 110–133.
35. Strichartz, R.S., Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. Vol. 44 (1977), 705–714.
36. Tsutsumi, Y., Global strong solutions for nonlinear Schrödinger equation. Nonlinear Anal. 11 (1987), 1143–1154.
37. 2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcialaj Ekvacioj Vol. 31 (1987), 115–125.
38. Tsutsumi, M., Fukuda, I., On solutions of the derivative nonlinear Schrödinger equation. Existence and Uniqueness Theorem. Funkcialaj Ekvacioj 23 (1980), 259–277.
39. Tsutsumi, M., Fukuda, I., On solutions of the derivative nonlinear Schröndinger equation. II. Funkcialaj Ekvacioj 24 (1981), 85–94.
40. Vega, L., Doctoral Thesis. 1987, Universidad Autonoma de Madrid, Spain.
41. Vega, L., The Schrödinger eqution: pointwise convergence to the initial date. Proc. Amer. Math. Soc. Vol. 102 (1988), 874–878.