Space-time resonances
Author:
Jalal Shatah
Journal:
Quart. Appl. Math. 68 (2010), 161-167
MSC (2000):
Primary 35B34; Secondary 35Q35
DOI:
https://doi.org/10.1090/S0033-569X-09-01175-3
Published electronically:
November 9, 2009
MathSciNet review:
2598888
Full-text PDF Free Access
References |
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Additional Information
References
- Thierry Cazenave and Fred B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), no. 1, 75–100. MR 1171761
- Ronald R. Coifman and Yves Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978 (French). With an English summary. MR 518170
- P. Germain, N. Masmoudi, J. Shatah, Global Solutions for $3$D Quadratic Schrödinger equations, Internat. Math. Res. Notices 2009, no. 3, 414–432.
- J. Ginibre and N. Hayashi, Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions, Math. Z. 219 (1995), no. 1, 119–140. MR 1340853, DOI https://doi.org/10.1007/BF02572354
- Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré 8 (2007), no. 7, 1303–1331. MR 2360438, DOI https://doi.org/10.1007/s00023-007-0336-6
- S. Gustafson, K. Nakanishi, T.-P. Tsai, Scattering theory for the Gross-Pitaevski equation in three dimensions, preprint.
- Nakao Hayashi, Tetsu Mizumachi, and Pavel I. Naumkin, Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D, Differential Integral Equations 16 (2003), no. 2, 159–179. MR 1947090
- Yuichiro Kawahara, Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in 3D, Differential Integral Equations 18 (2005), no. 2, 169–194. MR 2106101
- S. Klainerman and Gustavo Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), no. 1, 133–141. MR 680085, DOI https://doi.org/10.1002/cpa.3160360106
- Sergiu Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), no. 5, 631–641. MR 803252, DOI https://doi.org/10.1002/cpa.3160380512
- Kenji Nakanishi and Tohru Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 1, 45–68. MR 1891695, DOI https://doi.org/10.1007/s00030-002-8118-9
- Nakao Hayashi and Pavel I. Naumkin, On the quadratic nonlinear Schrödinger equation in three space dimensions, Internat. Math. Res. Notices 3 (2000), 115–132. MR 1741610, DOI https://doi.org/10.1155/S1073792800000088
- Jalal Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations 46 (1982), no. 3, 409–425. MR 681231, DOI https://doi.org/10.1016/0022-0396%2882%2990102-4
- Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. MR 803256, DOI https://doi.org/10.1002/cpa.3160380516
- Walter A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis 41 (1981), no. 1, 110–133. MR 614228, DOI https://doi.org/10.1016/0022-1236%2881%2990063-X
References
- T. Cazenave, F. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), no. 1, 75–100. MR 1171761 (93d:35150)
- R. Coifman, Y. Meyer, Au delà des opérateurs pseudo-différentiels. Astérisque, 57. Société Mathématique de France, Paris, 1978. MR 518170 (81b:47061)
- P. Germain, N. Masmoudi, J. Shatah, Global Solutions for $3$D Quadratic Schrödinger equations, Internat. Math. Res. Notices 2009, no. 3, 414–432.
- J. Ginibre, N. Hayashi, Almost global existence of small solutions to quadratic nonlinear Schrödinger equations in three space dimensions, Math. Z. 219 (1995), no. 1, 119–140. MR 1340853 (96e:35158)
- S. Gustafson, K. Nakanishi, T.-P. Tsai, Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions, Ann. Henri Poincaré 8 (2007), no. 7, 1303–1331. MR 2360438 (2008m:35335)
- S. Gustafson, K. Nakanishi, T.-P. Tsai, Scattering theory for the Gross-Pitaevski equation in three dimensions, preprint.
- N. Hayashi, T. Mizumachi, P. Naumkin, Time decay of small solutions to quadratic nonlinear Schrödinger equations in $3$D, Differential Integral Equations 16 (2003), no. 2, 159–179. MR 1947090 (2004a:35197)
- Y. Kawahara, Global existence and asymptotic behavior of small solutions to nonlinear Schrödinger equations in $3$D, Differential Integral Equations 18 (2005), no. 2, 169–194. MR 2106101 (2005i:35247)
- S. Klainerman, G. Ponce, Global, small amplitude solutions to nonlinear evolution equations, Comm. Pure Appl. Math. 36 (1983), no. 1, 133–141. MR 680085 (84a:35173)
- S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), no. 5, 631–641. MR 803252 (87e:35080)
- K. Nakanishi, T. Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 1, 45–68. MR 1891695 (2003a:35177)
- N. Hayashi, P. Naumkin, On the quadratic nonlinear Schrödinger equation in three space dimensions, Internat. Math. Res. Notices 2000, no. 3, 115–132. MR 1741610 (2001a:35165)
- J. Shatah, Global existence of small solutions to nonlinear evolution equations, J. Differential Equations 46 (1982), no. 3, 409–425. MR 681231 (84g:35036)
- J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. MR 803256 (87b:35160)
- W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), no. 1, 110–133. MR 614228 (83b:47074a)
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Additional Information
Jalal Shatah
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
MR Author ID:
160000
Email:
shatah@cims.nyu.edu
Received by editor(s):
December 31, 2008
Published electronically:
November 9, 2009
Additional Notes:
This research is funded in part by NSF DMS 0701272.
Article copyright:
© Copyright 2009
Brown University