Remarks on global a priori estimates for the nonlinear Schrödinger equation
Authors:
J. Colliander, M. Grillakis and N. Tzirakis
Journal:
Proc. Amer. Math. Soc. 138 (2010), 43594371
MSC (2010):
Primary 35Q55
Published electronically:
June 18, 2010
MathSciNet review:
2680061
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Abstract: We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schrödinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in two dimensions. We bound the restricted Strichartz norm of the solution on any curve in . For the specific case of a straight line we upgrade this estimate to a weighted Strichartz estimate valid in the full plane.
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 J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145171. MR 1626257 (99e:35208)
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 T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Math. 10, AMS, 2003. MR 2002047 (2004j:35266)
 3.
 J. Colliander, M. Grillakis, and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 62 (2009), no. 7, 920968. MR 2527809 (2010c:35175)
 4.
 J. Colliander, J. Holmer, M. Visan and X. Zhang, Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on , Commun. Pure Appl. Anal. 7 (2008), no. 3, 467489. MR 2379437 (2009c:35433)
 5.
 J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on , Comm. Pure Appl. Math. 57 (2004), no. 8, 9871014. MR 2053757 (2005b:35257)
 6.
 J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global wellposedness and scattering for the energycritical nonlinear Schrödinger equation in , Ann. of Math. (2) 167 (2008), no. 3, 767865. MR 2415387 (2009f:35315)
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 J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, in Nonlinear Wave Equations, Y. Guo, ed., Contemporary Mathematics, 263, AMS, 2000. MR 1777634 (2001g:35205)
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 J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pure Appl. 64 (1985), 363401. MR 839728 (87i:35171)
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 M. Grillakis, On nonlinear Schrödinger equations, Commun. Partial Differential Equations 25, no. 910 (2005), 18271844. MR 1778782 (2001g:35235)
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 J. E. Lin and W. A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal., 30 (1978), 245263. MR 515228 (80k:35056)
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 C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, Comm. Pure Appl. Math. 25 (1972), 131. MR 0303097 (46:2239)
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 K. Nakanishi, Energy scattering for nonlinear KleinGordon and Schrödinger equations in spatial dimensions and , J. Funct. Anal. 169 (1999), 201225. MR 1726753 (2000m:35141)
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 K. Nakanishi, Energy scattering for Hartree equations, Math. Res. Lett. 6, no. 1 (1999), 107118. MR 1682697 (2000d:35173)
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 F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 2, 261290. MR 2518079 (2010b:35441)
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 T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS 106, AMS, 2006. MR 2233925 (2008i:35211)
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 T. Tao, M. Visan, and X. Zhang, The nonlinear Schrödinger equation with combined powertype nonlinearities, Comm. Partial Differential Equations 32 (2007), no. 79, 12811343. MR 2354495 (2009f:35324)
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Additional Information
J. Colliander
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4
Email:
colliand@math.toronto.edu
M. Grillakis
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email:
mng@math.umd.edu
N. Tzirakis
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, Illinois, 61801
Email:
tzirakis@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S000299392010104872
Received by editor(s):
July 14, 2009
Received by editor(s) in revised form:
February 9, 2010
Published electronically:
June 18, 2010
Additional Notes:
The work of the third author was supported by NSF grant DMS0901222
Communicated by:
Hart F. Smith
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
