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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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), 4359-4371
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 $ L_t^4L_{\gamma}^4$ Strichartz norm of the solution on any curve $ \gamma$ in $ \mathbb{R}^2$. 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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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 Urbana-Champaign, Urbana, Illinois, 61801
Email: tzirakis@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10487-2
PII: S 0002-9939(2010)10487-2
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 DMS-0901222
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.