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 | References | Similar Articles | Additional Information

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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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:
https://doi.org/10.1090/S0002-9939-2010-10487-2

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.