Remarks on global a priori estimates for the nonlinear Schrödinger equation

Details Remarks on global a priori estimates for the nonlinear Schrödinger equation Remarks on global a priori estimates for the nonlinear Schrödinger equation

2009-08-05

arxiv.org/abs/0908.0644v1

Mathematics

Analysis of PDEs

Scientific paper

We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schr\"odinger 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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