# Remarks on global a priori estimates for the nonlinear Schrödinger equation - Mathematics > Analysis of PDEs

Abstract: We present a unified approach for obtaining global a priori estimates forsolutions of nonlinear defocusing Schr\-odinger equations with defocusingnonlinearities. The estimates are produced by contracting the local momentumconservation law with appropriate vector fields. The corresponding law iswritten for defocusing equations of tensored solutions. In particular, weobtain 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 thisestimate to a weighted Strichartz estimate valid in the full plane.

Author: J. Colliander, M. Grillakis, N. Tzirakis

Source: https://arxiv.org/