Remarks on global a priori estimates for the nonlinear Schrödinger equation

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 4 L γ 4 L_t^4L_{\gamma }^4 Strichartz norm of the solution on any curve γ \gamma in R 2 \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.