Remarks on global a priori estimates for the nonlinear Schr\"odinger equation

TL;DR

This paper introduces a unified method for deriving global a priori estimates for solutions to defocusing nonlinear Schrödinger equations, including a new estimate in two dimensions that bounds specific Strichartz norms.

Contribution

It develops a novel approach using local momentum conservation laws for defocusing equations, leading to new estimates, especially in two-dimensional settings.

Findings

Established a new estimate in two dimensions for the Schrödinger equation.

Bound the $L_t^4L_{ ext{γ}}^4$ Strichartz norm on any curve in $R^2$.

Upgraded the estimate to a weighted Strichartz estimate in the full plane.

Abstract

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.