A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates

,182 Memorial Dr, Cambridge, Ma 02142, Usa ,Felipe Hernandez

doi:10.3934/cpaa.2018034

Abstract

A new decomposition for frequency-localized solutions to the Schrodinger equation is given which describes the evolution of the wavefunction using a weighted sum of Lipschitz tubes. As an application of this decomposition, we provide a new proof of the bilinear Strichartz estimate as well as the multilinear restriction theorem for the paraboloid.

Highlights

This paper introduces a way of describing solutions to the Schrodinger equation, which bounds the evolution of the mass distribution of a solution using a weighted sum of Lipschitz tubes
Whereas in the usage of the wavepacket decomposition it is often necessary to induct on scales in order to resolve possible cancellations between tubes, all pieces of the decomposition introduced here are positive so there is no opportunity for cancellation
The lack of cancellation to exploit and the relatively loose control of the shape of the tubes means that this decomposition seems unable to prove many dispersive inequalities used in the literature on the Schrodinger equation

Summary

Introduction

This paper introduces a way of describing solutions to the Schrodinger equation, which bounds the evolution of the mass distribution of a solution using a weighted sum of Lipschitz tubes. We prove variants of the locally constant (LC) and finite speed (FS) properties for solutions ut to the Schrodinger equation satisfying supp u0 ⊂ B1.

Results

Conclusion