Pointwise Decay of solutions to the energy critical nonlinear Schr\"odinger equations

TL;DR

This paper proves pointwise decay over time for solutions to the 3D energy-critical nonlinear Schrödinger equations with initial data in L^1 and H^3, utilizing Hardy space bounds and fractional Leibniz rules.

Contribution

It introduces new decay estimates for solutions using Hardy space techniques and extends fractional chain rules to Hardy spaces, advancing analysis methods for nonlinear Schrödinger equations.

Findings

Solutions exhibit pointwise decay in time under specified initial data conditions.

Boundedness of Schrödinger propagators in Hardy space is established.

Fractional Leibniz and chain rules are extended to Hardy spaces.

Abstract

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schr\"odinger equations assuming data in $L^1\cap H^3$. The main ingredients are the boundness of the Schr\"odinger propagators in Hardy space due to Miyachi \cite{Miyachi} and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space.