Decay estimates for Nonlinear Schr\"odinger equation with the inverse-square potential

TL;DR

This paper establishes decay estimates for solutions to the 3D energy-critical nonlinear Schrödinger equation with an inverse-square potential, using Lorentz-Strichartz estimates and Sobolev norm equivalences.

Contribution

It introduces new decay estimates for NLS with inverse-square potential by deriving Lorentz-Strichartz estimates and proving uniform regularity in Sobolev spaces.

Findings

Solutions exhibit uniform ot;H^1(\u211d^3) regularity

Derived Lorentz-Strichartz estimates for the equation

Established decay estimates via bootstrap argument

Abstract

In this paper, we study the dispersive decay estimates for solution to the $3\mathrm{D}$ energy-critical nonlinear Schr\"odinger equation with an inverse-square operator $\mathcal{L}_a$ where the operator is denoted by $\mathcal{L}_{a}:=-\Delta+\frac{a}{|x|^2}$ with the constant $a\geq0$. Inspired by the work of \cite{KMVZZ1,K}, we first establish that the solutions exhibit $\dot{H}^1(\R^3)$ uniform regularity, derive the Lorentz-Strichartz estimates, and then obtain the desired decay estimates using the bootstrap argument. The key ingredients of our approach include the equivalence of Sobolev norms and the fractional product rule.