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A Decay estimate for cubic defocusing non-linear Schrödinger equation in three dimensions

Yi Sun

TL;DR

The paper analyzes the 3D cubic defocusing nonlinear Schrödinger equation at the $\dot{H}^{1/2}_x$-critical level and proves a decay estimate for solutions by leveraging a stronger scaling-critical Besov initial-data space $\dot{B}^{1/2}_{2,1}$. Through a bootstrap argument built on Besov-spacetime bounds, paraproduct decompositions, and perturbation theory, it derives a pointwise decay bound $\|u(t)\|_{L^{\infty}_x} \lesssim t^{-3/2}\|u_0\|_{L^1_x}$ under appropriate hypotheses and extends the result to a global, unconditional scenario in the radially symmetric Besov setting. The work connects scaling-critical Besov regularity with the full dispersive estimate, complements global well-posedness and scattering results, and provides a parallel pathway to established radial-decay results in the literature. It also discusses extensions to the radial case without perturbation theory, emphasizing the robustness of the Besov-space approach for decay in defocusing NLS. Overall, the paper advances decay theory for nonlinear dispersive equations by tying Besov norms to explicit time-decay rates in three dimensions.

Abstract

In this short note, we prove a decay estimate for non-linear solutions of 3D cubic defocusing non-linear Schrödinger equation.

A Decay estimate for cubic defocusing non-linear Schrödinger equation in three dimensions

TL;DR

The paper analyzes the 3D cubic defocusing nonlinear Schrödinger equation at the -critical level and proves a decay estimate for solutions by leveraging a stronger scaling-critical Besov initial-data space . Through a bootstrap argument built on Besov-spacetime bounds, paraproduct decompositions, and perturbation theory, it derives a pointwise decay bound under appropriate hypotheses and extends the result to a global, unconditional scenario in the radially symmetric Besov setting. The work connects scaling-critical Besov regularity with the full dispersive estimate, complements global well-posedness and scattering results, and provides a parallel pathway to established radial-decay results in the literature. It also discusses extensions to the radial case without perturbation theory, emphasizing the robustness of the Besov-space approach for decay in defocusing NLS. Overall, the paper advances decay theory for nonlinear dispersive equations by tying Besov norms to explicit time-decay rates in three dimensions.

Abstract

In this short note, we prove a decay estimate for non-linear solutions of 3D cubic defocusing non-linear Schrödinger equation.

Paper Structure

This paper contains 9 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Auxiliary Results
  4. Notations
  5. Preliminaries
  6. Proof of the main theorems
  7. Base case.
  8. Induction
  9. Remark on the proof of theorem \ref{['main-result-2']}

Key Result

Proposition 1.1

Suppose that $u(t)$ solves nls with initial data $u_0$. Assume that Then there exists $u_0^+$ such that and where $g:[0,\infty)\to [0,\infty)$ is an increasing function (c.f. corollary 5.3 in K-M-2010).

Theorems & Definitions (19)

  • Proposition 1.1: Theorem 1.1 in K-M-2010
  • Remark 1.1
  • Theorem 1.1
  • Proposition 1.2: Theorem 2 in D-2023
  • Theorem 1.2
  • Remark 1.2
  • Proposition 1.3: Perturbation theorem c.f. K-M-2010
  • Proposition 1.4: Strichartz estimate for Schrödinger, $d=3$, c.f. G-V-1989K-T-1998Y-1987
  • Lemma 2.1: Paraproduct decomposition
  • Proposition 2.1: Besov spacetime bounds
  • ...and 9 more