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Dispersive decay for the energy-critical nonlinear wave equation

Matthew Kowalski

TL;DR

We study pointwise-in-time dispersive decay for the energy-critical nonlinear wave equation in spatial dimension $d=3$ with data in the energy space $\dot{H}^1\times L^2$. Using a bootstrap framework built on frequency-localized dispersive bounds for the linear propagator $e^{\pm i t|\nabla|}$ and careful paraproduct-based nonlinear estimates, we obtain nonlinear analogues of linear dispersion: for $2<p<\infty$ the solution satisfies $\|u(t)\|_{L^p} \lesssim C(...) \;|t|^{-(1-\frac{2}{p})}$ with a data- dependent constant, and a Besov refinement yields $L^\infty$-decay under stronger Besov data. An edge theorem further strengthens decay in Besov spaces via an $\ell^1$ summation over frequencies. The results extend nonlinear dispersion theory for the energy-critical NLW by demonstrating precise, frequency-localized decay without requiring Lorentz-space spacetime bounds, and they provide a robust method for achieving pointwise decay in critical nonlinear wave models.

Abstract

We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear wave equation in spatial dimension $d = 3$.

Dispersive decay for the energy-critical nonlinear wave equation

TL;DR

We study pointwise-in-time dispersive decay for the energy-critical nonlinear wave equation in spatial dimension with data in the energy space . Using a bootstrap framework built on frequency-localized dispersive bounds for the linear propagator and careful paraproduct-based nonlinear estimates, we obtain nonlinear analogues of linear dispersion: for the solution satisfies with a data- dependent constant, and a Besov refinement yields -decay under stronger Besov data. An edge theorem further strengthens decay in Besov spaces via an summation over frequencies. The results extend nonlinear dispersion theory for the energy-critical NLW by demonstrating precise, frequency-localized decay without requiring Lorentz-space spacetime bounds, and they provide a robust method for achieving pointwise decay in critical nonlinear wave models.

Abstract

We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear wave equation in spatial dimension .
Paper Structure (4 sections, 6 theorems, 59 equations)

This paper contains 4 sections, 6 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Spacetime Bounds
  3. Proof of Theorem \ref{['theorem']}
  4. Proof of Theorem \ref{['edge theorem']}

Key Result

Theorem 1.1

Let there exist $(u_0, u_1) \in \dot{H}^1 \times L^2(\mathbb{R}^3)$. In the focusing case, suppose that $(u_0,u_1)$ satisfies where $W(x) = ( 1 + \tfrac{1}{3} |x|^2)^{-\frac{1}{2}}$ is a stationary solution to NLW. Then there exists a unique global solution $u \in C_t\dot{H}^1_x(\mathbb{R}_t \times \mathbb{R}_x^3)$ to NLW with initial data $(u_0,u_1)$ which satisfies Moreover, there exist scatte

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Lemma 1.5: Paraproduct decompositions
  • Definition 2.1: Wave-admissible
  • Theorem 2.2: Strichartz estimates
  • Proposition 2.3: Mixed spacetime bounds
  • proof : Proof of Theorem \ref{['theorem']}
  • proof : Proof of Theorem \ref{['edge theorem']}