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An extension of the $r^p$ method for wave equations with scale-critical potentials and first-order terms

Maxime Van de Moortel

TL;DR

The work addresses decay for wave equations with scale-critical potentials and first-order terms on asymptotically flat, spherically symmetric spacetimes. It extends the classical $r^p$ method by incorporating a Grönwall-based absorption that enlarges the admissible $p$-range, leveraging a spherical harmonic decomposition to push the range up to $p_{high} = 3 - O(\sqrt{\epsilon})$ for the spherical average. Central contributions include a detailed hierarchy for $\phi_0$ and $\phi_{\ge 1}$, Hardy-type controls for ingoing derivatives, and a commuted-energy framework that yields improved decay, particularly for the time-derivative under extra assumptions. The results yield energy decay $E[\phi](u) \lesssim u^{-3+O(\sqrt{\epsilon})}$ and pointwise decay $|\phi|(u,v,\omega) \lesssim u^{-2+O(\sqrt{\epsilon})}$ in the radiation zone, with sharper bounds when additional structure (e.g., bounds on $T$) is available, aligning with sharpness expectations up to arbitrarily small losses in $\epsilon$. These findings have implications for wave dynamics in spacetimes with scale-critical potentials, including Maxwell–charged scalar field-type models. All estimates are expressed with the radiative variable framework and remain robust under a broad class of time-dependent potentials satisfying the scale-critical constraints.$

Abstract

The $r^p$ method, first introduced in [DR10], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the $r^p$-weighted energy using a novel Grönwall argument, which allows a larger range of $p$ than the standard method. A spherically symmetric version of our strategy first appeared in [VdM22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.

An extension of the $r^p$ method for wave equations with scale-critical potentials and first-order terms

TL;DR

The work addresses decay for wave equations with scale-critical potentials and first-order terms on asymptotically flat, spherically symmetric spacetimes. It extends the classical method by incorporating a Grönwall-based absorption that enlarges the admissible -range, leveraging a spherical harmonic decomposition to push the range up to for the spherical average. Central contributions include a detailed hierarchy for and , Hardy-type controls for ingoing derivatives, and a commuted-energy framework that yields improved decay, particularly for the time-derivative under extra assumptions. The results yield energy decay and pointwise decay in the radiation zone, with sharper bounds when additional structure (e.g., bounds on ) is available, aligning with sharpness expectations up to arbitrarily small losses in . These findings have implications for wave dynamics in spacetimes with scale-critical potentials, including Maxwell–charged scalar field-type models. All estimates are expressed with the radiative variable framework and remain robust under a broad class of time-dependent potentials satisfying the scale-critical constraints.$

Abstract

The method, first introduced in [DR10], has become a robust strategy to prove decay for wave equations in the context of black holes and beyond. In this note, we propose an extension of this method, which is particularly suitable for proving decay for a general class of wave equations featuring a scale-critical time-dependent potential and/or first-order terms of small amplitude. Our approach consists of absorbing error terms in the -weighted energy using a novel Grönwall argument, which allows a larger range of than the standard method. A spherically symmetric version of our strategy first appeared in [VdM22] in the context of a weakly charged scalar field on a black hole whose equations also involve a scale-critical potential.

Paper Structure

This paper contains 24 sections, 26 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The principles of the $r^p$ method
  3. Our new strategy to address scale-critical potentials
  4. Examples of spacetimes and potentials satisfying the assumptions
  5. Decay rates in the assumptions
  6. Metric assumptions
  7. Integrated local decay and energy boundedness
  8. Potential assumptions for sharp energy and radiation field decay
  9. Potential assumptions for sharp point-wise scalar field decay
  10. Previous works on scale-critical potentials and sharpness of the decay
  11. Acknowledgments
  12. Preliminary
  13. Boundedness of the $r^p$-weighted energy and energy decay
  14. $r^p$-weighted energy and black box decay results
  15. $r^p$-weighted multipliers
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\phi$ be a solution of wave.main where $g$ and the potential terms satisfy H0 and H1 with (characteristic) initial data on the bifurcate null cones $([u_0,u_F] \times \mathbb{S}^2 )\cup( [v_0,+\infty)\times \mathbb{S}^2)$ with $u_0 \in \mathbb{R}$, $u_F \in \mathbb{R} \cup \{+\infty\}$, $v_0 \i Defining $E[\phi](u)$ to be the standard (unweighted) energy on a foliation radiating to infinity o

Figures (1)

  • Figure :

Theorems & Definitions (50)

  • Theorem 1.1
  • Lemma 3.1: Moi2, Lemma 6.3
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 40 more