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Decay estimates for the wave and Dirac equations with a magnetic potential

Piero D'Ancona, Luca Fanelli

TL;DR

The paper addresses dispersive decay for the 3D wave equation and the massless Dirac equation perturbed by small electromagnetic potentials, establishing a $t^{-1}$ dispersive bound for solutions with the decay constant controlled by a weighted $H^s$ norm of the initial data. The authors develop a method based on Paley–Littlewood frequency decompositions and resolvent-based estimates, leveraging free resolvent theory to obtain weighted-space decay and LAP-type results. A key contribution is proving a strong limiting absorption principle for the massless Dirac operator with a small, rough matrix potential, along with corresponding resolvent bounds that underpin the dispersive estimates. Additionally, they connect the Dirac analysis to a wave-type problem by squaring the equation (and via spectral calculus), obtaining $t^{-1}$ decay with quantified derivative losses and clarifying the role of perturbations through Neumann-series methods.

Abstract

We study the dispersive properties of the wave equation and the massless Dirac equation in three space dimensions, perturbed with electromagnetic potentials. The potentials are assumed to be small but may be rough. For both equations, we prove a dispersive estimate of the form |u(t,x)|< C/t. The constant C can be estimated in terms of a weighted H^s norm of the data, for suitable values of s. As a consequence of our method of proof, we establish the limiting absorption principle for the massless Dirac operator perturbed with a small, rough matrix potential.

Decay estimates for the wave and Dirac equations with a magnetic potential

TL;DR

The paper addresses dispersive decay for the 3D wave equation and the massless Dirac equation perturbed by small electromagnetic potentials, establishing a dispersive bound for solutions with the decay constant controlled by a weighted norm of the initial data. The authors develop a method based on Paley–Littlewood frequency decompositions and resolvent-based estimates, leveraging free resolvent theory to obtain weighted-space decay and LAP-type results. A key contribution is proving a strong limiting absorption principle for the massless Dirac operator with a small, rough matrix potential, along with corresponding resolvent bounds that underpin the dispersive estimates. Additionally, they connect the Dirac analysis to a wave-type problem by squaring the equation (and via spectral calculus), obtaining decay with quantified derivative losses and clarifying the role of perturbations through Neumann-series methods.

Abstract

We study the dispersive properties of the wave equation and the massless Dirac equation in three space dimensions, perturbed with electromagnetic potentials. The potentials are assumed to be small but may be rough. For both equations, we prove a dispersive estimate of the form |u(t,x)|< C/t. The constant C can be estimated in terms of a weighted H^s norm of the data, for suitable values of s. As a consequence of our method of proof, we establish the limiting absorption principle for the massless Dirac operator perturbed with a small, rough matrix potential.

Paper Structure

This paper contains 10 sections, 18 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. The self-adjointness of the perturbed operators
  3. The limiting absorption principle
  4. The limiting absorption principle for the magnetic laplacian
  5. The limiting absorption principle for the Dirac operator and its perturbation
  6. Resolvent Estimates
  7. Proof of Theorem \ref{['th.1']}
  8. Proof of Theorem \ref{['th.2']}
  9. Proof of Theorem \ref{['th.3']}
  10. Lorentz spaces

Key Result

Theorem 1.1

Assume the potentials A(x)\in\mathbb{R}^{3}, B(x)\in\mathbb{R} satisfy for some constant C_{0}>0 sufficiently small and some \beta>1. Then any solution of the Cauchy problem eq.onde, eq.onded satisfies the decay estimate where w_\beta(x):=|x|(|\log|x||+1)^\beta. If in addition we assume that, for some \epsilon>0, then u satisfies for any \delta>0 the estimate

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 31 more