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Weighted Decay Estimates for the Wave Equation with Radially Symmetric Data

Paschalis Karageorgis

TL;DR

This work studies the homogeneous wave equation with radially symmetric data in four or more spatial dimensions and derives weighted decay estimates for the solution using new integral representations of the Riemann operator. The authors develop sharp bounds for the key rational function $z(\lambda,r,t)$ and its derivatives, bounding the Riemann kernel and introducing odd/even dimension representations $L_o/L_e$ with linear operators $H_{ij}$ essential for differentiating the kernels. They present high-dimensional Riemann-operator estimates by region (interior $t \ge 2r$ and exterior), treating odd $n$ via a Legendre-based representation (no boundary terms) and even $n$ via the $U_{jm}, W_{jm}$ framework to manage endpoint singularities, obtaining derivative bounds for $L f_i$ and its time derivative with precise kernel decay. By combining these bounds, they bound the free solution $u_0$ through a Riemann-operator representation, prove decay for $D^\beta u_0$, establish uniqueness by an energy argument, and note refinements for small decay rates and parity-based improvements from Huygens’ principle.

Abstract

We study the homogeneous wave equation with radially symmetric data in four or higher space dimensions. Using some new integral representations for the Riemann operator, we establish weighted decay estimates for the solution.

Weighted Decay Estimates for the Wave Equation with Radially Symmetric Data

TL;DR

This work studies the homogeneous wave equation with radially symmetric data in four or more spatial dimensions and derives weighted decay estimates for the solution using new integral representations of the Riemann operator. The authors develop sharp bounds for the key rational function and its derivatives, bounding the Riemann kernel and introducing odd/even dimension representations with linear operators essential for differentiating the kernels. They present high-dimensional Riemann-operator estimates by region (interior and exterior), treating odd via a Legendre-based representation (no boundary terms) and even via the framework to manage endpoint singularities, obtaining derivative bounds for and its time derivative with precise kernel decay. By combining these bounds, they bound the free solution through a Riemann-operator representation, prove decay for , establish uniqueness by an energy argument, and note refinements for small decay rates and parity-based improvements from Huygens’ principle.

Abstract

We study the homogeneous wave equation with radially symmetric data in four or higher space dimensions. Using some new integral representations for the Riemann operator, we establish weighted decay estimates for the solution.

Paper Structure

This paper contains 4 sections, 15 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Basic Estimates and Facts
  3. The Riemann Operator in High Dimensions
  4. Estimates for the Free Solution

Key Result

Theorem 1.1

Let n\geq 4 be an integer and define a,m by am. Fix an integer 1\leq l\leq m and consider functions \varphi\in \mathcal{C}^{l+1}({\mathbb R}_+) and \psi\in \mathcal{C}^l({\mathbb R}_+) which are subject to data. Then the homogeneous equation he admits a unique solution u_0\in \mathcal{C}^l({\mathbb Besides, the constant C_0 that appears above depends solely on k and n.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 1.2
  • Lemma 1.3: The Riemann operator
  • Lemma 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Corollary 2.7
  • ...and 6 more