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Explicit formula of radiation fields of free waves with applications on channel of energy

Liang Li, Ruipeng Shen, Lijuan Wei

Abstract

In this work we give a few explicit formulas regarding the radiation fields of linear free waves. We then apply these formulas on the channel of energy theory. We characterize all the radial weakly non-radiative solutions in all dimensions and give a few new exterior energy estimates.

Explicit formula of radiation fields of free waves with applications on channel of energy

Abstract

In this work we give a few explicit formulas regarding the radiation fields of linear free waves. We then apply these formulas on the channel of energy theory. We characterize all the radial weakly non-radiative solutions in all dimensions and give a few new exterior energy estimates.

Paper Structure

This paper contains 23 sections, 21 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and topics
  3. Radiation field
  4. Channel of energy
  5. Main idea
  6. Main results
  7. Applications on channel of energy
  8. Structure of this work
  9. Notations
  10. From Radiation Profile to Solution
  11. Odd dimensions
  12. Even dimensions
  13. Universal formula
  14. Between Radiation Profiles
  15. Radial Weakly Non-radiative Solutions
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that $d\geq 3$ and let $u$ be a solution to the free wave equation $\partial_t^2 u - \Delta u = 0$ with initial data $(u_0,u_1) \in \dot{H}^1 \times L^2({\mathbb R}^d)$. Then and there exist two functions $G_\pm \in L^2({\mathbb R} \times \mathbb{S}^{d-1})$ so that In addition, the maps $(u_0,u_1) \rightarrow \sqrt{2} G_\pm$ are a bijective isometries form $\dot{H}^1 \times L^2({\mathbb R

Figures (1)

  • Figure 1: Change of variables

Theorems & Definitions (39)

  • Theorem 1.1: Radiation filed
  • Proposition 1.2: see Duyckaerts-Kenig-Merle dkmnonradial
  • Corollary 1.3
  • Theorem 1.4: See Kenig et al channel, the proof uses radial Fourier transform
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Corollary 1.8
  • Proposition 1.9: Radial weakly non-radiative solutions
  • Remark 1.10
  • ...and 29 more