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Asymptotic completeness for a scalar quasilinear wave equation satisfying the weak null condition

Dongxiao Yu

Abstract

In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in arXiv:2002.05355. Starting from a global solution $u$ to the quasilinear wave equation, we rigorously show that well chosen asymptotic variables solve the same reduced system with small error terms. This allows us to recover the scattering data for our system, as well as to construct a matching exact solution to the reduced system.

Asymptotic completeness for a scalar quasilinear wave equation satisfying the weak null condition

Abstract

In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in arXiv:2002.05355. Starting from a global solution to the quasilinear wave equation, we rigorously show that well chosen asymptotic variables solve the same reduced system with small error terms. This allows us to recover the scattering data for our system, as well as to construct a matching exact solution to the reduced system.

Paper Structure

This paper contains 45 sections, 74 theorems, 730 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The geometric reduced system
  4. Modified scattering theory: an overview
  5. Construction of an optical function
  6. The asymptotic equations and the scattering data
  7. An approximation
  8. The main theorem
  9. Acknowledgement
  10. Preliminaries
  11. Notations
  12. Commuting vector fields
  13. Several pointwise bounds
  14. A key theorem and a convention
  15. The null condition of a matrix
  16. ...and 30 more sections

Key Result

Theorem 1

Let $u$ be a smooth solution to the Cauchy problem qwe and init. Fix a constant $R>0$ such that $\text{supp }(u_0,u_1)\subset\{|x|\leq R\}$, so $u\equiv 0$ for $|x|\geq t+R$ by the finite speed of propagation. Set $T_0:=\exp(\delta/\varepsilon)$ for a fixed constant $\delta>0$. Then we have

Theorems & Definitions (153)

  • Definition
  • Theorem 1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 143 more