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Global smooth solutions of 2-D quadratic quasilinear wave equations with null conditions in exterior domains

Fei Hou, Huicheng Yin, Meng Yuan

TL;DR

The paper resolves the open problem of global existence for small-data solutions to 2-D quadratic quasilinear wave equations in exterior domains under the null condition. It integrates null-structure analysis, a new good unknown $V_a$, and divergence-form rewritings with refined pointwise and energy estimates, including ghost-weight energy and local energy decay in a two-dimensional exterior setting. The result demonstrates global smooth solvability with explicit decay for both the solution and its derivatives, advancing understanding of 2-D wave dynamics in exterior domains with boundaries. The methods overcome the lack of 2-D KSS-type estimates by leveraging refined pointwise bounds and boundary-localized energy analysis, with potential applicability to multi-component systems and related exterior-domain problems.

Abstract

For 3-D quadratic quasilinear wave equations with or without null conditions in exterior domains, when the compatible initial data and Dirichlet boundary values are given, the global existence or the maximal existence time of small data smooth solutions have been established in early references. For the Cauchy problem of 2-D quadratic quasilinear wave equations with null conditions, it has been shown that the small data smooth solutions exist globally. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. In the present paper, we solve this open problem through proving the global existence of small solutions in exterior domains. Our main ingredients include: deriving new precise pointwise estimates for the initial boundary value problem of 2-D linear wave equations in exterior domains; finding appropriate divergence structures of quasilinear wave equations under null conditions; introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity, and establishing some crucial pointwise spacetime decay estimates of solutions and their derivatives.

Global smooth solutions of 2-D quadratic quasilinear wave equations with null conditions in exterior domains

TL;DR

The paper resolves the open problem of global existence for small-data solutions to 2-D quadratic quasilinear wave equations in exterior domains under the null condition. It integrates null-structure analysis, a new good unknown , and divergence-form rewritings with refined pointwise and energy estimates, including ghost-weight energy and local energy decay in a two-dimensional exterior setting. The result demonstrates global smooth solvability with explicit decay for both the solution and its derivatives, advancing understanding of 2-D wave dynamics in exterior domains with boundaries. The methods overcome the lack of 2-D KSS-type estimates by leveraging refined pointwise bounds and boundary-localized energy analysis, with potential applicability to multi-component systems and related exterior-domain problems.

Abstract

For 3-D quadratic quasilinear wave equations with or without null conditions in exterior domains, when the compatible initial data and Dirichlet boundary values are given, the global existence or the maximal existence time of small data smooth solutions have been established in early references. For the Cauchy problem of 2-D quadratic quasilinear wave equations with null conditions, it has been shown that the small data smooth solutions exist globally. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. In the present paper, we solve this open problem through proving the global existence of small solutions in exterior domains. Our main ingredients include: deriving new precise pointwise estimates for the initial boundary value problem of 2-D linear wave equations in exterior domains; finding appropriate divergence structures of quasilinear wave equations under null conditions; introducing a good unknown to eliminate the resulting type nonlinearity, and establishing some crucial pointwise spacetime decay estimates of solutions and their derivatives.

Paper Structure

This paper contains 16 sections, 33 theorems, 272 equations, 1 table.

Table of Contents

  1. Introduction
  2. Null condition, divergence structure and a new good unknown
  3. Preliminaries
  4. Some useful lemmas
  5. Pointwise estimates of Cauchy problem
  6. Improved pointwise estimates of the Cauchy problem
  7. Estimates of the Cauchy problem: Proofs of Lemmas \ref{['lem:impr:pw']} and \ref{['lem:impr:dpw']}
  8. Pointwise estimates of the initial boundary value problem
  9. Energy estimates
  10. Bootstrap assumptions
  11. Energy estimates
  12. Decay estimates of local energy and improved energy estimates
  13. Improved pointwise estimates and proof of Theorem \ref{['thm1']}
  14. Decay estimates of the good derivatives
  15. Crucial pointwise estimates
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that the obstacle $\mathcal{O}$ is star-shaped and null:condition holds. It is assumed that and the compatibility conditions up to $(2N+1)-$order hold for the initial boundary values of QWE. Then there is a small constant $\varepsilon_0>0$ such that when $\varepsilon\le\varepsilon_0$, QWE has a global solution $u\in\bigcap\limits_{j=0}^{2N+1}C^{j}([0,\infty), H^{2N+1-j}(\mathcal{K}))$. Me

Theorems & Definitions (74)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 64 more