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Long time smooth solutions of 2-D quadratic quasilinear wave equations in exterior domains with Neumann boundary conditions

Fei Hou, Huicheng Yin, Meng Yuan

TL;DR

The authors address the long-time behavior of 2-D quadratic quasilinear wave equations in exterior domains with Neumann boundary conditions. They develop a divergence-form reformulation and a hierarchy of good unknowns to derive precise energy and pointwise decay estimates, including local energy decay, even in the presence of Neumann boundaries. Under convexity and compatibility assumptions, they prove a lifespan lower bound $T_ abla \ge 1/\varepsilon^{2-}$ for small data, and, when first and second null conditions hold, global existence with detailed decay rates and local energy decay, plus multiple applications to 2-D compressible Euler equations and membrane models. The work extends 3-D results to 2-D exterior Neumann problems, providing robust techniques—divergence structures, ghost weights, and refined energy methods—that are applicable to isentropic Euler dynamics, Chaplygin gases, and membrane equations in exterior settings.

Abstract

For the 3-D quadratic quasilinear wave equations in exterior domains with Dirichlet or Neumann boundary conditions, the global existence or the maximal existence time of small data smooth solutions have been established in the past. However, so far it is still open for the corresponding 2-D Neumann boundary value problem. In this paper, we investigate the long time existence of small data solutions to 2-D quadratic quasilinear wave equations with homogeneous Neumann boundary values. Our main ingredients include: establishing some new pointwise spacetime decay estimates for the 2-D initial boundary value problem of the divergence form wave equations, and introducing a series of good unknowns to derive the required energy estimates. The obtained results can be directly applied to the initial boundary value problem of 2-D isentropic and irrotational compressible Euler equations for both the polytropic gases and the Chaplygin gases in exterior domains with impermeable conditions, the 2-D relativistic membrane equations and 2-D membrane equations with homogeneous Neumann boundary values.

Long time smooth solutions of 2-D quadratic quasilinear wave equations in exterior domains with Neumann boundary conditions

TL;DR

The authors address the long-time behavior of 2-D quadratic quasilinear wave equations in exterior domains with Neumann boundary conditions. They develop a divergence-form reformulation and a hierarchy of good unknowns to derive precise energy and pointwise decay estimates, including local energy decay, even in the presence of Neumann boundaries. Under convexity and compatibility assumptions, they prove a lifespan lower bound for small data, and, when first and second null conditions hold, global existence with detailed decay rates and local energy decay, plus multiple applications to 2-D compressible Euler equations and membrane models. The work extends 3-D results to 2-D exterior Neumann problems, providing robust techniques—divergence structures, ghost weights, and refined energy methods—that are applicable to isentropic Euler dynamics, Chaplygin gases, and membrane equations in exterior settings.

Abstract

For the 3-D quadratic quasilinear wave equations in exterior domains with Dirichlet or Neumann boundary conditions, the global existence or the maximal existence time of small data smooth solutions have been established in the past. However, so far it is still open for the corresponding 2-D Neumann boundary value problem. In this paper, we investigate the long time existence of small data solutions to 2-D quadratic quasilinear wave equations with homogeneous Neumann boundary values. Our main ingredients include: establishing some new pointwise spacetime decay estimates for the 2-D initial boundary value problem of the divergence form wave equations, and introducing a series of good unknowns to derive the required energy estimates. The obtained results can be directly applied to the initial boundary value problem of 2-D isentropic and irrotational compressible Euler equations for both the polytropic gases and the Chaplygin gases in exterior domains with impermeable conditions, the 2-D relativistic membrane equations and 2-D membrane equations with homogeneous Neumann boundary values.
Paper Structure (26 sections, 40 theorems, 339 equations, 1 table)

This paper contains 26 sections, 40 theorems, 339 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main results and remarks
  3. Applications of main results
  4. Application to 2-D isentropic and irrotational compressible Euler equations
  5. Application to 3-D isentropic and irrotational supersonic Euler equations
  6. Application to 2-D relativistic membrane equation
  7. Application to 2-D membrane equation
  8. Previous results and sketches of proofs
  9. Preliminaries
  10. Several lemmas
  11. Some pointwise estimates of Cauchy problem
  12. Null conditions
  13. Some estimates of the initial boundary value problem
  14. Proof of Theorem \ref{['thm1']}
  15. Energy estimates
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose that the obstacle $\mathcal{O}$ is convex and YHCC-1-YHCC-3 hold. Then for any fixed constant $\delta\in(0,1)$, there is a constant $\varepsilon_0>0$ such that when $\varepsilon\le\varepsilon_0$ and problem QWE has a smooth solution $u\in\bigcap\limits_{j=0}^{2N+1}C^{j}([0,\varepsilon^{\delta-2}], H^{2N+1-j}(\mathcal{K}))$.

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1: Lemma 3.2 of Kubo13
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 73 more