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

TL;DR

This paper proves the long-time existence of small smooth solutions for 2-D quadratic quasilinear wave equations with Neumann boundary conditions in exterior domains, filling a gap in the understanding of such problems.

Contribution

It introduces new decay estimates and good unknowns to establish long-time solutions for 2-D quadratic quasilinear wave equations with Neumann boundary conditions.

Findings

Established pointwise spacetime decay estimates for 2-D wave equations.

Derived energy estimates using new good unknowns.

Applied results to Euler, membrane, and relativistic membrane equations.

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.