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

TL;DR

This paper proves the global existence of small smooth solutions for 2-D quadratic quasilinear wave equations with null conditions in exterior domains, addressing an open problem in the field.

Contribution

It introduces new pointwise estimates, divergence structures, and a good unknown technique to establish global solutions for 2-D exterior domain problems.

Findings

Established global existence of small solutions in exterior domains

Developed new decay estimates for solutions and derivatives

Identified divergence structures under null conditions

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.