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

TL;DR

This paper proves the global existence of small smooth solutions for 2-D quadratically quasilinear wave equations with null conditions in exterior domains, extending previous results to more general nonlinearities.

Contribution

It establishes the global existence for general null condition wave equations in exterior domains, using divergence structures, a good unknown, and new decay estimates.

Findings

Proved global solutions for general null condition wave equations in exterior domains.

Developed divergence structures and introduced a good unknown to handle nonlinearities.

Derived new pointwise decay estimates for solutions and derivatives.

Abstract

In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy problem of 2-D quadratically quasilinear wave equations with null conditions. However, for the corresponding 2-D initial boundary value problem in exterior domains, it is still open whether the global solutions exist. When the 2-D quadratic nonlinearity admits a special $Q_0$ type null form, the global small solution is shown in our previous article [Hou Fei, Yin Huicheng, Yuan Meng, Global smooth solutions of 2-D quadratically quasilinear wave equations with null conditions in exterior domains, arXiv:2411.06984]. In the present paper, we now solve this open problem through proving the global existence of small solutions to 2-D general quasilinear wave equations with null conditions in exterior domains. Our proof procedure is based on finding appropriate divergence structures of quasilinear wave equations under null conditions, introducing a good unknown to eliminate the resulting $Q_0$ type nonlinearity and deriving some new precise pointwise spacetime decay estimates of solutions and their derivatives.