Global existence for 2-D wave maps equation in exterior domains

TL;DR

This paper proves the global existence of small data solutions for the 2-D wave maps equation in exterior domains, extending previous results on semilinear wave equations with null conditions.

Contribution

It establishes global existence for 2-D wave maps with Dirichlet boundary conditions in exterior domains, using new pointwise decay estimates.

Findings

Global existence of small data solutions is proven.

Pointwise decay estimates are developed for the wave maps.

Extension of null condition results to wave maps in exterior domains.

Abstract

In the paper [H. Kubo, Global existence for exterior problems of semilinear wave equations with the null condition in 2D, Evol. Equ. Control Theory 2 (2013), no. 2, 319-335], for the 2-D semilinear wave equation system $(\partial_t^2-\Delta)v^I=Q^I(\partial_tv, \nabla_xv)$ ($1\le I\le M$) in the exterior domain with Dirichlet boundary condition, it is shown that the small data smooth solution $v=(v^1, \cdot\cdot\cdot, v^M)$ exists globally when the cubic nonlinearities $Q^I(\partial_tv, \nabla_xv)=O(|\partial_tv|^3+|\nabla_xv|^3)$ satisfy the null condition. We now focus on the global Dirichelt boundary value problem of 2-D wave maps equation with the form $\Box u^I=\sum_{J,K,L=1}^MC_{IJKL}u^JQ_0(u^K,u^L)$ $(1\le I\le M)$ and $Q_0(f,g)=\partial_tf\partial_tg-\sum_{j=1}^2\partial_jf\partial_jg$ in exterior domain. By establishing some crucial classes of pointwise spacetime decay estimates for the small data solution $u=(u^1, \cdot\cdot\cdot, u^M)$ and its derivatives, the global existence of $u$ is shown.