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

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.