 Solution

Global regularity for one-dimensional wave maps holds for arbitrary smooth localized compatible data. In retarded and advanced null coordinates $u=t-x$, $v=t+x$, the wave map equation says $D_v{\varphi }_u=D_u{\varphi }_v=0$, with $D$ the target covariant derivative. Hence $|{\varphi }_u{|}^2$ depends only on $u$, and $|{\varphi }_v{|}^2$ only on $v$. Initial derivative bounds persist. Compactness of the sphere and differentiated energy estimates then prevent finite-time loss of smoothness.