= Solution
\b[<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 $D_v\phi_u=D_u\phi_v=0$, with $D$ the target <covariant derivative>. Hence $|\phi_u|^2$ depends only on $u$, and $|\phi_v|^2$ only on $v$. Initial derivative bounds persist. Compactness of the <sphere> and differentiated <energy estimates> then prevent finite-time loss of smoothness.
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