Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 10 / 4 / 7 / Solution

Small data global regularity for wave maps also holds in three spatial dimensions. For localized compatible data small in high weighted Sobolev norms, the key is the classical null condition for wave equations. Each derivative contraction is a null form for wave equations, vanishing for parallel null derivatives. The vector field method for wave equations exploits derivatives tangent to the light cone and weighted energy estimates to close the bootstrap argument; ordinary $(1+t{)}^{-1}$ decay alone is insufficient.