Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 10 / 4 / 6

Small data global regularity for wave maps holds in four spatial dimensions. Smallness is measured in sufficiently high weighted Sobolev norms relative to a constant map, with localized compatible Cauchy data. The vector field method for wave equations gives derivative decay $(1+t{)}^{-3/2}$. This is time-integrable, so commuted wave energy estimates close a small-data bootstrap argument for the derivative-quadratic semilinear wave equation. Higher regularity persists. No smallness of energy alone is asserted.