Wave map energy ($E=\frac{1}{2}\int (|{\varphi }_t{|}^2+|\nabla \varphi {|}^2)dx$)

The conserved energy of a wave map from Minkowski spacetime to a Riemannian target is $E=\frac{1}{2}\int (|{\varphi }_t{|}^2+|\nabla \varphi {|}^2)dx$, with norms measured in the target metric. Contraction of the wave map equation with the velocity and integration by parts gives conservation.

Under ${\varphi }_{\lambda }(t,x)=\varphi (t/\lambda ,x/\lambda )$, the conserved wave map energy scales as ${\lambda }^{n-2}E$. Energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\ge 3$. The critical derivative index of the homogeneous Sobolev space is $s_c=n/2$. Small energy alone is not an appropriate general small-data regularity hypothesis in supercritical dimensions.