Wave map energy and criticality
= Wave map energy and criticality
{title2=$E(\phi_\lambda)=\lambda^{n-2}E(\phi)$}
Under $\phi_\lambda(t,x)=\phi(t/\lambda,x/\lambda)$, the conserved <wave map energy> scales as $\lambda^{n-2}E$. Energy is subcritical for $n=1$, critical for $n=2$, and supercritical for $n\geq3$. 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.