Nonexistence of blow-up solutions with smooth radiation for energy-critical equivariant wave maps

We study $k$-equivariant energy critical wave maps ${\mathbb{R}}^{1+2}\to {\mathbb{S}}^2$, for any equivariance degree $k\ge 2$. We prove that the radiation associated with any finite-energy blow-up solution cannot satisfy a certain regularity condition; in particular, it cannot be smooth. The assumption $k\ge 2$ is necessary, since for $k=1$ such solutions are known to exist. The starting point of our analysis is the soliton resolution theorem. The key ingredient is a novel application of the modulation method, in which we compare the effects of the radiation and inner bubbles to study the dynamic behavior of the widest bubble.