Characterization of large energy solutions of the equivariant wave map problem: I

Abstract

We consider $1$-equivariant wave maps from ${\Bbb R}^{1+2}\to{\Bbb S}^2$. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, $Q$, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are pre-compact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either $0$ or $Q$ up to a rescaling. This gives a proof in the equivariant case of a refined version of the {\it threshold conjecture} adapted to the degree zero theory where the true threshold is $2{\mathcal E}(Q)$, not ${\mathcal E}(Q)$. The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting. For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of $Q$. Under this restriction on the energy, we show that a blow-up solution of degree one is essentially the sum of a rescaled $Q$ plus a remainder term of topological degree zero of energy less than twice the energy of $Q$. This result reveals the universal character of the known blow-up constructions for degree one, $1$-equivariant wave maps of Krieger, the fourth author, and Tataru as well as Rapha\"{e}l and Rodnianski.