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Characterization of large energy solutions of the equivariant wave map problem: I
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 137, Number 1, February 2015
- pp. 139-207
- 10.1353/ajm.2015.0002
- Article
- Additional Information
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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.