We consider the Schrödinger map initial value problem
with
a smooth
map from the Euclidean
space
to the sphere
with subthreshold
() energy. Assuming an
a priori
boundedness
condition on the solution
,
we prove that the Schrödinger map system admits a unique global smooth solution
provided that
the initial data
is sufficiently energy-dispersed, i.e., sufficiently small in the critical Besov space
.
Also shown are global-in-time bounds on certain Sobolev norms of
.
Toward these ends we establish improved local smoothing and bilinear Strichartz
estimates, adapting the Planchon–Vega approach to such estimates to the nonlinear
setting of Schrödinger maps.
Keywords
Schrödinger maps, global regularity, energy-critical,
critical Besov spaces, subthreshold
