CHAPTER 1
Introduction
In this article we consider the Schr¨ odinger map equation in R2+1 with values
into S2,
(1.1) ut = u × Δu, u(0) = u0
This equation admits a conserved energy,
E(u) =
1
2
R2
|∇u|2dx
and is invariant with respect to the dimensionless scaling
u(t, x)
u(λ2t,
λx).
The energy is invariant with respect to the above scaling, therefore the Schr¨odinger
map equation in
R2+1
is energy critical.
Local solutions for regular large initial data have been constructed in [25] and
[18]. Low regularity small data Schr¨ odinger maps were studied in several works,
see [1], [2], [3], [10], [11], [13], [14], [15], [20], [21], [22]. The definitive result for
the small data problem was obtained by the authors and collaborators in [4]. There
global well-posedness and scattering are proved for initial data which is small in
the energy space
˙
H
1.
However, such a result cannot hold for large data. In particular there exists a
collection of families
Qm
of finite energy stationary solutions, indexed by integers
m 1. To describe these families we begin with the maps
Qm
defined in polar
coordinates by
Qm(r,
θ) =
emθR
¯m(r),
Q
¯m(r)
Q =


h1
m(r)
0
h3
m(r)


, m Z \ {0}
with
h1
m(r)
=
2rm
r2m + 1
, h3
m(r)
=
r2m
1
r2m + 1
.
Here R is the generator of horizontal rotations, which can be interpreted as a matrix
or, equivalently, as the operator below
R =


0 −1 0
1 0 0
0 0 0


, Ru =

k × u
The families
Qm
are constructed from
Qm
via the symmetries of the problem,
namely scaling and isometries of the base space
R2
and of the target space
S2.
Q−m
generates the same family
Qm.
The elements of
Qm
are harmonic maps
from
R2
into
S2,
and admit a variational characterization as the unique energy
1
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