We consider the wave maps problem with domain
R
2
+
1
\mathbb {R}^{2+1}
and target
S
2
\mathbb {S}^{2}
in the 1-equivariant, topological degree one setting. In this setting, we recall that the soliton is a harmonic map from
R
2
\mathbb {R}^{2}
to
S
2
\mathbb {S}^{2}
, with polar angle equal to
Q
1
(
r
)
=
2
arctan
(
r
)
Q_{1}(r) = 2 \arctan (r)
. By applying the scaling symmetry of the equation,
Q
λ
(
r
)
=
Q
1
(
r
λ
)
Q_{\lambda }(r) = Q_{1}(r \lambda )
is also a harmonic map, and the family of all such
Q
λ
Q_{\lambda }
are the unique minimizers of the harmonic map energy among finite energy, 1-equivariant, topological degree one maps. In this work, we construct infinite time blowup solutions along the
Q
λ
Q_{\lambda }
family. More precisely, for
b
>
0
b>0
, and for all
λ
0
,
0
,
b
∈
C
∞
(
[
100
,
∞
)
)
\lambda _{0,0,b} \in C^{\infty }([100,\infty ))
satisfying, for some
C
l
,
C
m
,
k
>
0
C_{l}, C_{m,k}>0
,
C
l
log
b
(
t
)
≤
λ
0
,
0
,
b
(
t
)
≤
C
m
log
b
(
t
)
,
|
λ
0
,
0
,
b
(
k
)
(
t
)
|
≤
C
m
,
k
t
k
log
b
+
1
(
t
)
,
k
≥
1
t
≥
100
\begin{equation*} \frac {C_{l}}{\log ^{b}(t)} \leq \lambda _{0,0,b}(t) \leq \frac {C_{m}}{\log ^{b}(t)}, \quad |\lambda _{0,0,b}^{(k)}(t)| \leq \frac {C_{m,k}}{t^{k} \log ^{b+1}(t) }, k\geq 1 \quad t \geq 100 \end{equation*}
there exists a wave map with the following properties. If
u
b
u_{b}
denotes the polar angle of the wave map into
S
2
\mathbb {S}^{2}
, we have
u
b
(
t
,
r
)
=
Q
1
λ
b
(
t
)
(
r
)
+
v
2
(
t
,
r
)
+
v
e
(
t
,
r
)
,
t
≥
T
0
\begin{equation*} u_{b}(t,r) = Q_{\frac {1}{\lambda _{b}(t)}}(r) + v_{2}(t,r) + v_{e}(t,r), \quad t \geq T_{0} \end{equation*}
where
−
∂
t
t
v
2
+
∂
r
r
v
2
+
1
r
∂
r
v
2
−
v
2
r
2
=
0
\begin{equation*} -\partial _{tt}v_{2}+\partial _{rr}v_{2}+\frac {1}{r}\partial _{r}v_{2}-\frac {v_{2}}{r^{2}}=0 \end{equation*}
|
|
∂
t
(
Q
1
λ
b
(
t
)
+
v
e
)
|
|
L
2
(
r
d
r
)
2
+
|
|
v
e
r
|
|
L
2
(
r
d
r
)
2
+
|
|
∂
r
v
e
|
|
L
2
(
r
d
r
)
2
≤
C
t
2
log
2
b
(
t
)
,
t
≥
T
0
\begin{equation*} ||\partial _{t}(Q_{\frac {1}{\lambda _{b}(t)}}+v_{e})||_{L^{2}(r dr)}^{2}+||\frac {v_{e}}{r}||_{L^{2}(r dr)}^{2} + ||\partial _{r}v_{e}||_{L^{2}(r dr)}^{2} \leq \frac {C}{t^{2} \log ^{2b}(t)}, \quad t \geq T_{0} \end{equation*}
and
λ
b
(
t
)
=
λ
0
,
0
,
b
(
t
)
+
O
(
1
log
b
(
t
)
log
(
log
(
t
)
)
)
\begin{equation*} \lambda _{b}(t) = \lambda _{0,0,b}(t) + O\left (\frac {1}{\log ^{b}(t) \sqrt {\log (\log (t))}}\right ) \end{equation*}