Infinite Time Blow-Up Solutions to the Energy Critical Wave Maps Equation
Authors: Pillai Mohandas 
Published: April 2023

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

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction upper C Subscript l Baseline Over log Superscript b Baseline left-parenthesis t right-parenthesis EndFraction less-than-or-equal-to lamda Subscript 0 comma 0 comma b Baseline left-parenthesis t right-parenthesis less-than-or-equal-to StartFraction upper C Subscript m Baseline Over log Superscript b Baseline left-parenthesis t right-parenthesis EndFraction comma StartAbsoluteValue lamda Subscript 0 comma 0 comma b Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis t right-parenthesis EndAbsoluteValue less-than-or-equal-to StartFraction upper C Subscript m comma k Baseline Over t Superscript k Baseline log Superscript b plus 1 Baseline left-parenthesis t right-parenthesis EndFraction comma k greater-than-or-equal-to 1 t greater-than-or-equal-to 100">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript b Baseline left-parenthesis t comma r right-parenthesis equals upper Q Subscript StartFraction 1 Over lamda Sub Subscript b Subscript left-parenthesis t right-parenthesis EndFraction Baseline left-parenthesis r right-parenthesis plus v 2 left-parenthesis t comma r right-parenthesis plus v Subscript e Baseline left-parenthesis t comma r right-parenthesis comma t greater-than-or-equal-to upper T 0">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus partial-differential Subscript t t Baseline v 2 plus partial-differential Subscript r r Baseline v 2 plus StartFraction 1 Over r EndFraction partial-differential Subscript r Baseline v 2 minus StartFraction v 2 Over r squared EndFraction equals 0">

−

∂

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*}

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue partial-differential Subscript t Baseline left-parenthesis upper Q Subscript StartFraction 1 Over lamda Sub Subscript b Subscript left-parenthesis t right-parenthesis EndFraction Baseline plus v Subscript e Baseline right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript upper L squared left-parenthesis r d r right-parenthesis Superscript 2 Baseline plus StartAbsoluteValue EndAbsoluteValue StartFraction v Subscript e Baseline Over r EndFraction StartAbsoluteValue EndAbsoluteValue Subscript upper L squared left-parenthesis r d r right-parenthesis Superscript 2 Baseline plus StartAbsoluteValue EndAbsoluteValue partial-differential Subscript r Baseline v Subscript e Baseline StartAbsoluteValue EndAbsoluteValue Subscript upper L squared left-parenthesis r d r right-parenthesis Superscript 2 Baseline less-than-or-equal-to StartFraction upper C Over t squared log Superscript 2 b Baseline left-parenthesis t right-parenthesis EndFraction comma t greater-than-or-equal-to upper T 0">

|

|

∂

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript b Baseline left-parenthesis t right-parenthesis equals lamda Subscript 0 comma 0 comma b Baseline left-parenthesis t right-parenthesis plus upper O left-parenthesis StartFraction 1 Over log Superscript b Baseline left-parenthesis t right-parenthesis StartRoot log left-parenthesis log left-parenthesis t right-parenthesis right-parenthesis EndRoot EndFraction right-parenthesis">

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*}