Universality of blow up for small wave maps
H. Jia
The talk is based on joint works with Duyckaerts, Kenig, Merle
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Channel of energy inequalities for outgoing waves
Let us begin with the channel of energy inequality for outgoing waves.
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Channel of energy inequalities for outgoing waves
Let us begin with the channel of energy inequality for outgoing waves.
Fix β ∈ (0, 1), for “outgoing” initial data (u0 , u1 ), i.e.,
k∂θ u0 kL2 + k(u0 , u1 )kḢ 1 ×L2 (B1+δ \B1−δ ) + k∂r u0 + u1 kL2 ≤ δk(u0 , u1 )kḢ 1 ×L2 ,
with a sufficiently small δ > 0, then solution u to the linear wave
equation with initial data (u0 , u1 ) satisfies
Z
|∇x,t u|2 (x, t) dx ≥ βk(u0 , u1 )k2Ḣ 1 ×L2 .
(1)
|x|≥β+t
Qualitative version DJKM 2016 for d ≥ 3; this version for d = 2 DJKM 2016 Dec.
Outgoing condition appears naturally.
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Channel of energy for outgoing waves
We refer to the paper “universality of blow up...” for the simple proof.
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Wave maps
Let us now turn to wave maps.
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Wave maps
Wave map u from R 2+1 with Minkowski metric to the standard 2-sphere
S 2 ⊂ R 3 satisfies
∂tt u − ∆u = (|∇u|2 − |∂t u|2 )u, in R 2 × [0, ∞),
(2)
−
with initial data →
u (0) := (u, ∂t u)(0) = (u0 , u1 ).
The initial data (u0 , u1 ) must satisfy the “compatibility condition” that
|u0 | ≡ 1 and u0 · u1 ≡ 0.
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Wave maps into the sphere, Cauchy problem
Equation (2) is invariant under the natural scaling
u → uλ (x, t) = u(λx, λt).
(3)
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Wave maps into the sphere, Cauchy problem
Equation (2) is invariant under the natural scaling
u → uλ (x, t) = u(λx, λt).
(3)
The conserved energy
−
E(→
u ) :=
Z
R2
|∂t u|2
|∇u|2
+
2
2
(x, t) dx
(4)
is invariant under the scaling (3).
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Wave maps into the sphere, Cauchy problem
Equation (2) is invariant under the natural scaling
u → uλ (x, t) = u(λx, λt).
(3)
The conserved energy
−
E(→
u ) :=
Z
R2
|∂t u|2
|∇u|2
+
2
2
(x, t) dx
(4)
is invariant under the scaling (3).
Scale-invariance of the equation plays an essential role in both the
Cauchy problem and the dynamics of solutions.
Energy is invariant under scaling → the equation is called energy critical.
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Channel of Energy for small wave map
The first main result is the following channel of energy for small wave
maps.
Theorem
Fix β ∈ (0, 1). There exists a small δ = δ(β) > 0 and sufficiently small
0 = 0 (β) > 0, such that if u is a classical wave map with energy
−
E(→
u ) < 0 satisfying the “outgoing condition”, then for all t ≥ 0, we
have
Z
2
|∇x,t u|2 (x, t) dx ≥ β k(u0 , u1 )kḢ
(5)
1 ×L2 .
|x|>β+t
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Universality for small blow up wave maps
As an application, we obtain a classification of finite time blow up wave
maps u with energy
−
E(→
u ) < E(Q, 0) + 20 .
(6)
Q is the harmonic map with the least energy (which is equal to 4π).
Denote M1 as the space of degree one harmonic maps, these harmonic
maps are all co-rotational.
Let
M`,1 := {Q` : Q ∈ M} ,
where
Q` (x, t) = Q
`·x
`·x
|`|2 ` − `t
x − 2 `+ p
|`|
1 − |`|2
!
,
(7)
is the Lorentz transformation of the harmonic map Q.
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Universality for small blow up wave maps
Then we have
Theorem
There exists velocity ` ∈ R 2 with |`| 1, position x(t) ∈ R 2 , scale
λ(t) > 0 with
lim
t→T+
x(t)
= `, λ(t) = o (T+ − t) ,
T+ − t
and (v0 , v1 ) ∈ Ḣ 1 × L2 ∩ C ∞ (R 2 \{0})
such that as t → T+
−
(i)
inf →
u (t) − (v0 , v1 ) − (Q` , ∂t Q` )Ḣ 1 ×L2 : Q` ∈ M`,1 → 0,
(ii)
→
−
u (t) − (v0 , v1 )
→ 0.
Ḣ 1 ×L2 (R 2 \Bλ(t) (x(t)))
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Universality for small blow up time
In comparison, the famous Sterbenz-Tataru result, in this restricted
setting, says that along a sequence of times and locally in space in
suitable region, this is the case.
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Wave maps into the sphere, Cauchy problem
To put our result in perspective, let us very briefly recall the history of
the study of energy critical wave maps.
The Cauchy problem of (2) has been studied for a long time and there
are many deep results.
Firstly, if (u0 , u1 ) ∈ Ḣ s × H s−1 with sufficiently large s, then by standard
energy method, one can prove local existence of solutions.
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Wave maps into the sphere, Cauchy problem
To put our result in perspective, let us very briefly recall the history of
the study of energy critical wave maps.
The Cauchy problem of (2) has been studied for a long time and there
are many deep results.
Firstly, if (u0 , u1 ) ∈ Ḣ s × H s−1 with sufficiently large s, then by standard
energy method, one can prove local existence of solutions.
Klainerman and Machedon, Klainerman and Selberg, and Selberg
estabilished local wellposedness for (u0 , u1 ) ∈ Ḣ s × H s−1 with s > 1,
using bilinear estimates in X s,b type spaces.
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Wave maps into the sphere, Cauchy problem
Denote F(u) as the spacetime Fourier transform of u. For s, b ∈ R, and
tempered distribution u ∈ S 0 (R 3 ), define
Z
kukX s,b (R 3 ) :=
2 s
2b
2
(1 + |ξ| ) (1 + ||ξ| − |τ ||) |F(u)(ξ, τ )| dξdτ
21
,
R3
(8)
and set
X s,b (R 3 ) := u ∈ S 0 (R 3 ) : kukX s,b (R 3 ) < ∞ .
(9)
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Wave maps into the sphere, Cauchy problem
Denote F(u) as the spacetime Fourier transform of u. For s, b ∈ R, and
tempered distribution u ∈ S 0 (R 3 ), define
Z
kukX s,b (R 3 ) :=
2 s
2b
2
(1 + |ξ| ) (1 + ||ξ| − |τ ||) |F(u)(ξ, τ )| dξdτ
21
,
R3
(8)
and set
X s,b (R 3 ) := u ∈ S 0 (R 3 ) : kukX s,b (R 3 ) < ∞ .
(9)
Basic idea: use the fact that is elliptic for frequencies |τ | =
6 |ξ|.
For instance, consider u = f , intead of requiring f ∈ L1t L2x we only need
L2t,x to solve u, if f has frequency support away from the lightcone. The
X s,b space takes into account of this Fourier support information. Here
one needs s > 1, b > 21 .
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Wave maps into the sphere, Cauchy problem
It remained an outstanding open question to prove wellposedness in the
critical, energy space, until the breakthrough work of Tataru and Tao.
In this case, since the initial data space is critical, one expects a small
data global existence result.
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Wave maps into the sphere, Cauchy problem
It remained an outstanding open question to prove wellposedness in the
critical, energy space, until the breakthrough work of Tataru and Tao.
In this case, since the initial data space is critical, one expects a small
data global existence result.
The X s,b space technique does not work, because simply the free wave is
not in X s,b .
The deadlock was firstly broken by Tataru, who introduced new spaces,
the null frame spaces, and new sets of estimates for the problem.
Roughly speaking, he observed that free waves can be decomposed as
sums of traveling waves with respect to various null directions, and one
can obtain good energy estimates with respect to a null direction as long
as the frequency support forms an angle with that direction.
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Null frames illustrated
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Wellposedness in the critical space
With these new estimates and spaces, Tataru proved global wellposedess
if
X
kPk (u0 , u1 )kḢ 1 ×L2 < ,
k
for sufficiently small .
This norm is stronger than the energy norm, and consequently the global
wellposedness for small energy data was not covered by Tataru’s result.
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Wellposedness in the critical space
With these new estimates and spaces, Tataru proved global wellposedess
if
X
kPk (u0 , u1 )kḢ 1 ×L2 < ,
k
for sufficiently small .
This norm is stronger than the energy norm, and consequently the global
wellposedness for small energy data was not covered by Tataru’s result.
Tao made another breakthrough. He found that in order to transform the
nonlinearity to a manageable perturbative, one needs perform a gauge
transform.
To illustrate the idea, we need to use Littlewood-Paley decompositions.
Let Pk denote the standard projection of frequency to |ξ| ∼ 2k , and P<k
projection to |ξ| . 2k .
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Wellposedness in the critical space
Let
ψ := Pm u.
Then ψ verifies
∂tt ψ − ∆ψ
→
−
ψ (0)
= Pm u ∂ α u † ∂α u
= (Pm u0 , Pm u1 ).
(10)
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Wellposedness in the critical space
Let
ψ := Pm u.
Then ψ verifies
∂tt ψ − ∆ψ
→
−
ψ (0)
= Pm u ∂ α u † ∂α u
= (Pm u0 , Pm u1 ).
(10)
It turns out that the above equation can be re-written as
†
†
∂α ψ,
∂tt ψ − ∆ψ = error + 2 u<m−10 ∂ α u<m−10
− ∂ α u<m−10 u<m−10
(11)
†
†
An important feature is that u<m−10 ∂ α u<m−10
− ∂ α u<m−10 u<m−10
is
anti-symmetric matrices.
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The gauge transform
To illustrate the usefulness of a gauge transform, let us consider the
model problem
∂tt u − ∆u + A1 ∂x1 u = 0,
where we assume u is a vector and A1 is a constant anti-symmetric
metrix.
This equation is moderately complicated as the lower order term involves
derivatives.
But if we introduce
A
u = e 2 x1 v ,
then v verifies
∂tt v − ∆v +
A2
v = 0.
2
This is a much simpler equation, as the lower order term no longer
contains derivatives.
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The gauge transform
To illustrate the usefulness of a gauge transform, let us consider the
model problem
∂tt u − ∆u + A1 ∂x1 u = 0,
where we assume u is a vector and A1 is a constant anti-symmetric
metrix.
This equation is moderately complicated as the lower order term involves
derivatives.
But if we introduce
A
u = e 2 x1 v ,
then v verifies
∂tt v − ∆v +
A2
v = 0.
2
This is a much simpler equation, as the lower order term no longer
contains derivatives.
Tao introduced a “micro-local” gauge to achieve something like this–the
details are too technical to describe here.
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The gauge transform
After “rotating” the waves by a (nearly) orthogonal matrix, the waves
then behave almost linearly.
This in fact also explains why the channel of energy argument also works
for (small) wave maps. There are two things to check:
1. the outgoing condition is “preserved” under frequency projections;
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The gauge transform
After “rotating” the waves by a (nearly) orthogonal matrix, the waves
then behave almost linearly.
This in fact also explains why the channel of energy argument also works
for (small) wave maps. There are two things to check:
1. the outgoing condition is “preserved” under frequency projections;
2. the gauge transform does not change the energy distribution very
much.
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The gauge transform
After “rotating” the waves by a (nearly) orthogonal matrix, the waves
then behave almost linearly.
This in fact also explains why the channel of energy argument also works
for (small) wave maps. There are two things to check:
1. the outgoing condition is “preserved” under frequency projections;
2. the gauge transform does not change the energy distribution very
much.
The first condition can be checked, by relatively straightforward (but
technical and lengthy) calculations.
The second condition holds because the gauge is “almost orthogonal”
and has “small derivatives”. Thus
|∇x,t (Uψ)| = |∇x,t U ψ + U∇x,t ψ| ≈ |U∇x,t ψ| ≈ |∇x,t ψ|.
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Blow up Wave maps
Going beyond small energy wave maps, one encounter a variety of new
phenomenon.
1 The large data theory was studied by Tao, Sterbenz-Tataru, Krieger-Schlag
independently.
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Blow up Wave maps
Going beyond small energy wave maps, one encounter a variety of new
phenomenon.
Firstly there are harmonic maps, which are stationary solutions to the
wave map equations.
1 The large data theory was studied by Tao, Sterbenz-Tataru, Krieger-Schlag
independently.
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Blow up Wave maps
Going beyond small energy wave maps, one encounter a variety of new
phenomenon.
Firstly there are harmonic maps, which are stationary solutions to the
wave map equations.
Then it is known, by Krieger-Schlag-Tataru, Rodnianski-Sterbenz,
Raphael-Rodnianski, that in the equivariant setting, when energy is
higher than the smallest energy harmonic maps, solutions can blow up.
The blow up is through concentrating harmonic maps.
1 The large data theory was studied by Tao, Sterbenz-Tataru, Krieger-Schlag
independently.
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Blow up Wave maps
Going beyond small energy wave maps, one encounter a variety of new
phenomenon.
Firstly there are harmonic maps, which are stationary solutions to the
wave map equations.
Then it is known, by Krieger-Schlag-Tataru, Rodnianski-Sterbenz,
Raphael-Rodnianski, that in the equivariant setting, when energy is
higher than the smallest energy harmonic maps, solutions can blow up.
The blow up is through concentrating harmonic maps.
By the small data theory, if a wave map blows up, then there must be
energy concentration near the blow up point. The question is then how
the energy is concentrated. 1
1 The large data theory was studied by Tao, Sterbenz-Tataru, Krieger-Schlag
independently.
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Blow up Wave maps
In the large data theory, global control provided by Morawetz estimates
play an essential role in understanding the way the energy is concentrated.
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Blow up wave map with restriced -size- energy
The vanishing of ∂t u + xt · ∇u turns out to make the behavior of the
wave map in the interior easy to understand. Basically, one can recover
strong “compactness”.
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Blow up wave map with restriced -size- energy
The vanishing of ∂t u + xt · ∇u turns out to make the behavior of the
wave map in the interior easy to understand. Basically, one can recover
strong “compactness”.
The main difficulty is to understand what happens at the boundary.
Sterbenz-Tataru, in a long and difficult paper, showed that if the wave
map blows up, then not all energy can concentrate on the boundary.
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Blow up wave map with restriced -size- energy
The vanishing of ∂t u + xt · ∇u turns out to make the behavior of the
wave map in the interior easy to understand. Basically, one can recover
strong “compactness”.
The main difficulty is to understand what happens at the boundary.
Sterbenz-Tataru, in a long and difficult paper, showed that if the wave
map blows up, then not all energy can concentrate on the boundary.
This implies that there is energy inside, and combined this fact with the
“compactness property”, they show that there is a concentration of
traveling waves inside.
Indeed, if ∂t u + ` · ∇u ≡ 0 and u is a wave map, then u must be a
traveling wave. This happens when x is highly concentrated near `t in
the vanishing of ∂t u + xt · ∇u.
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Elimination of dispersive energy, illustrated
Our main contribution is to rule out completely concentration of energy
near the boundary.
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Universal blow up behavior and possible extension to larger
energies
Then our main theorem follows from (expected) coercive property of
energy near traveling waves.
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Universal blow up behavior and possible extension to larger
energies
Then our main theorem follows from (expected) coercive property of
energy near traveling waves.
We expect that the same idea works for arbitrarily large outgoing wave
maps, but at this moment I did not know how to do this.
23 / 24
Universal blow up behavior and possible extension to larger
energies
Then our main theorem follows from (expected) coercive property of
energy near traveling waves.
We expect that the same idea works for arbitrarily large outgoing wave
maps, but at this moment I did not know how to do this.
The main issue is that in the large data case, the current perturbative
theory does not allow easy way to use the “outgoing condition” to gain
smallness.
23 / 24
Universal blow up behavior and possible extension to larger
energies
Then our main theorem follows from (expected) coercive property of
energy near traveling waves.
We expect that the same idea works for arbitrarily large outgoing wave
maps, but at this moment I did not know how to do this.
The main issue is that in the large data case, the current perturbative
theory does not allow easy way to use the “outgoing condition” to gain
smallness.
This is in sharp contrast to the semilinear case
∂tt u − ∆u = u 5 ,
where the outgoing condition on the initial data immediately implies that
u L (the free evolution) satisfies
ku L kL5t L10
3
x (R ×[0,∞))
is small.
23 / 24
Universal blow up behavior and possible extension to larger
energies
Then our main theorem follows from (expected) coercive property of
energy near traveling waves.
We expect that the same idea works for arbitrarily large outgoing wave
maps, but at this moment I did not know how to do this.
The main issue is that in the large data case, the current perturbative
theory does not allow easy way to use the “outgoing condition” to gain
smallness.
This is in sharp contrast to the semilinear case
∂tt u − ∆u = u 5 ,
where the outgoing condition on the initial data immediately implies that
u L (the free evolution) satisfies
ku L kL5t L10
3
x (R ×[0,∞))
is small.
There is no such an easy way to gain smallness for wave maps, at this
moment.
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Thank you!
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