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Dynamics of the focusing critical wave equation
Thomas Duyckaerts1 (with H. Jia, C. Kenig and F. Merle)
1Institut Galilée, Université Paris 13, Sorbonne Paris Cité Université and IUF
PDE/Analysis Mini-schoolUNC Chapel HillFebruary 2017
Thomas Duyckaerts (Paris 13) Critical waves February 2017 1 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 2 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 2 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry
4 Profile decompositionDefect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 2 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 2 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 3 / 52
Focusing critical wave equation
(NLW)
{∂2
t u −∆u = |u|4
N−2 u, x ∈ RN
~u�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)
N ≥ 3.
Conservation laws: energy
E(~u) =12
∫RN|∇xu(t)|2 +
12
∫RN|∂tu(t)|2 − N − 2
2N
∫|u(t)|
2NN−2
and momentumP(~u) =
∫RN∇xu∂tu.
(both invariant by the scaling).Local well-posedness in the scale invariant-space H = H1 × L2.Global dynamics (without size restriction)?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 4 / 52
Focusing critical wave equation
(NLW)
{∂2
t u −∆u = |u|4
N−2 u, x ∈ RN
~u�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)
N ≥ 3.Conservation laws: energy
E(~u) =12
∫RN|∇xu(t)|2 +
12
∫RN|∂tu(t)|2 − N − 2
2N
∫|u(t)|
2NN−2
and momentumP(~u) =
∫RN∇xu∂tu.
(both invariant by the scaling).
Local well-posedness in the scale invariant-space H = H1 × L2.Global dynamics (without size restriction)?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 4 / 52
Focusing critical wave equation
(NLW)
{∂2
t u −∆u = |u|4
N−2 u, x ∈ RN
~u�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)
N ≥ 3.Conservation laws: energy
E(~u) =12
∫RN|∇xu(t)|2 +
12
∫RN|∂tu(t)|2 − N − 2
2N
∫|u(t)|
2NN−2
and momentumP(~u) =
∫RN∇xu∂tu.
(both invariant by the scaling).Local well-posedness in the scale invariant-space H = H1 × L2.
Global dynamics (without size restriction)?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 4 / 52
Focusing critical wave equation
(NLW)
{∂2
t u −∆u = |u|4
N−2 u, x ∈ RN
~u�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)
N ≥ 3.Conservation laws: energy
E(~u) =12
∫RN|∇xu(t)|2 +
12
∫RN|∂tu(t)|2 − N − 2
2N
∫|u(t)|
2NN−2
and momentumP(~u) =
∫RN∇xu∂tu.
(both invariant by the scaling).Local well-posedness in the scale invariant-space H = H1 × L2.Global dynamics (without size restriction)?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 4 / 52
Linear wave equation
The linear wave equation
(LW)
{∂2
t uL −∆uL = 0, x ∈ RN
~uL�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)
is dispersive and satisfies finite speed of propagation.
Asymptotics: let /∂ be the tangential derivative. Then
limt→+∞
∫1|x |2|uL(t , x)|2 + |/∂uL(t , x)|2 + |uL(t , x)|6 dx = 0
and [Friedlander] there exists G+ ∈ L2(R× SN−1) such that
limt→+∞
∫ +∞
0
∫SN−1
∣∣∣r N−12 ∂r uL(t , rω)−G+(r − t , ω)
∣∣∣2+∣∣∣r N−1
2 ∂tuL(t , rω) + G+(r − t , ω)∣∣∣2 drdω.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 5 / 52
Linear wave equation
The linear wave equation
(LW)
{∂2
t uL −∆uL = 0, x ∈ RN
~uL�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)
is dispersive and satisfies finite speed of propagation.Asymptotics: let /∂ be the tangential derivative. Then
limt→+∞
∫1|x |2|uL(t , x)|2 + |/∂uL(t , x)|2 + |uL(t , x)|6 dx = 0
and [Friedlander] there exists G+ ∈ L2(R× SN−1) such that
limt→+∞
∫ +∞
0
∫SN−1
∣∣∣r N−12 ∂r uL(t , rω)−G+(r − t , ω)
∣∣∣2+∣∣∣r N−1
2 ∂tuL(t , rω) + G+(r − t , ω)∣∣∣2 drdω.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 5 / 52
Linear wave equation
The linear wave equation
(LW)
{∂2
t uL −∆uL = 0, x ∈ RN
~uL�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)
is dispersive and satisfies finite speed of propagation.Asymptotics: let /∂ be the tangential derivative. Then
limt→+∞
∫1|x |2|uL(t , x)|2 + |/∂uL(t , x)|2 + |uL(t , x)|6 dx = 0
and [Friedlander] there exists G+ ∈ L2(R× SN−1) such that
limt→+∞
∫ +∞
0
∫SN−1
∣∣∣r N−12 ∂r uL(t , rω)−G+(r − t , ω)
∣∣∣2+∣∣∣r N−1
2 ∂tuL(t , rω) + G+(r − t , ω)∣∣∣2 drdω.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 5 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 6 / 52
Local well-posedness
(Assume N = 3 to fix ideas).Theorem. Let ~u0 = (u0,u1) ∈ H. Then there exists a unique maximalsolution u of (NLW) s.t.
~u ∈ C0((T−,T+),H), u ∈ L5loc((T−,T+),L10).
(blow-up criterion) If T+ <∞, then u /∈ L5 ((0,T+),L10)).(small data) if ‖SL(t)(u0,u1)‖L5(R,L10) ≤ ε, then u is global,‖u‖L5(R,L10) ≤ 2ε.
By Strichartz estimates, for small data,∑t∈R
∥∥SL(t)(u0,u1)− ~u(t)∥∥H . ε
5.
Furthermore, small data in H are small in the above sense.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 7 / 52
Local well-posedness
(Assume N = 3 to fix ideas).Theorem. Let ~u0 = (u0,u1) ∈ H. Then there exists a unique maximalsolution u of (NLW) s.t.
~u ∈ C0((T−,T+),H), u ∈ L5loc((T−,T+),L10).
(blow-up criterion) If T+ <∞, then u /∈ L5 ((0,T+),L10)).(small data) if ‖SL(t)(u0,u1)‖L5(R,L10) ≤ ε, then u is global,‖u‖L5(R,L10) ≤ 2ε.
By Strichartz estimates, for small data,∑t∈R
∥∥SL(t)(u0,u1)− ~u(t)∥∥H . ε
5.
Furthermore, small data in H are small in the above sense.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 7 / 52
Scattering solutions
Let T+(u) be the maximal time of existence of u.
u scatters when T+(u) = +∞ and
limt→+∞
‖~u(t)− ~uL(t)‖H = 0
for a solution uL of ∂2t uL −∆uL = 0.
Equivalently u scatters when u ∈ L5((T−,T+),L10).All solutions scatter in the defocusing case: [Grillakis 90, 92], [ShatahStruwe, 93, 94], [Kapitanski 94], [Ginibre Velo 95], [Nakanishi 95],[Bahouri Shatah 98], [Bahouri Gérard 99], [Tao 06]In the focusing case:
Scattering for small solutions.Stability: the set of scattering solutions is open in the energytopology.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 8 / 52
Scattering solutions
Let T+(u) be the maximal time of existence of u.u scatters when T+(u) = +∞ and
limt→+∞
‖~u(t)− ~uL(t)‖H = 0
for a solution uL of ∂2t uL −∆uL = 0.
Equivalently u scatters when u ∈ L5((T−,T+),L10).All solutions scatter in the defocusing case: [Grillakis 90, 92], [ShatahStruwe, 93, 94], [Kapitanski 94], [Ginibre Velo 95], [Nakanishi 95],[Bahouri Shatah 98], [Bahouri Gérard 99], [Tao 06]In the focusing case:
Scattering for small solutions.Stability: the set of scattering solutions is open in the energytopology.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 8 / 52
Scattering solutions
Let T+(u) be the maximal time of existence of u.u scatters when T+(u) = +∞ and
limt→+∞
‖~u(t)− ~uL(t)‖H = 0
for a solution uL of ∂2t uL −∆uL = 0.
Equivalently u scatters when u ∈ L5((T−,T+),L10).
All solutions scatter in the defocusing case: [Grillakis 90, 92], [ShatahStruwe, 93, 94], [Kapitanski 94], [Ginibre Velo 95], [Nakanishi 95],[Bahouri Shatah 98], [Bahouri Gérard 99], [Tao 06]In the focusing case:
Scattering for small solutions.Stability: the set of scattering solutions is open in the energytopology.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 8 / 52
Scattering solutions
Let T+(u) be the maximal time of existence of u.u scatters when T+(u) = +∞ and
limt→+∞
‖~u(t)− ~uL(t)‖H = 0
for a solution uL of ∂2t uL −∆uL = 0.
Equivalently u scatters when u ∈ L5((T−,T+),L10).All solutions scatter in the defocusing case: [Grillakis 90, 92], [ShatahStruwe, 93, 94], [Kapitanski 94], [Ginibre Velo 95], [Nakanishi 95],[Bahouri Shatah 98], [Bahouri Gérard 99], [Tao 06]
In the focusing case:
Scattering for small solutions.Stability: the set of scattering solutions is open in the energytopology.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 8 / 52
Scattering solutions
Let T+(u) be the maximal time of existence of u.u scatters when T+(u) = +∞ and
limt→+∞
‖~u(t)− ~uL(t)‖H = 0
for a solution uL of ∂2t uL −∆uL = 0.
Equivalently u scatters when u ∈ L5((T−,T+),L10).All solutions scatter in the defocusing case: [Grillakis 90, 92], [ShatahStruwe, 93, 94], [Kapitanski 94], [Ginibre Velo 95], [Nakanishi 95],[Bahouri Shatah 98], [Bahouri Gérard 99], [Tao 06]In the focusing case:
Scattering for small solutions.
Stability: the set of scattering solutions is open in the energytopology.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 8 / 52
Scattering solutions
Let T+(u) be the maximal time of existence of u.u scatters when T+(u) = +∞ and
limt→+∞
‖~u(t)− ~uL(t)‖H = 0
for a solution uL of ∂2t uL −∆uL = 0.
Equivalently u scatters when u ∈ L5((T−,T+),L10).All solutions scatter in the defocusing case: [Grillakis 90, 92], [ShatahStruwe, 93, 94], [Kapitanski 94], [Ginibre Velo 95], [Nakanishi 95],[Bahouri Shatah 98], [Bahouri Gérard 99], [Tao 06]In the focusing case:
Scattering for small solutions.Stability: the set of scattering solutions is open in the energytopology.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 8 / 52
Type I blow-up
Type I blow-up: T+(u) <∞ and
limt→T+(u)
‖~u(t)‖H = +∞.
Example: blow-up solutions of the ODE y ′′ = yN+2N−2 such as
y0(t) =
(N(N − 2)
4
)N−24 1
(T − t)N−2
2
.
Numerical evidences that generic blow-up solutions behave likey0(t): [Bizon Chmaj Tabor 04]Stability of y0 in light cones, in the energy topology: [Donninger2015] (previous results in stronger topology [DonningerSchörkhuber]).No general classification (see [Merle Zaag] for pseudo-conformallysubcritical equation). No general stability result.
What about bounded, non-scattering solutions?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 9 / 52
Type I blow-up
Type I blow-up: T+(u) <∞ and
limt→T+(u)
‖~u(t)‖H = +∞.
Example: blow-up solutions of the ODE y ′′ = yN+2N−2 such as
y0(t) =
(N(N − 2)
4
)N−24 1
(T − t)N−2
2
.
Numerical evidences that generic blow-up solutions behave likey0(t): [Bizon Chmaj Tabor 04]Stability of y0 in light cones, in the energy topology: [Donninger2015] (previous results in stronger topology [DonningerSchörkhuber]).No general classification (see [Merle Zaag] for pseudo-conformallysubcritical equation). No general stability result.
What about bounded, non-scattering solutions?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 9 / 52
Type I blow-up
Type I blow-up: T+(u) <∞ and
limt→T+(u)
‖~u(t)‖H = +∞.
Example: blow-up solutions of the ODE y ′′ = yN+2N−2 such as
y0(t) =
(N(N − 2)
4
)N−24 1
(T − t)N−2
2
.
Numerical evidences that generic blow-up solutions behave likey0(t): [Bizon Chmaj Tabor 04]
Stability of y0 in light cones, in the energy topology: [Donninger2015] (previous results in stronger topology [DonningerSchörkhuber]).No general classification (see [Merle Zaag] for pseudo-conformallysubcritical equation). No general stability result.
What about bounded, non-scattering solutions?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 9 / 52
Type I blow-up
Type I blow-up: T+(u) <∞ and
limt→T+(u)
‖~u(t)‖H = +∞.
Example: blow-up solutions of the ODE y ′′ = yN+2N−2 such as
y0(t) =
(N(N − 2)
4
)N−24 1
(T − t)N−2
2
.
Numerical evidences that generic blow-up solutions behave likey0(t): [Bizon Chmaj Tabor 04]Stability of y0 in light cones, in the energy topology: [Donninger2015] (previous results in stronger topology [DonningerSchörkhuber]).
No general classification (see [Merle Zaag] for pseudo-conformallysubcritical equation). No general stability result.
What about bounded, non-scattering solutions?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 9 / 52
Type I blow-up
Type I blow-up: T+(u) <∞ and
limt→T+(u)
‖~u(t)‖H = +∞.
Example: blow-up solutions of the ODE y ′′ = yN+2N−2 such as
y0(t) =
(N(N − 2)
4
)N−24 1
(T − t)N−2
2
.
Numerical evidences that generic blow-up solutions behave likey0(t): [Bizon Chmaj Tabor 04]Stability of y0 in light cones, in the energy topology: [Donninger2015] (previous results in stronger topology [DonningerSchörkhuber]).No general classification (see [Merle Zaag] for pseudo-conformallysubcritical equation). No general stability result.
What about bounded, non-scattering solutions?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 9 / 52
Type I blow-up
Type I blow-up: T+(u) <∞ and
limt→T+(u)
‖~u(t)‖H = +∞.
Example: blow-up solutions of the ODE y ′′ = yN+2N−2 such as
y0(t) =
(N(N − 2)
4
)N−24 1
(T − t)N−2
2
.
Numerical evidences that generic blow-up solutions behave likey0(t): [Bizon Chmaj Tabor 04]Stability of y0 in light cones, in the energy topology: [Donninger2015] (previous results in stronger topology [DonningerSchörkhuber]).No general classification (see [Merle Zaag] for pseudo-conformallysubcritical equation). No general stability result.
What about bounded, non-scattering solutions?Thomas Duyckaerts (Paris 13) Critical waves February 2017 9 / 52
Solitons
Stationary solutions:
(E) −∆Q = |Q|4
N−2 Q, Q ∈ H1(RN).
“Unique” radial solution of (E) (ground state):
W =1(
1 + |x |2N(N−2)
)N2−1
.
The energy of W is a threshold for the dynamics [Kenig Merle 2008].See [Krieger Nakanishi Schlag 2015] for the dynamics around W .
Existence of solutions of (E) with arbitrary large energies: [W.Y. Ding1986], [Del Pino, Musso, Pacard, Pistoia 2013].Solitary waves or solitons: if p = |p| < 1:
Qp(t , x) = Q
((− t√
1− p2+
1p2
(1√
1− p2− 1
)p · x
)p + x
)Qp(t , x) = Qp(0, x − tp).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 10 / 52
Solitons
Stationary solutions:
(E) −∆Q = |Q|4
N−2 Q, Q ∈ H1(RN).
“Unique” radial solution of (E) (ground state):
W =1(
1 + |x |2N(N−2)
)N2−1
.
The energy of W is a threshold for the dynamics [Kenig Merle 2008].See [Krieger Nakanishi Schlag 2015] for the dynamics around W .Existence of solutions of (E) with arbitrary large energies: [W.Y. Ding1986], [Del Pino, Musso, Pacard, Pistoia 2013].
Solitary waves or solitons: if p = |p| < 1:
Qp(t , x) = Q
((− t√
1− p2+
1p2
(1√
1− p2− 1
)p · x
)p + x
)Qp(t , x) = Qp(0, x − tp).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 10 / 52
Solitons
Stationary solutions:
(E) −∆Q = |Q|4
N−2 Q, Q ∈ H1(RN).
“Unique” radial solution of (E) (ground state):
W =1(
1 + |x |2N(N−2)
)N2−1
.
The energy of W is a threshold for the dynamics [Kenig Merle 2008].See [Krieger Nakanishi Schlag 2015] for the dynamics around W .Existence of solutions of (E) with arbitrary large energies: [W.Y. Ding1986], [Del Pino, Musso, Pacard, Pistoia 2013].Solitary waves or solitons: if p = |p| < 1:
Qp(t , x) = Q
((− t√
1− p2+
1p2
(1√
1− p2− 1
)p · x
)p + x
)Qp(t , x) = Qp(0, x − tp).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 10 / 52
Other global, non-scattering solutions
One parameter family of solution with energy E(W ,0):
Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.
where a ∈ R, Y eigenfunction for the linearized operator [DuyckaertsMerle 2008].
Global solutions of the form:
~u(t) =
(1
λ(t)N−2
2
W(
xλ(t)
),0
)+ ~vL(t) + o(1), t → +∞,
vL small solution of the linear wave equation.λ(t) = 1 [Krieger Schlag 2007].λ(t) = tα, α ∈ R, α small [Donninger Krieger 2013]Open questions: solutions with other stationary profile than W?What are the vL(t) admissible?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 11 / 52
Other global, non-scattering solutions
One parameter family of solution with energy E(W ,0):
Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.
where a ∈ R, Y eigenfunction for the linearized operator [DuyckaertsMerle 2008].Global solutions of the form:
~u(t) =
(1
λ(t)N−2
2
W(
xλ(t)
),0
)+ ~vL(t) + o(1), t → +∞,
vL small solution of the linear wave equation.
λ(t) = 1 [Krieger Schlag 2007].λ(t) = tα, α ∈ R, α small [Donninger Krieger 2013]Open questions: solutions with other stationary profile than W?What are the vL(t) admissible?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 11 / 52
Other global, non-scattering solutions
One parameter family of solution with energy E(W ,0):
Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.
where a ∈ R, Y eigenfunction for the linearized operator [DuyckaertsMerle 2008].Global solutions of the form:
~u(t) =
(1
λ(t)N−2
2
W(
xλ(t)
),0
)+ ~vL(t) + o(1), t → +∞,
vL small solution of the linear wave equation.λ(t) = 1 [Krieger Schlag 2007].
λ(t) = tα, α ∈ R, α small [Donninger Krieger 2013]Open questions: solutions with other stationary profile than W?What are the vL(t) admissible?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 11 / 52
Other global, non-scattering solutions
One parameter family of solution with energy E(W ,0):
Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.
where a ∈ R, Y eigenfunction for the linearized operator [DuyckaertsMerle 2008].Global solutions of the form:
~u(t) =
(1
λ(t)N−2
2
W(
xλ(t)
),0
)+ ~vL(t) + o(1), t → +∞,
vL small solution of the linear wave equation.λ(t) = 1 [Krieger Schlag 2007].λ(t) = tα, α ∈ R, α small [Donninger Krieger 2013]
Open questions: solutions with other stationary profile than W?What are the vL(t) admissible?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 11 / 52
Other global, non-scattering solutions
One parameter family of solution with energy E(W ,0):
Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.
where a ∈ R, Y eigenfunction for the linearized operator [DuyckaertsMerle 2008].Global solutions of the form:
~u(t) =
(1
λ(t)N−2
2
W(
xλ(t)
),0
)+ ~vL(t) + o(1), t → +∞,
vL small solution of the linear wave equation.λ(t) = 1 [Krieger Schlag 2007].λ(t) = tα, α ∈ R, α small [Donninger Krieger 2013]Open questions: solutions with other stationary profile than W?What are the vL(t) admissible?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 11 / 52
Type II blow-up
These are solutions bounded in H such that T+(u) <∞. Knownexamples:
~u(t) =
(1
λ(t)N−2
2
W(·
λ(t)
),0
)+ (v0, v1), t → T+,
where (v0, v1) ∈ H, andN = 3, λ(t) = (T+ − t)α, α > 1 [Krieger Schlag Tataru 2009],[Krieger Schlag 2014] (instability: [Krieger Nahas 2013]).
N = 5, λ(t) = (T+ − t)α, α > 9, [Jendrej 2015].
N = 4, λ(t) ≈ (T+ − t)e−√| log(T+−t)|, (v0, v1) smooth [Hillairet
Raphaël 2012].N = 5, λ(t) ≈ (T+ − t)4, (v0, v1) is any smooth solution withv0(0) > 0 [Jendrej 2015].N = 3, λ(t) = (T+ − t)α exp(ε0 sin(log(t))), α > 4, [DonningerHuang Krieger Schlag 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 12 / 52
Type II blow-up
These are solutions bounded in H such that T+(u) <∞. Knownexamples:
~u(t) =
(1
λ(t)N−2
2
W(·
λ(t)
),0
)+ (v0, v1), t → T+,
where (v0, v1) ∈ H, andN = 3, λ(t) = (T+ − t)α, α > 1 [Krieger Schlag Tataru 2009],[Krieger Schlag 2014] (instability: [Krieger Nahas 2013]).N = 5, λ(t) = (T+ − t)α, α > 9, [Jendrej 2015].
N = 4, λ(t) ≈ (T+ − t)e−√| log(T+−t)|, (v0, v1) smooth [Hillairet
Raphaël 2012].N = 5, λ(t) ≈ (T+ − t)4, (v0, v1) is any smooth solution withv0(0) > 0 [Jendrej 2015].N = 3, λ(t) = (T+ − t)α exp(ε0 sin(log(t))), α > 4, [DonningerHuang Krieger Schlag 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 12 / 52
Type II blow-up
These are solutions bounded in H such that T+(u) <∞. Knownexamples:
~u(t) =
(1
λ(t)N−2
2
W(·
λ(t)
),0
)+ (v0, v1), t → T+,
where (v0, v1) ∈ H, andN = 3, λ(t) = (T+ − t)α, α > 1 [Krieger Schlag Tataru 2009],[Krieger Schlag 2014] (instability: [Krieger Nahas 2013]).N = 5, λ(t) = (T+ − t)α, α > 9, [Jendrej 2015].
N = 4, λ(t) ≈ (T+ − t)e−√| log(T+−t)|, (v0, v1) smooth [Hillairet
Raphaël 2012].
N = 5, λ(t) ≈ (T+ − t)4, (v0, v1) is any smooth solution withv0(0) > 0 [Jendrej 2015].N = 3, λ(t) = (T+ − t)α exp(ε0 sin(log(t))), α > 4, [DonningerHuang Krieger Schlag 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 12 / 52
Type II blow-up
These are solutions bounded in H such that T+(u) <∞. Knownexamples:
~u(t) =
(1
λ(t)N−2
2
W(·
λ(t)
),0
)+ (v0, v1), t → T+,
where (v0, v1) ∈ H, andN = 3, λ(t) = (T+ − t)α, α > 1 [Krieger Schlag Tataru 2009],[Krieger Schlag 2014] (instability: [Krieger Nahas 2013]).N = 5, λ(t) = (T+ − t)α, α > 9, [Jendrej 2015].
N = 4, λ(t) ≈ (T+ − t)e−√| log(T+−t)|, (v0, v1) smooth [Hillairet
Raphaël 2012].N = 5, λ(t) ≈ (T+ − t)4, (v0, v1) is any smooth solution withv0(0) > 0 [Jendrej 2015].
N = 3, λ(t) = (T+ − t)α exp(ε0 sin(log(t))), α > 4, [DonningerHuang Krieger Schlag 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 12 / 52
Type II blow-up
These are solutions bounded in H such that T+(u) <∞. Knownexamples:
~u(t) =
(1
λ(t)N−2
2
W(·
λ(t)
),0
)+ (v0, v1), t → T+,
where (v0, v1) ∈ H, andN = 3, λ(t) = (T+ − t)α, α > 1 [Krieger Schlag Tataru 2009],[Krieger Schlag 2014] (instability: [Krieger Nahas 2013]).N = 5, λ(t) = (T+ − t)α, α > 9, [Jendrej 2015].
N = 4, λ(t) ≈ (T+ − t)e−√| log(T+−t)|, (v0, v1) smooth [Hillairet
Raphaël 2012].N = 5, λ(t) ≈ (T+ − t)4, (v0, v1) is any smooth solution withv0(0) > 0 [Jendrej 2015].N = 3, λ(t) = (T+ − t)α exp(ε0 sin(log(t))), α > 4, [DonningerHuang Krieger Schlag 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 12 / 52
Multi-solitons
Existence of a solution such that T+ = +∞ and
~u(t , x) =
(W (x) +
1λ(t)2 W
(xλ(t)
),0)
+ o(1), t → +∞,
where N = 6, λ(t) =√
4/5e−√
5/4t . [Jendrej 2016].
Existence of solutions such that T+ = +∞ and
~u(t , x) =J∑
j=1
ιj
λ32j
~Wpj
(tλj,x − xj
λj
)+ o(1), t → +∞,
where N = 5, ιj ∈ {±1}, λj > 0, xj ∈ R5, |pj | < 1 (collinears if J ≥ 3)and
j 6= k =⇒ pj 6= pk .
[Martel Merle 2015].Open: finite time-blow-up case (see [Côte Zaag 2013] for subcriticalequations in one space dimension).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 13 / 52
Multi-solitons
Existence of a solution such that T+ = +∞ and
~u(t , x) =
(W (x) +
1λ(t)2 W
(xλ(t)
),0)
+ o(1), t → +∞,
where N = 6, λ(t) =√
4/5e−√
5/4t . [Jendrej 2016].Existence of solutions such that T+ = +∞ and
~u(t , x) =J∑
j=1
ιj
λ32j
~Wpj
(tλj,x − xj
λj
)+ o(1), t → +∞,
where N = 5, ιj ∈ {±1}, λj > 0, xj ∈ R5, |pj | < 1 (collinears if J ≥ 3)and
j 6= k =⇒ pj 6= pk .
[Martel Merle 2015].
Open: finite time-blow-up case (see [Côte Zaag 2013] for subcriticalequations in one space dimension).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 13 / 52
Multi-solitons
Existence of a solution such that T+ = +∞ and
~u(t , x) =
(W (x) +
1λ(t)2 W
(xλ(t)
),0)
+ o(1), t → +∞,
where N = 6, λ(t) =√
4/5e−√
5/4t . [Jendrej 2016].Existence of solutions such that T+ = +∞ and
~u(t , x) =J∑
j=1
ιj
λ32j
~Wpj
(tλj,x − xj
λj
)+ o(1), t → +∞,
where N = 5, ιj ∈ {±1}, λj > 0, xj ∈ R5, |pj | < 1 (collinears if J ≥ 3)and
j 6= k =⇒ pj 6= pk .
[Martel Merle 2015].Open: finite time-blow-up case (see [Côte Zaag 2013] for subcriticalequations in one space dimension).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 13 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 14 / 52
Soliton resolution conjecture
Conjecture. Let u be a non scattering solution such thatT+(u) = +∞.
Then there exists J ≥ 1, a linear wave vL, solitary waves Qjpj
,j = 1 . . . J, and parameters xj(t) ∈ RN , λj(t) > 0, such that
u(t) = vL(t) +J∑
j=1
1
λN−2
2j (t)
Qjpj
(0,
x − xj(t)λj(t)
)+ r(t)
wherelim
t→+∞
∥∥~r(t)∥∥H = 0
∀j , limt→+∞
xj(t)t
= pj , limt→+∞
λj(t)t
= 0
∀j , k , j 6= k =⇒ limt→+∞
|xj(t)− xk (t)|λj(t)
+λj(t)λk (t)
+λk (t)λj(t)
= +∞.
(Analogous conjecture for type II blow-up solutions).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 15 / 52
Soliton resolution conjecture
Conjecture. Let u be a non scattering solution such thatT+(u) = +∞.Then there exists J ≥ 1, a linear wave vL, solitary waves Qj
pj,
j = 1 . . . J, and parameters xj(t) ∈ RN , λj(t) > 0, such that
u(t) = vL(t) +J∑
j=1
1
λN−2
2j (t)
Qjpj
(0,
x − xj(t)λj(t)
)+ r(t)
wherelim
t→+∞
∥∥~r(t)∥∥H = 0
∀j , limt→+∞
xj(t)t
= pj , limt→+∞
λj(t)t
= 0
∀j , k , j 6= k =⇒ limt→+∞
|xj(t)− xk (t)|λj(t)
+λj(t)λk (t)
+λk (t)λj(t)
= +∞.
(Analogous conjecture for type II blow-up solutions).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 15 / 52
Soliton resolution conjecture
Conjecture. Let u be a non scattering solution such thatT+(u) = +∞.Then there exists J ≥ 1, a linear wave vL, solitary waves Qj
pj,
j = 1 . . . J, and parameters xj(t) ∈ RN , λj(t) > 0, such that
u(t) = vL(t) +J∑
j=1
1
λN−2
2j (t)
Qjpj
(0,
x − xj(t)λj(t)
)+ r(t)
wherelim
t→+∞
∥∥~r(t)∥∥H = 0
∀j , limt→+∞
xj(t)t
= pj , limt→+∞
λj(t)t
= 0
∀j , k , j 6= k =⇒ limt→+∞
|xj(t)− xk (t)|λj(t)
+λj(t)λk (t)
+λk (t)λj(t)
= +∞.
(Analogous conjecture for type II blow-up solutions).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 15 / 52
Soliton resolution conjecture
Conjecture. Let u be a non scattering solution such thatT+(u) = +∞.Then there exists J ≥ 1, a linear wave vL, solitary waves Qj
pj,
j = 1 . . . J, and parameters xj(t) ∈ RN , λj(t) > 0, such that
u(t) = vL(t) +J∑
j=1
1
λN−2
2j (t)
Qjpj
(0,
x − xj(t)λj(t)
)+ r(t)
wherelim
t→+∞
∥∥~r(t)∥∥H = 0
∀j , limt→+∞
xj(t)t
= pj , limt→+∞
λj(t)t
= 0
∀j , k , j 6= k =⇒ limt→+∞
|xj(t)− xk (t)|λj(t)
+λj(t)λk (t)
+λk (t)λj(t)
= +∞.
(Analogous conjecture for type II blow-up solutions).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 15 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 16 / 52
Korteweg-de-Vries
(KdV)
{∂tu − 6u∂xu + ∂3
x u = 0, x ∈ R, t ∈ Ru�t=0 = u0.
The equation has (stable) solitons, moving to the right:
Qκ(t , x) = − 2κ2 cosh2 (κ(x − 4κ2t)
) .There also exist multi-solitons of all order, given by completeintegrability.
Then [Eckaus, Schuur], if u0 is smooth and decays at infinity:
limt→+∞
∫x≥−νt1/3
∣∣∣u(t , x)−J∑
j=1
Qκj (t , x − xj)∣∣∣dx = 0.
J ≥ 0, κJ < κJ−1 < . . . < κ1, xj ∈ R.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 17 / 52
Korteweg-de-Vries
(KdV)
{∂tu − 6u∂xu + ∂3
x u = 0, x ∈ R, t ∈ Ru�t=0 = u0.
The equation has (stable) solitons, moving to the right:
Qκ(t , x) = − 2κ2 cosh2 (κ(x − 4κ2t)
) .There also exist multi-solitons of all order, given by completeintegrability.Then [Eckaus, Schuur], if u0 is smooth and decays at infinity:
limt→+∞
∫x≥−νt1/3
∣∣∣u(t , x)−J∑
j=1
Qκj (t , x − xj)∣∣∣dx = 0.
J ≥ 0, κJ < κJ−1 < . . . < κ1, xj ∈ R.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 17 / 52
Nonlinear focusing Schrödinger equations
(NLS)
{i∂tu + ∆u = −|u|p−1u
u�t=0 = u0 ∈ H1(RN)
where x ∈ RN , N ≥ 1, 1 + 4N < p < N+2
N−2 .
If p ≥ 1 + 4N , there exist finite time blow-up solutions, and small
solutions (in H1) are global and scatter to linear solutions. There alsoexist solitary waves of the form:
eiξ·x/2e−i|ξ|2t/4eiωtQ(x − ξt) =: Qξ(t , x),
where ξ ∈ RN , ω > 0 and
∆Q + |Q|p−1Q − ωQ = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 18 / 52
Nonlinear focusing Schrödinger equations
(NLS)
{i∂tu + ∆u = −|u|p−1u
u�t=0 = u0 ∈ H1(RN)
where x ∈ RN , N ≥ 1, 1 + 4N < p < N+2
N−2 .If p ≥ 1 + 4
N , there exist finite time blow-up solutions, and smallsolutions (in H1) are global and scatter to linear solutions.
There alsoexist solitary waves of the form:
eiξ·x/2e−i|ξ|2t/4eiωtQ(x − ξt) =: Qξ(t , x),
where ξ ∈ RN , ω > 0 and
∆Q + |Q|p−1Q − ωQ = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 18 / 52
Nonlinear focusing Schrödinger equations
(NLS)
{i∂tu + ∆u = −|u|p−1u
u�t=0 = u0 ∈ H1(RN)
where x ∈ RN , N ≥ 1, 1 + 4N < p < N+2
N−2 .If p ≥ 1 + 4
N , there exist finite time blow-up solutions, and smallsolutions (in H1) are global and scatter to linear solutions. There alsoexist solitary waves of the form:
eiξ·x/2e−i|ξ|2t/4eiωtQ(x − ξt) =: Qξ(t , x),
where ξ ∈ RN , ω > 0 and
∆Q + |Q|p−1Q − ωQ = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 18 / 52
Compact attractor for NLS
Theorem [Tao 2007]. Assume N ≥ 5. Let A > 0. There exists acompact subset KA of H1, invariant by the flow of (NLS) such that if uis a radial solution of (NLS) on [0,+∞) with lim supt→∞ ‖u(t)‖H1 < A,then there exists u+ ∈ H1 such that
limt→+∞
dH1
(u(t)− eit∆u+,KA
)= 0.
(Analogous theorem without symmetry assumptions).Next step:Prove the Rigidity conjecture. Let u be a solution such that there existsx(t) ∈ RN with
{u(t , · − x(t)), t ∈ R
}has compact closure in H1(RN).
Then u is a soliton.Proved for KdV [C. Laurent, Y. Martel 2003], and conditionally for(NLW) [TD, Kenig, Merle 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 19 / 52
Compact attractor for NLS
Theorem [Tao 2007]. Assume N ≥ 5. Let A > 0. There exists acompact subset KA of H1, invariant by the flow of (NLS) such that if uis a radial solution of (NLS) on [0,+∞) with lim supt→∞ ‖u(t)‖H1 < A,then there exists u+ ∈ H1 such that
limt→+∞
dH1
(u(t)− eit∆u+,KA
)= 0.
(Analogous theorem without symmetry assumptions).
Next step:Prove the Rigidity conjecture. Let u be a solution such that there existsx(t) ∈ RN with
{u(t , · − x(t)), t ∈ R
}has compact closure in H1(RN).
Then u is a soliton.Proved for KdV [C. Laurent, Y. Martel 2003], and conditionally for(NLW) [TD, Kenig, Merle 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 19 / 52
Compact attractor for NLS
Theorem [Tao 2007]. Assume N ≥ 5. Let A > 0. There exists acompact subset KA of H1, invariant by the flow of (NLS) such that if uis a radial solution of (NLS) on [0,+∞) with lim supt→∞ ‖u(t)‖H1 < A,then there exists u+ ∈ H1 such that
limt→+∞
dH1
(u(t)− eit∆u+,KA
)= 0.
(Analogous theorem without symmetry assumptions).Next step:Prove the Rigidity conjecture. Let u be a solution such that there existsx(t) ∈ RN with
{u(t , · − x(t)), t ∈ R
}has compact closure in H1(RN).
Then u is a soliton.
Proved for KdV [C. Laurent, Y. Martel 2003], and conditionally for(NLW) [TD, Kenig, Merle 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 19 / 52
Compact attractor for NLS
Theorem [Tao 2007]. Assume N ≥ 5. Let A > 0. There exists acompact subset KA of H1, invariant by the flow of (NLS) such that if uis a radial solution of (NLS) on [0,+∞) with lim supt→∞ ‖u(t)‖H1 < A,then there exists u+ ∈ H1 such that
limt→+∞
dH1
(u(t)− eit∆u+,KA
)= 0.
(Analogous theorem without symmetry assumptions).Next step:Prove the Rigidity conjecture. Let u be a solution such that there existsx(t) ∈ RN with
{u(t , · − x(t)), t ∈ R
}has compact closure in H1(RN).
Then u is a soliton.Proved for KdV [C. Laurent, Y. Martel 2003], and conditionally for(NLW) [TD, Kenig, Merle 2014].
Thomas Duyckaerts (Paris 13) Critical waves February 2017 19 / 52
Wave maps and bubble theorems
Conjecture for the (energy-critical) wave maps equation on R2, withtarget space S2: any solution is asymptotically the finite sums ofLorentz transforms of harmonic maps. Note that all these harmonicmaps are known.For these equations, the soliton resolution conjecture is not known butresults exist for sequence of times.
[Sterbenz and Tataru 2010] proved the following bubble theorem forwave maps: for any wave map blowing-up in finite time Φ on R2, thereexist sequences tn → T+, {xn}n, λn such that Φ(tn + λnt , xn + λnx)converges in H1
loc to the Lorentz transform of a Harmonic map. See[Christodoulou, Tahvildar-Zadeh 1993], [Struwe 2003] for theequivariant case.
In the equivariant case and close to the ground state soliton, thesoliton resolution is known along a sequence of times. See [Côte,Kenig, Lawrie, Schlag 2015], [Côte 2015], [TD, Jia, Kenig, Merle],using techniques developed for (NLW).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 20 / 52
Wave maps and bubble theorems
Conjecture for the (energy-critical) wave maps equation on R2, withtarget space S2: any solution is asymptotically the finite sums ofLorentz transforms of harmonic maps. Note that all these harmonicmaps are known.For these equations, the soliton resolution conjecture is not known butresults exist for sequence of times.
[Sterbenz and Tataru 2010] proved the following bubble theorem forwave maps: for any wave map blowing-up in finite time Φ on R2, thereexist sequences tn → T+, {xn}n, λn such that Φ(tn + λnt , xn + λnx)converges in H1
loc to the Lorentz transform of a Harmonic map. See[Christodoulou, Tahvildar-Zadeh 1993], [Struwe 2003] for theequivariant case.
In the equivariant case and close to the ground state soliton, thesoliton resolution is known along a sequence of times. See [Côte,Kenig, Lawrie, Schlag 2015], [Côte 2015], [TD, Jia, Kenig, Merle],using techniques developed for (NLW).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 20 / 52
Wave maps and bubble theorems
Conjecture for the (energy-critical) wave maps equation on R2, withtarget space S2: any solution is asymptotically the finite sums ofLorentz transforms of harmonic maps. Note that all these harmonicmaps are known.For these equations, the soliton resolution conjecture is not known butresults exist for sequence of times.
[Sterbenz and Tataru 2010] proved the following bubble theorem forwave maps: for any wave map blowing-up in finite time Φ on R2, thereexist sequences tn → T+, {xn}n, λn such that Φ(tn + λnt , xn + λnx)converges in H1
loc to the Lorentz transform of a Harmonic map. See[Christodoulou, Tahvildar-Zadeh 1993], [Struwe 2003] for theequivariant case.
In the equivariant case and close to the ground state soliton, thesoliton resolution is known along a sequence of times. See [Côte,Kenig, Lawrie, Schlag 2015], [Côte 2015], [TD, Jia, Kenig, Merle],using techniques developed for (NLW).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 20 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 21 / 52
Global solutions
Theorem [Duyckaerts Kenig Merle 2012]. Assume N = 3. Let u be aradial solution of (NLW) such that T+(u) = +∞.
Then there existsJ ≥ 0 and:
vL such that ∂2t vL −∆vL = 0,
signs ιj ∈ {±1}, j = 1 . . . J,parameters λj(t), 0 < λ1(t)� λ2(t)� . . .� λJ(t)� t ,
such that:
u(t) = vL(t) +J∑
j=1
ιj
λ12j (t)
W(
xλj(t)
)+ r(t),
where: limt→+∞
∥∥~r(t)∥∥H = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 22 / 52
Global solutions
Theorem [Duyckaerts Kenig Merle 2012]. Assume N = 3. Let u be aradial solution of (NLW) such that T+(u) = +∞. Then there existsJ ≥ 0 and:
vL such that ∂2t vL −∆vL = 0,
signs ιj ∈ {±1}, j = 1 . . . J,parameters λj(t), 0 < λ1(t)� λ2(t)� . . .� λJ(t)� t ,
such that:
u(t) = vL(t) +J∑
j=1
ιj
λ12j (t)
W(
xλj(t)
)+ r(t),
where: limt→+∞
∥∥~r(t)∥∥H = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 22 / 52
Global solutions
Theorem [Duyckaerts Kenig Merle 2012]. Assume N = 3. Let u be aradial solution of (NLW) such that T+(u) = +∞. Then there existsJ ≥ 0 and:
vL such that ∂2t vL −∆vL = 0,
signs ιj ∈ {±1}, j = 1 . . . J,parameters λj(t), 0 < λ1(t)� λ2(t)� . . .� λJ(t)� t ,
such that:
u(t) = vL(t) +J∑
j=1
ιj
λ12j (t)
W(
xλj(t)
)+ r(t),
where: limt→+∞
∥∥~r(t)∥∥H = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 22 / 52
Type II blow-up solutions
Theorem. Assume N = 3. Let u be a radial solution of (NLW) suchthat T+(u) < +∞. Then
limt→T+(u)
‖~u‖H = +∞
or there exists J ≥ 1 and:vL such that ∂2
t vL −∆vL = 0,signs ιj ∈ {±1}, j = 1 . . . J,parameters λj(t), 0 < λ1(t)� λ2(t)� . . .� λJ(t)� T+ − t ,
such that:
u(t) = vL(t) +J∑
j=1
ιj
λ12j (t)
W(
xλj(t)
)+ r(t),
where: limt→T+
∥∥~r(t)∥∥H = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 23 / 52
Type II blow-up solutions
Theorem. Assume N = 3. Let u be a radial solution of (NLW) suchthat T+(u) < +∞. Then
limt→T+(u)
‖~u‖H = +∞
or there exists J ≥ 1 and:vL such that ∂2
t vL −∆vL = 0,signs ιj ∈ {±1}, j = 1 . . . J,parameters λj(t), 0 < λ1(t)� λ2(t)� . . .� λJ(t)� T+ − t ,
such that:
u(t) = vL(t) +J∑
j=1
ιj
λ12j (t)
W(
xλj(t)
)+ r(t),
where: limt→T+
∥∥~r(t)∥∥H = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 23 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 24 / 52
Exterior energy for the linear wave equation
Theorem [TD, Kenig, Merle 2012]. Assume that N is odd. Let uL be asolution of the linear wave equation. Then the following holds for allt ≥ 0 for all t ≤ 0:∫
|x |≥|t ||∇t ,xuL(t , x)|2 dx ≥ 1
2
∫RN|∇t ,xuL(0, x)|2 dx .
Proof by symmetry argument, using the explicit formula of the solution.Does not hold in even dimension [Côte, Kenig, Schlag 2014].
Question: what are the solutions of the nonlinear wave equation suchthat there exists η > 0.
∀t ≥ 0 or ∀t ≤ 0,∫|x |≥|t |
|∇t ,xu(t , x)|2 dx ≥ η?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 25 / 52
Exterior energy for the linear wave equation
Theorem [TD, Kenig, Merle 2012]. Assume that N is odd. Let uL be asolution of the linear wave equation. Then the following holds for allt ≥ 0 for all t ≤ 0:∫
|x |≥|t ||∇t ,xuL(t , x)|2 dx ≥ 1
2
∫RN|∇t ,xuL(0, x)|2 dx .
Proof by symmetry argument, using the explicit formula of the solution.
Does not hold in even dimension [Côte, Kenig, Schlag 2014].
Question: what are the solutions of the nonlinear wave equation suchthat there exists η > 0.
∀t ≥ 0 or ∀t ≤ 0,∫|x |≥|t |
|∇t ,xu(t , x)|2 dx ≥ η?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 25 / 52
Exterior energy for the linear wave equation
Theorem [TD, Kenig, Merle 2012]. Assume that N is odd. Let uL be asolution of the linear wave equation. Then the following holds for allt ≥ 0 for all t ≤ 0:∫
|x |≥|t ||∇t ,xuL(t , x)|2 dx ≥ 1
2
∫RN|∇t ,xuL(0, x)|2 dx .
Proof by symmetry argument, using the explicit formula of the solution.Does not hold in even dimension [Côte, Kenig, Schlag 2014].
Question: what are the solutions of the nonlinear wave equation suchthat there exists η > 0.
∀t ≥ 0 or ∀t ≤ 0,∫|x |≥|t |
|∇t ,xu(t , x)|2 dx ≥ η?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 25 / 52
Exterior energy for the linear wave equation
Theorem [TD, Kenig, Merle 2012]. Assume that N is odd. Let uL be asolution of the linear wave equation. Then the following holds for allt ≥ 0 for all t ≤ 0:∫
|x |≥|t ||∇t ,xuL(t , x)|2 dx ≥ 1
2
∫RN|∇t ,xuL(0, x)|2 dx .
Proof by symmetry argument, using the explicit formula of the solution.Does not hold in even dimension [Côte, Kenig, Schlag 2014].
Question: what are the solutions of the nonlinear wave equation suchthat there exists η > 0.
∀t ≥ 0 or ∀t ≤ 0,∫|x |≥|t |
|∇t ,xu(t , x)|2 dx ≥ η?
Thomas Duyckaerts (Paris 13) Critical waves February 2017 25 / 52
Linear exterior energy for radial data in 3D
Proposition. Let vL be a radial solution of the linear wave equation, in3 space dimensions. Let r0 > 0. Then
∀t ≥ 0 or ∀t ≤ 0,∫ +∞
r0+|t |(∂t ,r (ruL(t , r)))2 dr ≥ 1
2
∫ +∞
r0
(∂t ,r (ruL(0, r)))2 dr .
Generalization to other odd dimensions: [Kenig, Lawrie, Baoping Liu,Schlag 2015]
Thomas Duyckaerts (Paris 13) Critical waves February 2017 26 / 52
Linear exterior energy for radial data in 3D
Proposition. Let vL be a radial solution of the linear wave equation, in3 space dimensions. Let r0 > 0. Then
∀t ≥ 0 or ∀t ≤ 0,∫ +∞
r0+|t |(∂t ,r (ruL(t , r)))2 dr ≥ 1
2
∫ +∞
r0
(∂t ,r (ruL(0, r)))2 dr .
Generalization to other odd dimensions: [Kenig, Lawrie, Baoping Liu,Schlag 2015]
Thomas Duyckaerts (Paris 13) Critical waves February 2017 26 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 27 / 52
Statement of the theorem
Theorem. Assume N = 3. Let u be a global, radial solution of (NLW).Assume
∀r0 > 0, lim inft→±∞
∫|x |≥r0+|t |
|∇t ,xu|2 dx = 0.
Then u = 0 or there exists λ > 0, ι ∈ {±1} such thatu(t , x) = ι
λ1/2 W( xλ
).
Recall that W (x) = 1(1+ |x|
23
) 12
, so that
ι
λ1/2 W(xλ
)≈√
3λ1/2
|x |, |x | → ∞.
First step of the proof: there exists ` ∈ R such that
limr→∞
ru0(r) = `.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 28 / 52
Statement of the theorem
Theorem. Assume N = 3. Let u be a global, radial solution of (NLW).Assume
∀r0 > 0, lim inft→±∞
∫|x |≥r0+|t |
|∇t ,xu|2 dx = 0.
Then u = 0 or there exists λ > 0, ι ∈ {±1} such thatu(t , x) = ι
λ1/2 W( xλ
).
Recall that W (x) = 1(1+ |x|
23
) 12
, so that
ι
λ1/2 W(xλ
)≈√
3λ1/2
|x |, |x | → ∞.
First step of the proof: there exists ` ∈ R such that
limr→∞
ru0(r) = `.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 28 / 52
A technical lemma
Lemma. There exists δ0,C0 > 0 with the following property. AssumeN = 3. Let u be a global, radial solution of (NLW). Assume
∀r0 > 0, lim inft→±∞
∫|x |≥r0+|t |
|∇t ,xu|2 dx = 0.
Assume furthermore
(1)∫ +∞
r0
((∂r u0)2 + u2
1
)r2 dr = δ ≤ δ0.
Then, letting (v0, v1) = (ru0, ru1),
(2)∫ +∞
r0
(∂r (v0))2 + v21 dr ≤ C0
v100
r50.
Furthermore, if r0 ≤ r ≤ 2r0,
(3) |v0(r)− v0(r0)| ≤√
C0|v0(r0)|5
r20
≤√
C0δ2|v0(r0)|.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 29 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 30 / 52
Boundedness along a sequence of times
Proposition. Let u be a solution of (NLW) such that T+(u) = +∞.Then there exists tn → +∞ such that
lim supn→+∞
‖~u(tn)‖H <∞.
Remark. This works in any dimension N ≥ 3 and without symmetryassumption.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 31 / 52
Boundedness along a sequence of times
Proposition. Let u be a solution of (NLW) such that T+(u) = +∞.Then there exists tn → +∞ such that
lim supn→+∞
‖~u(tn)‖H <∞.
Remark. This works in any dimension N ≥ 3 and without symmetryassumption.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 31 / 52
Extraction of the radiation term
Proposition. Let u be a solution such that T+(u) = +∞. Then thereexists a solution vL of the linear wave equation such that
∀A ∈ R, limt→+∞
∫|x |≥t+A
|∇t ,x (u − vL)|2 dx = 0.
Simple in the radial case: solutions of the nonlinear wave equationslocalized close to |x | = t are dispersive.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 32 / 52
Extraction of the radiation term
Proposition. Let u be a solution such that T+(u) = +∞. Then thereexists a solution vL of the linear wave equation such that
∀A ∈ R, limt→+∞
∫|x |≥t+A
|∇t ,x (u − vL)|2 dx = 0.
Simple in the radial case: solutions of the nonlinear wave equationslocalized close to |x | = t are dispersive.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 32 / 52
Soliton resolution along a sequence of times
Theorem. Let tn → +∞ such that (~u(tn))n is bounded in H. Then(after extraction of a subsequence in n) there exists J ≥ 0, and, forj = 1, . . . , J, ιj ∈ {±1}, a sequence (λj,n)n such that
limn→∞
∥∥∥∥∥∥~u(tn)− ~vL(tn)−J∑
j=1
ιj ~Wλj,n
∥∥∥∥∥∥H
= 0,
where
~Wλj,n (x) =
1
λ1/2j,n
W(
xλj,n
),0
.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 33 / 52
Proof along a sequence of times
Let tn → +∞ such that (~u(tn))n is bounded in H. Extractingsubsequences, we obtain solutions U j of (NLW), solutions wJ
L,n of (LW),sequences (λj,n)n, (tj,n)n such that, on appropriate intervals In, letting
U jn(t , x) =
1
λ1/2j,n
U j(
t − tj,nλj,n
,xλj,n
).
U jn is defined on In for all j and
εJn(t , x) = un(t , x)−
J∑j=1
U jn(t , x)− wJ
L,n(t , x),
we have
(4) limJ→+∞
lim supn→+∞
(supt∈In‖~εJ
n(t , x)‖H +∥∥∥εJ
n
∥∥∥L5(In,L10)
)= 0.
Use a channels of energy argument.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 34 / 52
Proof along a sequence of times
Let tn → +∞ such that (~u(tn))n is bounded in H. Extractingsubsequences, we obtain solutions U j of (NLW), solutions wJ
L,n of (LW),sequences (λj,n)n, (tj,n)n such that, on appropriate intervals In, letting
U jn(t , x) =
1
λ1/2j,n
U j(
t − tj,nλj,n
,xλj,n
).
U jn is defined on In for all j and
εJn(t , x) = un(t , x)−
J∑j=1
U jn(t , x)− wJ
L,n(t , x),
we have
(4) limJ→+∞
lim supn→+∞
(supt∈In‖~εJ
n(t , x)‖H +∥∥∥εJ
n
∥∥∥L5(In,L10)
)= 0.
Use a channels of energy argument.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 34 / 52
Proof along a sequence of times
Let tn → +∞ such that (~u(tn))n is bounded in H. Extractingsubsequences, we obtain solutions U j of (NLW), solutions wJ
L,n of (LW),sequences (λj,n)n, (tj,n)n such that, on appropriate intervals In, letting
U jn(t , x) =
1
λ1/2j,n
U j(
t − tj,nλj,n
,xλj,n
).
U jn is defined on In for all j and
εJn(t , x) = un(t , x)−
J∑j=1
U jn(t , x)− wJ
L,n(t , x),
we have
(4) limJ→+∞
lim supn→+∞
(supt∈In‖~εJ
n(t , x)‖H +∥∥∥εJ
n
∥∥∥L5(In,L10)
)= 0.
Use a channels of energy argument.Thomas Duyckaerts (Paris 13) Critical waves February 2017 34 / 52
End of the proof
The first step is to prove:
limt→+∞
‖∇(u − vL)(t)‖2L2 = J‖∇W‖2L2
limt→+∞
‖∂t (u − vL)(t)‖2L2 = 0.
It remains to choose the translation parameters.Define
Bj := (j − 1)‖∇W‖2L2 +
∫|x |≤1
|∇W (x)|2 dx
and
λj(t) := inf
{λ > 0 s.t.
∫|x |≤λ
|∇(u − vL)(t , x)|2 dx ≥ Bj
}.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 35 / 52
End of the proof
The first step is to prove:
limt→+∞
‖∇(u − vL)(t)‖2L2 = J‖∇W‖2L2
limt→+∞
‖∂t (u − vL)(t)‖2L2 = 0.
It remains to choose the translation parameters.Define
Bj := (j − 1)‖∇W‖2L2 +
∫|x |≤1
|∇W (x)|2 dx
and
λj(t) := inf
{λ > 0 s.t.
∫|x |≤λ
|∇(u − vL)(t , x)|2 dx ≥ Bj
}.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 35 / 52
General case
Theorem [TD, Jia, Kenig, Merle, 2016]. Assume N = 3,4,5. Let u bea solution such that T+(u) = +∞ and
lim supt→+∞
‖~u(t)‖H <∞.
Then there exists tn → +∞, J ≥ 1, a linear wave vL, solitary wavesQj
pj, j = 1 . . . J, and parameters xj,n ∈ RN , λj,n > 0, such that
u(tn) = vL(tn) +J∑
j=1
1
λN−2
2j,n
Qjpj
(0,
x − xj,n
λj,n
)+ εn
(+ analogous expansion for the time derivative) wherelim
n→+∞‖~εn‖H = 0
∀j , limn→+∞
xj,n
tn= pj , lim
n→+∞
λj,n
tn= 0
∀j , k , j 6= k =⇒ limn→+∞
|xj,n − xk ,n|λj,n
+λj,n
λk ,n+λk ,n
λj,n= +∞.
Analogous theorem for Type II Blow-up solutions (see also [Jia 2015]).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 36 / 52
General case
Theorem [TD, Jia, Kenig, Merle, 2016]. Assume N = 3,4,5. Let u bea solution such that T+(u) = +∞ and
lim supt→+∞
‖~u(t)‖H <∞.
Then there exists tn → +∞, J ≥ 1, a linear wave vL, solitary wavesQj
pj, j = 1 . . . J, and parameters xj,n ∈ RN , λj,n > 0, such that
u(tn) = vL(tn) +J∑
j=1
1
λN−2
2j,n
Qjpj
(0,
x − xj,n
λj,n
)+ εn
(+ analogous expansion for the time derivative) wherelim
n→+∞‖~εn‖H = 0
∀j , limn→+∞
xj,n
tn= pj , lim
n→+∞
λj,n
tn= 0
∀j , k , j 6= k =⇒ limn→+∞
|xj,n − xk ,n|λj,n
+λj,n
λk ,n+λk ,n
λj,n= +∞.
Analogous theorem for Type II Blow-up solutions (see also [Jia 2015]).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 36 / 52
General case
Theorem [TD, Jia, Kenig, Merle, 2016]. Assume N = 3,4,5. Let u bea solution such that T+(u) = +∞ and
lim supt→+∞
‖~u(t)‖H <∞.
Then there exists tn → +∞, J ≥ 1, a linear wave vL, solitary wavesQj
pj, j = 1 . . . J, and parameters xj,n ∈ RN , λj,n > 0, such that
u(tn) = vL(tn) +J∑
j=1
1
λN−2
2j,n
Qjpj
(0,
x − xj,n
λj,n
)+ εn
(+ analogous expansion for the time derivative) wherelim
n→+∞‖~εn‖H = 0
∀j , limn→+∞
xj,n
tn= pj , lim
n→+∞
λj,n
tn= 0
∀j , k , j 6= k =⇒ limn→+∞
|xj,n − xk ,n|λj,n
+λj,n
λk ,n+λk ,n
λj,n= +∞.
Analogous theorem for Type II Blow-up solutions (see also [Jia 2015]).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 36 / 52
General case
Theorem [TD, Jia, Kenig, Merle, 2016]. Assume N = 3,4,5. Let u bea solution such that T+(u) = +∞ and
lim supt→+∞
‖~u(t)‖H <∞.
Then there exists tn → +∞, J ≥ 1, a linear wave vL, solitary wavesQj
pj, j = 1 . . . J, and parameters xj,n ∈ RN , λj,n > 0, such that
u(tn) = vL(tn) +J∑
j=1
1
λN−2
2j,n
Qjpj
(0,
x − xj,n
λj,n
)+ εn
(+ analogous expansion for the time derivative) wherelim
n→+∞‖~εn‖H = 0
∀j , limn→+∞
xj,n
tn= pj , lim
n→+∞
λj,n
tn= 0
∀j , k , j 6= k =⇒ limn→+∞
|xj,n − xk ,n|λj,n
+λj,n
λk ,n+λk ,n
λj,n= +∞.
Analogous theorem for Type II Blow-up solutions (see also [Jia 2015]).Thomas Duyckaerts (Paris 13) Critical waves February 2017 36 / 52
Ideas of proof
Ingredients of the proof:Profile decomposition [Bahouri Gérard 99]).
Monotonicity formulas, including a Morawetz-type inequality,similar to the one used for energy-critical wave maps. This ismade possible by new bounds of the solution on the boundary{|x | = t} of the wave cone.A bound from below of the exterior energy for well-prepared data.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 37 / 52
Ideas of proof
Ingredients of the proof:Profile decomposition [Bahouri Gérard 99]).Monotonicity formulas, including a Morawetz-type inequality,similar to the one used for energy-critical wave maps. This ismade possible by new bounds of the solution on the boundary{|x | = t} of the wave cone.
A bound from below of the exterior energy for well-prepared data.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 37 / 52
Ideas of proof
Ingredients of the proof:Profile decomposition [Bahouri Gérard 99]).Monotonicity formulas, including a Morawetz-type inequality,similar to the one used for energy-critical wave maps. This ismade possible by new bounds of the solution on the boundary{|x | = t} of the wave cone.A bound from below of the exterior energy for well-prepared data.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 37 / 52
Extraction of the radiation term
Theorem [TD, Kenig, Merle, 2013,2016]. Let u be a nonscatteringsolution such that T+(u) = +∞ and
lim supt→∞
‖~u(t)‖H <∞.
Then there exists a linear wave vL such that
∀A ∈ R, limt→+∞
∫|x |≥t+A
|∇t ,x (u − vL)|2 dx = 0.
Proof more complicated than in the radial case: solutions localizedclose to |x | = t are essentially linear.In the nonradial case, one must exclude profiles that are localizedaround a point of the form x = tp + x0, |p| = 1.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 38 / 52
Extraction of the radiation term
Theorem [TD, Kenig, Merle, 2013,2016]. Let u be a nonscatteringsolution such that T+(u) = +∞ and
lim supt→∞
‖~u(t)‖H <∞.
Then there exists a linear wave vL such that
∀A ∈ R, limt→+∞
∫|x |≥t+A
|∇t ,x (u − vL)|2 dx = 0.
Proof more complicated than in the radial case: solutions localizedclose to |x | = t are essentially linear.
In the nonradial case, one must exclude profiles that are localizedaround a point of the form x = tp + x0, |p| = 1.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 38 / 52
Extraction of the radiation term
Theorem [TD, Kenig, Merle, 2013,2016]. Let u be a nonscatteringsolution such that T+(u) = +∞ and
lim supt→∞
‖~u(t)‖H <∞.
Then there exists a linear wave vL such that
∀A ∈ R, limt→+∞
∫|x |≥t+A
|∇t ,x (u − vL)|2 dx = 0.
Proof more complicated than in the radial case: solutions localizedclose to |x | = t are essentially linear.In the nonradial case, one must exclude profiles that are localizedaround a point of the form x = tp + x0, |p| = 1.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 38 / 52
Morawetz inequality
Lemma. Let u be a solution such that T+(u) = +∞ and
lim supt→∞
‖~u(t)‖H <∞.
then there exists C > 0 such that, for 0 < 10t1 < t2,∫ t2
t1
∫|x |<t
(∂tu +
xt· ∇u +
(N2− 1)
ut
)2
dxdtt≤ C log
(t2t1
) 12
.
Corollary. There exists tn → +∞ such that
limn→∞
∫|x |<tn
(∂tu(tn) +
xtn· ∇u(tn) +
(N2− 1)
u(tn)
tn
)2
dx = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 39 / 52
Morawetz inequality
Lemma. Let u be a solution such that T+(u) = +∞ and
lim supt→∞
‖~u(t)‖H <∞.
then there exists C > 0 such that, for 0 < 10t1 < t2,∫ t2
t1
∫|x |<t
(∂tu +
xt· ∇u +
(N2− 1)
ut
)2
dxdtt≤ C log
(t2t1
) 12
.
Corollary. There exists tn → +∞ such that
limn→∞
∫|x |<tn
(∂tu(tn) +
xtn· ∇u(tn) +
(N2− 1)
u(tn)
tn
)2
dx = 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 39 / 52
Estimates on the boundary of the wave cone
(assume N = 3 to fix ideas).Proposition. Let u be a solution such that T+(u) = +∞ and
lim supt→∞
‖~u(t)‖H <∞.
Then the following quantity are well-defined and finite:∫ +∞
0
∫|x |=t|u(t , x)|6 dσ(5) ∫ +∞
1
∫|x |=t|/∂u(t , x)|2 +
∣∣∣xt∇u + ∂tu
∣∣∣2 dσ(6) ∫ +∞
1
∫|x |=t
1t2 |u(t , x)|2 dσ(7)
Thomas Duyckaerts (Paris 13) Critical waves February 2017 40 / 52
Self-similar change of variables
We let t = es, x = esy , and
w(s, y) = es2 u(es,esy).
Then:
∂2s w =
1ρ
div (ρ∇w − ρ(y · ∇w)y) + w5 − 34
w + 2y · ∇∂sw + 2∂sw ,
where ρ = (1− |y |2)−1/2.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 41 / 52
Self-similar change of variables
We let t = es, x = esy , and
w(s, y) = es2 u(es,esy).
Then:
∂2s w =
1ρ
div (ρ∇w − ρ(y · ∇w)y) + w5 − 34
w + 2y · ∇∂sw + 2∂sw ,
where ρ = (1− |y |2)−1/2.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 41 / 52
Consequence of the Morawetz estimate
Theorem Assume N = 3 (to fix ideas) . Let u be a solution such thatT+(u) = +∞ and
lim supt→+∞
‖~u(t)‖H <∞.
Then there exists tn → +∞, J ≥ 1, a linear wave vL, solitary wavesQj
pj, j = 1 . . . J, and parameters xj,n ∈ RN , λj,n > 0, such that
u(tn) = vL(tn) +J∑
j=1
1
λN−2
2j,n
Qjpj
(0,
x − xj,n
λj,n
)+ εn
limn→+∞
‖εn‖L6 = 0
∀j , limn→+∞
xj,n
tn= pj , lim
n→+∞
λj,n
tn= 0
∀j , k , j 6= k =⇒ limn→+∞
|xj,n − xk ,n|λj,n
+λj,n
λk ,n+λk ,n
λj,n= +∞.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 42 / 52
Consequence of the Morawetz estimate
Theorem Assume N = 3 (to fix ideas) . Let u be a solution such thatT+(u) = +∞ and
lim supt→+∞
‖~u(t)‖H <∞.
Then there exists tn → +∞, J ≥ 1, a linear wave vL, solitary wavesQj
pj, j = 1 . . . J, and parameters xj,n ∈ RN , λj,n > 0, such that
u(tn) = vL(tn) +J∑
j=1
1
λN−2
2j,n
Qjpj
(0,
x − xj,n
λj,n
)+ εn
limn→+∞
‖εn‖L6 = 0
∀j , limn→+∞
xj,n
tn= pj , lim
n→+∞
λj,n
tn= 0
∀j , k , j 6= k =⇒ limn→+∞
|xj,n − xk ,n|λj,n
+λj,n
λk ,n+λk ,n
λj,n= +∞.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 42 / 52
Consequence of the Morawetz estimate
Theorem Assume N = 3 (to fix ideas) . Let u be a solution such thatT+(u) = +∞ and
lim supt→+∞
‖~u(t)‖H <∞.
Then there exists tn → +∞, J ≥ 1, a linear wave vL, solitary wavesQj
pj, j = 1 . . . J, and parameters xj,n ∈ RN , λj,n > 0, such that
u(tn) = vL(tn) +J∑
j=1
1
λN−2
2j,n
Qjpj
(0,
x − xj,n
λj,n
)+ εn
limn→+∞
‖εn‖L6 = 0
∀j , limn→+∞
xj,n
tn= pj , lim
n→+∞
λj,n
tn= 0
∀j , k , j 6= k =⇒ limn→+∞
|xj,n − xk ,n|λj,n
+λj,n
λk ,n+λk ,n
λj,n= +∞.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 42 / 52
Bound from below of the exterior energy
Lemma. Let (wn)n be a sequence of solutions of the linear waveequation, with initial data (w0,n,w1,n) ∈ H such that
∀n,∫|∇w0,n|2 + w2
1,n dx = 1(8)
∀n, |x | ≤ 1 on supp (w0,n,w1,n)(9)
∀λ < 1, limn→∞
∫|x |<λ
|∇w0,n|2 + w21,n dx = 0(10)
limn→∞
∫|/∂w0,n|2 dx +
∫|w1,n + x · ∇w0,n|2 dx = 0.(11)
Then for all ε > 0, and all large n,
(12) lim inft→+∞
∫|x |≥1−ε+t
|∇t ,xwn(t , x)|2 dx ≥ 12
Thomas Duyckaerts (Paris 13) Critical waves February 2017 43 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 44 / 52
Strichartz estimates
Our goal is to describe a sequence of solution un of the nonlinear waveequation, at least close to its initial data. In application we will haveun(t , x) = u(tn + t , x).
Strichartz estimates [Ginibre, Velo] (see also [Lindblad,Sogge]). Letv be a solution of {
∂2t v −∆v = f , x ∈ R3
~v�t=0 = (v0, v1) ∈ H.
Then
‖v‖L4(R,L12) + supt∈R‖~v(t)‖H . ‖f‖L1(R,L2) + ‖(v0, v1)‖H.
(note that this implies a bound on the L5L10 norm).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 45 / 52
Strichartz estimates
Our goal is to describe a sequence of solution un of the nonlinear waveequation, at least close to its initial data. In application we will haveun(t , x) = u(tn + t , x).
Strichartz estimates [Ginibre, Velo] (see also [Lindblad,Sogge]). Letv be a solution of {
∂2t v −∆v = f , x ∈ R3
~v�t=0 = (v0, v1) ∈ H.
Then
‖v‖L4(R,L12) + supt∈R‖~v(t)‖H . ‖f‖L1(R,L2) + ‖(v0, v1)‖H.
(note that this implies a bound on the L5L10 norm).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 45 / 52
Strichartz estimates
Our goal is to describe a sequence of solution un of the nonlinear waveequation, at least close to its initial data. In application we will haveun(t , x) = u(tn + t , x).
Strichartz estimates [Ginibre, Velo] (see also [Lindblad,Sogge]). Letv be a solution of {
∂2t v −∆v = f , x ∈ R3
~v�t=0 = (v0, v1) ∈ H.
Then
‖v‖L4(R,L12) + supt∈R‖~v(t)‖H . ‖f‖L1(R,L2) + ‖(v0, v1)‖H.
(note that this implies a bound on the L5L10 norm).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 45 / 52
Perturbation in a Strichartz space
Proposition. Let u be a solution of (NLW) with initial data (u0,u1),I ⊂ (T−(u),T+(u)) such that u ∈ L5(I,L10). Let (u0,n,u1,n)n be abounded sequence in H, un the corresponding nonlinear solutions anddenote
wn(t , x) = SL(t , x)((u0,u1)− (u0,n,u1,n
).
Assumelim
n→∞‖wn‖L5(I,L10) = 0.
Thenlim
n→∞supt∈I‖un(t)− u(t)− wn(t)‖H = 0.
Denote‖(u0,u1)‖S(I) = ‖SL(·)(u0,u1)‖L5(I,L10).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 46 / 52
Perturbation in a Strichartz space
Proposition. Let u be a solution of (NLW) with initial data (u0,u1),I ⊂ (T−(u),T+(u)) such that u ∈ L5(I,L10). Let (u0,n,u1,n)n be abounded sequence in H, un the corresponding nonlinear solutions anddenote
wn(t , x) = SL(t , x)((u0,u1)− (u0,n,u1,n
).
Assumelim
n→∞‖wn‖L5(I,L10) = 0.
Thenlim
n→∞supt∈I‖un(t)− u(t)− wn(t)‖H = 0.
Denote‖(u0,u1)‖S(I) = ‖SL(·)(u0,u1)‖L5(I,L10).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 46 / 52
Perturbation in a Strichartz space
Proposition. Let u be a solution of (NLW) with initial data (u0,u1),I ⊂ (T−(u),T+(u)) such that u ∈ L5(I,L10). Let (u0,n,u1,n)n be abounded sequence in H, un the corresponding nonlinear solutions anddenote
wn(t , x) = SL(t , x)((u0,u1)− (u0,n,u1,n
).
Assumelim
n→∞‖wn‖L5(I,L10) = 0.
Thenlim
n→∞supt∈I‖un(t)− u(t)− wn(t)‖H = 0.
Denote‖(u0,u1)‖S(I) = ‖SL(·)(u0,u1)‖L5(I,L10).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 46 / 52
Description of the defect of compactness
Denote‖(u0,u1)‖S(I) = ‖SL(·)(u0,u1)‖L5(I,L10).
Theorem [Bahouri-Gérard 1999]. Let (u0,n,u1,n) be a boundedsequence in H,
uL,n(t , x) = SL(t)(u0,n,u1,n)(x).
Assume that for all sequences (tn)n, λn, xn,(λ
1/2n uL,n(tn, λn ·+xn), λ
3/2n uL,n(tn, λn ·+xn)
)−−−⇀n→∞
0
weakly in H. Then (u0,n,u1,n)n converges strongly in S(R).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 47 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 48 / 52
Linear profile decomposition
Let (u0,n,u1,n)n be a bounded sequence in H,uL,n(t) = SL(t)(u0,n,u1,n). Extracting subsequences if necessary, thereexists:
Solutions U jL (j ≥ 1) of the linear wave equation.
Sequences (λj,n)n, (xj,n)n, (tj,n)n with
(13) j 6= k =⇒ limn→∞
λj,n
λk ,n+λk ,n
λj,n+|tj,n − tk ,n|
λj,n+|xj,n − xk ,n|
λj,n= +∞.
Such that, denoting U jL,n(t , x) = 1
λ1/2j,n
U jL
(t−tj,nλj,n
,x−xj,nλj,n
), and
wJL,n(t , x) = uL,n(t , x)−
J∑j=1
U jL,n(t , x)
we have:
(14) limJ→+∞
lim supn→+∞
∥∥∥wJL,n
∥∥∥L5(R,L10)
= 0,
Thomas Duyckaerts (Paris 13) Critical waves February 2017 49 / 52
Outline
1 IntroductionEnergy-critical wave equationExamples of solutions for the critical wave equationsSoliton resolution conjecture for energy-critical waveSoliton resolution conjecture for dispersive equations
2 Radial case, space dimension 3Statement of the resultLinear estimatesRigidity theoremProof of the soliton resolution
3 General case, without symmetry4 Profile decomposition
Defect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition
Thomas Duyckaerts (Paris 13) Critical waves February 2017 50 / 52
Nonlinear profiles
Assume that the following limit exists:
limn→∞
−tj,nλj,n
= τj ∈ R ∪ {−∞} ∪ {+∞}.
Let U j be the solution of the nonlinear wave equation such that
limt→τj
∥∥∥~U j(t)− ~U jL(t)
∥∥∥H
= 0.
Let
U jn(t , x) =
1
λ1/2j,n
U j(
t − tj,nλj,n
,x − xj,n
λj,n
).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 51 / 52
Nonlinear profiles
Assume that the following limit exists:
limn→∞
−tj,nλj,n
= τj ∈ R ∪ {−∞} ∪ {+∞}.
Let U j be the solution of the nonlinear wave equation such that
limt→τj
∥∥∥~U j(t)− ~U jL(t)
∥∥∥H
= 0.
Let
U jn(t , x) =
1
λ1/2j,n
U j(
t − tj,nλj,n
,x − xj,n
λj,n
).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 51 / 52
Nonlinear profiles
Assume that the following limit exists:
limn→∞
−tj,nλj,n
= τj ∈ R ∪ {−∞} ∪ {+∞}.
Let U j be the solution of the nonlinear wave equation such that
limt→τj
∥∥∥~U j(t)− ~U jL(t)
∥∥∥H
= 0.
Let
U jn(t , x) =
1
λ1/2j,n
U j(
t − tj,nλj,n
,x − xj,n
λj,n
).
Thomas Duyckaerts (Paris 13) Critical waves February 2017 51 / 52
Nonlinear profile decomposition
Let (u0,n,u1,n)n be a bounded sequence in H, un the solution of (NLW)with data (u0,n,u1,n). Let In be a sequence of intervals such that
lim supn‖un‖L5(In,L10) <∞.
Then for large n, all the U jn are defined on In and letting
εJn(t , x) = un(t , x)−
J∑j=1
U jn(t , x)− wJ
L,n(t , x),
we have
(15) limJ→+∞
lim supn→+∞
(supt∈In‖~εJ
n(t , x)‖H +∥∥∥εJ
n
∥∥∥L5(In,L10)
)= 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 52 / 52
Nonlinear profile decomposition
Let (u0,n,u1,n)n be a bounded sequence in H, un the solution of (NLW)with data (u0,n,u1,n). Let In be a sequence of intervals such that
lim supn‖un‖L5(In,L10) <∞.
Then for large n, all the U jn are defined on In and letting
εJn(t , x) = un(t , x)−
J∑j=1
U jn(t , x)− wJ
L,n(t , x),
we have
(15) limJ→+∞
lim supn→+∞
(supt∈In‖~εJ
n(t , x)‖H +∥∥∥εJ
n
∥∥∥L5(In,L10)
)= 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 52 / 52
Nonlinear profile decomposition
Let (u0,n,u1,n)n be a bounded sequence in H, un the solution of (NLW)with data (u0,n,u1,n). Let In be a sequence of intervals such that
lim supn‖un‖L5(In,L10) <∞.
Then for large n, all the U jn are defined on In and letting
εJn(t , x) = un(t , x)−
J∑j=1
U jn(t , x)− wJ
L,n(t , x),
we have
(15) limJ→+∞
lim supn→+∞
(supt∈In‖~εJ
n(t , x)‖H +∥∥∥εJ
n
∥∥∥L5(In,L10)
)= 0.
Thomas Duyckaerts (Paris 13) Critical waves February 2017 52 / 52