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

Outline

3 General case, without symmetry

4 Profile decompositionDefect of compactness for the Strichartz estimateLinear profilesNonlinear profile decomposition

Outline

Focusing critical wave equation

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 +

∫RN|∂tu(t)|2 − N − 2

∫|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)?

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 +

∫RN|∂tu(t)|2 − N − 2

∫|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)?

t u −∆u = |u|4

N−2 u, x ∈ RN

~u�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)

E(~u) =12

∫RN|∇xu(t)|2 +

∫RN|∂tu(t)|2 − N − 2

∫|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)?

t u −∆u = |u|4

N−2 u, x ∈ RN

~u�t=0 = (u0,u1) ∈ H = H1(RN)× L2(RN)

E(~u) =12

∫RN|∇xu(t)|2 +

∫RN|∂tu(t)|2 − N − 2

∫|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)?

Linear wave equation

The linear wave equation

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→+∞

∫ +∞

∫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.

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→+∞

∫ +∞

∫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.

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→+∞

∫ +∞

∫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.

Outline

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

Furthermore, small data in H are small in the above sense.

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

Furthermore, small data in H are small in the above sense.

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.

Let T+(u) be the maximal time of existence of u.u scatters when T+(u) = +∞ and

limt→+∞

‖~u(t)− ~uL(t)‖H = 0

limt→+∞

‖~u(t)− ~uL(t)‖H = 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:

limt→+∞

‖~u(t)− ~uL(t)‖H = 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:

limt→+∞

‖~u(t)− ~uL(t)‖H = 0

Scattering for small solutions.

Stability: the set of scattering solutions is open in the energytopology.

limt→+∞

‖~u(t)− ~uL(t)‖H = 0

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)

)N−24 1

(T − t)N−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?

Type I blow-up

limt→T+(u)

‖~u(t)‖H = +∞.

y0(t) =

(N(N − 2)

)N−24 1

(T − t)N−2

Type I blow-up

limt→T+(u)

‖~u(t)‖H = +∞.

y0(t) =

(N(N − 2)

)N−24 1

(T − t)N−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.

Type I blow-up

limt→T+(u)

‖~u(t)‖H = +∞.

y0(t) =

(N(N − 2)

)N−24 1

(T − t)N−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.

Type I blow-up

limt→T+(u)

‖~u(t)‖H = +∞.

y0(t) =

(N(N − 2)

)N−24 1

(T − t)N−2

Type I blow-up

limt→T+(u)

‖~u(t)‖H = +∞.

y0(t) =

(N(N − 2)

)N−24 1

(T − t)N−2

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):

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+

1− p2− 1

)p · x

)p + x

)Qp(t , x) = Qp(0, x − tp).

Solitons

(E) −∆Q = |Q|4

N−2 Q, Q ∈ H1(RN).

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+

1− p2− 1

)p · x

)p + x

)Qp(t , x) = Qp(0, x − tp).

Solitons

(E) −∆Q = |Q|4

N−2 Q, Q ∈ H1(RN).

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+

1− p2− 1

)p · x

)p + x

)Qp(t , x) = Qp(0, x − tp).

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) =

λ(t)N−2

xλ(t)

)+ ~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?

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) =

λ(t)N−2

xλ(t)

)+ ~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?

Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.

~u(t) =

λ(t)N−2

xλ(t)

)+ ~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?

Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.

~u(t) =

λ(t)N−2

xλ(t)

)+ ~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?

Wa(t) = W + ae−ωtY +O(e−2ωt ), t → +∞.

~u(t) =

λ(t)N−2

xλ(t)

)+ ~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?

Type II blow-up

These are solutions bounded in H such that T+(u) <∞. Knownexamples:

~u(t) =

λ(t)N−2

)+ (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].

Type II blow-up

~u(t) =

λ(t)N−2

)+ (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].

Type II blow-up

~u(t) =

λ(t)N−2

)+ (v0, v1), t → T+,

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].

Type II blow-up

~u(t) =

λ(t)N−2

)+ (v0, v1), t → T+,

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].

Type II blow-up

~u(t) =

λ(t)N−2

)+ (v0, v1), t → T+,

Multi-solitons

Existence of a solution such that T+ = +∞ and

~u(t , x) =

(W (x) +

1λ(t)2 W

(xλ(t)

+ 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∑

(tλj,x − xj

)+ 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).

Multi-solitons

~u(t , x) =

(W (x) +

1λ(t)2 W

(xλ(t)

+ o(1), t → +∞,

4/5e−√

5/4t . [Jendrej 2016].Existence of solutions such that T+ = +∞ and

~u(t , x) =J∑

(tλj,x − xj

)+ o(1), t → +∞,

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).

Multi-solitons

~u(t , x) =

(W (x) +

1λ(t)2 W

(xλ(t)

+ o(1), t → +∞,

4/5e−√

5/4t . [Jendrej 2016].Existence of solutions such that T+ = +∞ and

~u(t , x) =J∑

(tλj,x − xj

)+ o(1), t → +∞,

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).

Outline

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∑

λN−2

2j (t)

x − xj(t)λj(t)

)+ r(t)

wherelim

t→+∞

∥∥~r(t)∥∥H = 0

∀j , limt→+∞

xj(t)t

= pj , limt→+∞

λj(t)t

∀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).

Conjecture. Let u be a non scattering solution such thatT+(u) = +∞.Then there exists J ≥ 1, a linear wave vL, solitary waves Qj

j = 1 . . . J, and parameters xj(t) ∈ RN , λj(t) > 0, such that

u(t) = vL(t) +J∑

λN−2

2j (t)

x − xj(t)λj(t)

)+ r(t)

wherelim

t→+∞

∥∥~r(t)∥∥H = 0

∀j , limt→+∞

xj(t)t

= pj , limt→+∞

λj(t)t

∀j , k , j 6= k =⇒ limt→+∞

+λj(t)λk (t)

+λk (t)λj(t)

= +∞.

u(t) = vL(t) +J∑

λN−2

2j (t)

x − xj(t)λj(t)

)+ r(t)

wherelim

t→+∞

∥∥~r(t)∥∥H = 0

∀j , limt→+∞

xj(t)t

= pj , limt→+∞

λj(t)t

∀j , k , j 6= k =⇒ limt→+∞

+λj(t)λk (t)

+λk (t)λj(t)

= +∞.

u(t) = vL(t) +J∑

λN−2

2j (t)

x − xj(t)λj(t)

)+ r(t)

wherelim

t→+∞

∥∥~r(t)∥∥H = 0

∀j , limt→+∞

xj(t)t

= pj , limt→+∞

λj(t)t

∀j , k , j 6= k =⇒ limt→+∞

+λj(t)λk (t)

+λk (t)λj(t)

= +∞.

Outline

Korteweg-de-Vries

{∂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∑

Qκj (t , x − xj)∣∣∣dx = 0.

J ≥ 0, κJ < κJ−1 < . . . < κ1, xj ∈ R.

Korteweg-de-Vries

{∂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∑

Qκj (t , x − xj)∣∣∣dx = 0.

J ≥ 0, κJ < κJ−1 < . . . < κ1, xj ∈ R.

Nonlinear focusing Schrödinger equations

{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.

{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:

∆Q + |Q|p−1Q − ωQ = 0.

{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:

∆Q + |Q|p−1Q − ωQ = 0.

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→+∞

(u(t)− eit∆u+,KA

(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].

limt→+∞

(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

limt→+∞

{u(t , · − x(t)), t ∈ R

Then u is a soliton.

Proved for KdV [C. Laurent, Y. Martel 2003], and conditionally for(NLW) [TD, Kenig, Merle 2014].

limt→+∞

{u(t , · − x(t)), t ∈ R

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).

Outline

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∑

λ12j (t)

xλj(t)

)+ r(t),

where: limt→+∞

∥∥~r(t)∥∥H = 0.

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:

such that:

u(t) = vL(t) +J∑

λ12j (t)

xλj(t)

)+ r(t),

where: limt→+∞

∥∥~r(t)∥∥H = 0.

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:

such that:

u(t) = vL(t) +J∑

λ12j (t)

xλj(t)

)+ r(t),

where: limt→+∞

∥∥~r(t)∥∥H = 0.

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∑

λ12j (t)

xλj(t)

)+ r(t),

where: limt→T+

∥∥~r(t)∥∥H = 0.

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∑

λ12j (t)

xλj(t)

)+ r(t),

where: limt→T+

∥∥~r(t)∥∥H = 0.

Outline

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

∫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?

|x |≥|t ||∇t ,xuL(t , x)|2 dx ≥ 1

∫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].

∀t ≥ 0 or ∀t ≤ 0,∫|x |≥|t |

|∇t ,xu(t , x)|2 dx ≥ η?

|x |≥|t ||∇t ,xuL(t , x)|2 dx ≥ 1

∫RN|∇t ,xuL(0, x)|2 dx .

∀t ≥ 0 or ∀t ≤ 0,∫|x |≥|t |

|∇t ,xu(t , x)|2 dx ≥ η?

|x |≥|t ||∇t ,xuL(t , x)|2 dx ≥ 1

∫RN|∇t ,xuL(0, x)|2 dx .

∀t ≥ 0 or ∀t ≤ 0,∫|x |≥|t |

|∇t ,xu(t , x)|2 dx ≥ η?

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

∫ +∞

(∂t ,r (ruL(0, r)))2 dr .

Generalization to other odd dimensions: [Kenig, Lawrie, Baoping Liu,Schlag 2015]

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

∫ +∞

(∂t ,r (ruL(0, r)))2 dr .

Generalization to other odd dimensions: [Kenig, Lawrie, Baoping Liu,Schlag 2015]

Outline

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|

, so that

λ1/2 W(xλ

)≈√

3λ1/2

|x |, |x | → ∞.

First step of the proof: there exists ` ∈ R such that

limr→∞

ru0(r) = `.

Statement of the theorem

Theorem. Assume N = 3. Let u be a global, radial solution of (NLW).Assume

∫|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|

, so that

λ1/2 W(xλ

)≈√

3λ1/2

|x |, |x | → ∞.

First step of the proof: there exists ` ∈ R such that

limr→∞

ru0(r) = `.

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

∫|x |≥r0+|t |

|∇t ,xu|2 dx = 0.

Assume furthermore

(1)∫ +∞

((∂r u0)2 + u2

)r2 dr = δ ≤ δ0.

Then, letting (v0, v1) = (ru0, ru1),

(2)∫ +∞

(∂r (v0))2 + v21 dr ≤ C0

Furthermore, if r0 ≤ r ≤ 2r0,

(3) |v0(r)− v0(r0)| ≤√

C0|v0(r0)|5

≤√

C0δ2|v0(r0)|.

Outline

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.

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.

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.

Proposition. Let u be a solution such that T+(u) = +∞. Then thereexists a solution vL of the linear wave equation such that

∫|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.

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∑

ιj ~Wλj,n

∥∥∥∥∥∥H

~Wλj,n (x) =

λ1/2j,n

xλj,n

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) =

λ1/2j,n

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

∥∥∥L5(In,L10)

Use a channels of energy argument.

U jn(t , x) =

λ1/2j,n

t − tj,nλj,n

,xλj,n

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

∥∥∥L5(In,L10)

Use a channels of energy argument.

U jn(t , x) =

λ1/2j,n

t − tj,nλj,n

,xλj,n

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

∥∥∥L5(In,L10)

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

λj(t) := inf

{λ > 0 s.t.

∫|x |≤λ

|∇(u − vL)(t , x)|2 dx ≥ Bj

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

λj(t) := inf

{λ > 0 s.t.

∫|x |≤λ

|∇(u − vL)(t , x)|2 dx ≥ Bj

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∑

λN−2

x − xj,n

)+ εn

(+ analogous expansion for the time derivative) wherelim

n→+∞‖~εn‖H = 0

∀j , limn→+∞

tn= pj , lim

n→+∞

∀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]).

General case

lim supt→+∞

‖~u(t)‖H <∞.

λN−2

x − xj,n

)+ εn

n→+∞‖~εn‖H = 0

∀j , limn→+∞

tn= pj , lim

n→+∞

∀j , k , j 6= k =⇒ limn→+∞

+λj,n

λk ,n+λk ,n

λj,n= +∞.

General case

lim supt→+∞

‖~u(t)‖H <∞.

λN−2

x − xj,n

)+ εn

n→+∞‖~εn‖H = 0

∀j , limn→+∞

tn= pj , lim

n→+∞

∀j , k , j 6= k =⇒ limn→+∞

+λj,n

λk ,n+λk ,n

λj,n= +∞.

General case

lim supt→+∞

‖~u(t)‖H <∞.

λN−2

x − xj,n

)+ εn

n→+∞‖~εn‖H = 0

∀j , limn→+∞

tn= pj , lim

n→+∞

∀j , k , j 6= k =⇒ limn→+∞

+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.

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.

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.

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

∫|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.

lim supt→∞

‖~u(t)‖H <∞.

∫|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.

lim supt→∞

‖~u(t)‖H <∞.

∫|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.

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

∫|x |<t

(∂tu +

xt· ∇u +

(N2− 1)

dxdtt≤ C log

Corollary. There exists tn → +∞ such that

limn→∞

∫|x |<tn

(∂tu(tn) +

xtn· ∇u(tn) +

(N2− 1)

dx = 0.

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

∫|x |<t

(∂tu +

xt· ∇u +

(N2− 1)

dxdtt≤ C log

Corollary. There exists tn → +∞ such that

limn→∞

∫|x |<tn

(∂tu(tn) +

xtn· ∇u(tn) +

(N2− 1)

dx = 0.

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:∫ +∞

∫|x |=t|u(t , x)|6 dσ(5) ∫ +∞

∫|x |=t|/∂u(t , x)|2 +

∣∣∣xt∇u + ∂tu

∣∣∣2 dσ(6) ∫ +∞

∫|x |=t

1t2 |u(t , x)|2 dσ(7)

Self-similar change of variables

We let t = es, x = esy , and

w(s, y) = es2 u(es,esy).

∂2s w =

div (ρ∇w − ρ(y · ∇w)y) + w5 − 34

w + 2y · ∇∂sw + 2∂sw ,

where ρ = (1− |y |2)−1/2.

Self-similar change of variables

We let t = es, x = esy , and

w(s, y) = es2 u(es,esy).

∂2s w =

div (ρ∇w − ρ(y · ∇w)y) + w5 − 34

w + 2y · ∇∂sw + 2∂sw ,

where ρ = (1− |y |2)−1/2.

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 <∞.

λN−2

x − xj,n

)+ εn

limn→+∞

‖εn‖L6 = 0

∀j , limn→+∞

tn= pj , lim

n→+∞

∀j , k , j 6= k =⇒ limn→+∞

+λj,n

λk ,n+λk ,n

λj,n= +∞.

lim supt→+∞

‖~u(t)‖H <∞.

λN−2

x − xj,n

)+ εn

limn→+∞

‖εn‖L6 = 0

∀j , limn→+∞

tn= pj , lim

n→+∞

∀j , k , j 6= k =⇒ limn→+∞

+λj,n

λk ,n+λk ,n

λj,n= +∞.

lim supt→+∞

‖~u(t)‖H <∞.

λN−2

x − xj,n

)+ εn

limn→+∞

‖εn‖L6 = 0

∀j , limn→+∞

tn= pj , lim

n→+∞

∀j , k , j 6= k =⇒ limn→+∞

+λj,n

λk ,n+λk ,n

λj,n= +∞.

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

Outline

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.

‖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).

∂2t v −∆v = f , x ∈ R3

~v�t=0 = (v0, v1) ∈ H.

∂2t v −∆v = f , x ∈ R3

~v�t=0 = (v0, v1) ∈ H.

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).

wn(t , x) = SL(t , x)((u0,u1)− (u0,n,u1,n

Assumelim

n→∞‖wn‖L5(I,L10) = 0.

Thenlim

wn(t , x) = SL(t , x)((u0,u1)− (u0,n,u1,n

Assumelim

n→∞‖wn‖L5(I,L10) = 0.

Thenlim

Description of the defect of compactness

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→∞

weakly in H. Then (u0,n,u1,n)n converges strongly in S(R).

Outline

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→∞

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

(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)

Outline

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

U jn(t , x) =

λ1/2j,n

t − tj,nλj,n

,x − xj,n

Nonlinear profiles

limn→∞

−tj,nλj,n

= τj ∈ R ∪ {−∞} ∪ {+∞}.

limt→τj

∥∥∥~U j(t)− ~U jL(t)

∥∥∥H

U jn(t , x) =

λ1/2j,n

t − tj,nλj,n

,x − xj,n

Nonlinear profiles

limn→∞

−tj,nλj,n

= τj ∈ R ∪ {−∞} ∪ {+∞}.

limt→τj

∥∥∥~U j(t)− ~U jL(t)

∥∥∥H

U jn(t , x) =

λ1/2j,n

t − tj,nλj,n

,x − xj,n

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

∥∥∥L5(In,L10)

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

∥∥∥L5(In,L10)

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

∥∥∥L5(In,L10)

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