SOLITON RESOLUTION FOR CRITICAL CO-ROTATIONAL WAVE … · 2021. 3. 3. · arXiv:2103.01293v1...

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SOLITON RESOLUTION FOR CRITICAL CO-ROTATIONAL

WAVE MAPS AND RADIAL CUBIC WAVE EQUATION

THOMAS DUYCKAERTS1, CARLOS KENIG2, YVAN MARTEL3, AND FRANK MERLE4

Abstract. In this paper we prove the soliton resolution conjecture for alltimes, for all solutions in the energy space, of the co-rotational wave mapequation. To our knowledge this is the first such result for all initial data inthe energy space for a wave-type equation. We also prove the correspondingresults for radial solutions, which remain bounded in the energy norm, of thecubic (energy-critical) nonlinear wave equation in space dimension 4.

Contents

1. Introduction 21.1. Background on the soliton resolution conjecture 31.2. Background on rigidity theorems for dispersive equations 61.3. Main results and ideas of proofs 71.4. Outline of the paper 112. Preliminaries 112.1. Notations 112.2. Strichartz estimates and well-posedness 122.3. Solutions outside a wave cone 133. Exterior energy estimates 143.1. Exterior energy estimates for the linear equation 153.2. Exterior energy estimates over channels 183.3. Exterior energy in space dimension 6 293.4. Exterior energy for the linear inhomogeneous wave equation 304. Rigidity theorem part I: compactly supported pertubations and constant

sign solutions 314.1. From co-rotational wave maps to four dimensional waves 314.2. Compactly supported perturbation of a stationary solution 334.3. Constant sign solutions 345. Rigidity theorem, part II: odd solutions 405.1. Gain of regularity 415.2. Approximate non-radiative solution 435.3. Proof of the rigidity result 446. Soliton resolution for the wave equation 47

Key words and phrases. Wave maps, focusing wave equation, dynamics, soliton resolution,global solutions, blow-up.

1LAGA (UMR 7539), Universite Sorbonne Paris Nord, and Institut Universitaire de France.2University of Chicago. Partially supported by NSF Grants DMS-14363746 and DMS-1800082.3CMLS (UMR 7640), Ecole Polytechnique.4AGM (UMR 8088), CY Cergy Paris Universite and IHES.

1

http://arxiv.org/abs/2103.01293v1

2 T. DUYCKAERTS, C. KENIG, Y. MARTEL, AND F. MERLE

6.1. Profile decomposition 486.2. Finite time blow-up case 496.3. Comments on the proof in the global case 567. Soliton resolution for corotional wave maps 577.1. Preliminaries on wave maps 577.2. Proof of the soliton resolution 67Appendix A. Radiation term for the free wave equation 73Appendix B. Pseudo-ortogonality and channels of energy 75B.1. Linear wave equation 75B.2. Pythagorean expansion of the energy for wave maps 77Appendix C. Boundedness of integral operators on LpLq spaces 79References 81

1. Introduction

In this paper we first consider co-rotational wave maps, from Minkowski spaceinto the two-sphere, which in spherical coordinates for the two-sphere, correspondsto solutions of the following equation:

(1.1) ∂2t ψ − ∂2rψ − 1

r∂rψ +

sin(2ψ)

2r2= 0,

where r > 0 and t ∈ R, with initial data

(1.2) ~ψt=0 = (ψ0, ψ1) ∈ E,

where ~ψ = (ψ, ∂tψ) (see for example [72] and the introduction of [8]). Here E isthe space of initial data such that the conserved energy:

EM (ψ0, ψ1) =1

2

∫ ∞

0

(ψ1)2rdr +

1

2

∫ ∞

0

(∂rψ0)2rdr +

1

2

∫ ∞

0

sin2 ψ0

r2rdr

is finite.If (ψ0, ψ1) in E, there exists (ℓ,m) ∈ Z

2 such that

(1.3) limr→0

ψ0(r) = ℓπ, limr→∞

ψ0(r) = mπ.

We denote by Hℓ,m the set of (ψ0, ψ1) in E such that (1.3) holds. This is an affine

space, parallel to the Hilbert spaceH = H0,0 of functions (ψ0, ψ1) ∈(L2loc

((0,∞)

))2such that

(1.4) ‖(ψ0, ψ1)‖2H =

∫ ∞

0

((∂rψ0)

2 +1

r2(ψ0(r))

2

)rdr +

∫ ∞

0

ψ21rdr <∞.

It is well known ([72]) that (1.1) is locally well-posed in E and that the spaces Hℓ,m

are stable by the flow (see for instance [8]). The equation (1.1) has the followingscaling invariance: if ψ is a solution of (1.1), then ψ(λ ·, λ ·) is also a solution of(1.1), with same energy.

The bubble

(1.5) Q(r) = 2 arctan r

is a stationary solution of (1.1) such that (Q, 0) ∈ H0,1. Other stationary solutionsof (1.1) are given by ℓπ±Q(λ ·), λ > 0. These are the only finite energy, stationarysolutions of (1.1).

SOLITON RESOLUTION FOR CRITICAL WAVE EQUATIONS 3

The stationary solution Q has several important properties. It is stable up tothe symmetries (including scaling) as a solution of (1.1), but also as a general wavemap from the Minkowski space into the two-sphere (this is, for example, a simpleconsequence of Theorem 6.1 in [16] and the conservation laws for general wavemaps). In addition, Q is up to symmetries the static solution of least energy amongall the (general) static wave maps from Minkowski space into the two-sphere. Thus,Q is the “ground state”.

The equation (1.1) is the case k = 1 of the equation obtained by consideringk-equivariant wave maps from Minkowski space into the two-sphere:

(1.6) ∂2t ψ − ∂2rψ − 1

r∂rψ + k2

sin(2ψ)

2r2= 0,

The co-rotational case k = 1 is distinguished among the k-equivariant ones, becauseit is the one satisfied by the ground state Q (of the whole wave map equation).

The linearized equation of (1.6) at ψ = 0 is

∂2t ψL − ∂2rψL − 1

r∂rψL +

k2

r2ψL = 0,

which can be reduced to the radial wave equation in dimension 2k + 2 by thechange of unknown function rkuL = ψL. The linearized equation for (1.1) is thusessentially the 4D wave equation.

In this paper we will also consider a twin problem, the wave equation on R4,

with the energy-critical focusing nonlinearity:

(1.7) ∂2t u−∆u = u3,

where t ∈ R and x ∈ R4, with initial data

(1.8) ~ut=0 = (u0, u1) ∈ H,where ~u = (u, ∂tu), and H = H1(R4)×L2(R4). We will only consider radial initial

data, i.e. data depending only on r = |x| =√x21 + x22 + x23 + x24.

We denote by

W (x) =1

1 + |x|28

the ground state of (1.7).The equation (1.7) is a special case of the energy-critical wave equation

(1.9) ∂2t u−∆u = |u| 4N−2u

in general space dimension N ≥ 3. In this general case, the ground state stationary

solution is given by W (x) =(1 + |x|2

N(N−2)

)1−N2

.

1.1. Background on the soliton resolution conjecture. The main results ofthis paper are the proofs of soliton resolution, without size constraints, and for alltimes, for solutions of (1.1) and for radial solutions of (1.7). In particular, wewill obtain the complete classification of the solutions of (1.1), which is to ourknowledge the first result of soliton resolution for all initial data in the energyspace for a wave-type equation. In this work we only consider co-rotational wavemaps and radial solutions of (1.7). The general problems, without any symmetryassumption, seem out of reach by current methods.

To put these results in perspective, we start with a general discussion of thesoliton resolution conjecture for nonlinear dispersive equations. This conjecture


predicts that any global in time solution of this type of equation evolves asymp-totically as a sum of decoupled solitons (traveling wave solutions, which are well-localized and traveling at a fixed speed), a radiative term (typically a solution toa linear equation) and a term going to zero in the energy space. For finite timeblow-up solutions, a similar decomposition should hold, depending on the natureof the blow-up.

This conjecture arose in the 1970’s from numerical simulations and the theoryof integrable equations, in order to explain a “puzzling paradox” stemming fromthe birth of scientific computation, in a numerical simulation carried out at LosAlamos in the early 1950’s (see [32]) by Fermi, Pasta and Ulam. In 1965 Kruskalfound that as the spacial mesh in the discretization of the Fermi-Pasta-Ulam modeltends to 0, the solutions of the Fermi-Pasta-Ulam problem converge to solutions ofthe KdV equation (see [53] for an account). The fact that the KdV equation hassoliton solutions provided an explanation for the “puzzling paradox”. Followingthis discovery, Zabusky and Kruskal [82] conducted another influential numericalsimulation, which showed numerically the emergence of solitons and multisolitonsin the KdV equation. This experiment led to the soliton resolution conjecture, andto the theory of complete integrability, to explain the inelastic collision of solitonsthat was observed.

The first theoretical results in the direction of soliton resolution were obtainedfor the completely integrable KdV, mKdV and 1-dimensional cubic NLS, using themethod of inverse scattering ([54], [29],[28], [70], [71], [61], [3]). Very few completeresults exist for equations that are not completely integrable. For non-integrablenonlinear dispersive equations, such as nonlinear Schrodinger and Klein-Gordonequations, complete results seem out of reach. Known results include scatteringbelow a threshold given by the ground state of the equation (see e.g. [47], [14], [39],[13]), local study close to the ground state soliton [60]), and in some special casesthe existence of a global compact attractor (e.g. [79]). We refer to the introductionof [15] for a more complete discussion and more references on the subject.

The soliton resolution conjecture is believed to hold unconditionally for energy-critical wave maps into the two-sphere, and for energy-bounded solutions of theenergy-critical nonlinear wave equation. For the case of wave maps, a first result inthis direction was that any solution that blows up in finite time converges locally inspace, up to the symmetries of the equation, for a sequence of times, to a travelingwave solution (see [5], [78] for the equivariant case, [74], [75] for the general case).Similar results hold for solutions that exist for infinite time but don’t scatter (see[75]). In the equivariant case, using techniques developed by the first, second andfourth authors, one can prove stronger statements, namely that the soliton reso-lution holds in the equivariant case with a condition on the energy ruling out amultisoliton configuration [8], [9]) and that it holds for a sequence of times withoutthis condition in the co-rotational case [7] and for the k = 2 equivariant case [45](and for all equivariant cases, modulo a technical condition regarding the local well-posedness theory, also in [45]). The limiting case of a pure two-soliton was treatedin [41]), where it is shown, in particular, that the collision between the two solitonsis inelastic. Subsequent work on the case of two solitons is in [42], [44] and [69],while in [43], the full resolution is proved in the equivariant case, in the two-solitonsetting. For wave maps into the two-sphere without any symmetry assumption,the resolution is known only close to the ground state, in work of the first, second


and fourth authors, with Hao Jia [16] (see also [37] for a weaker version of thedecomposition, without size constraint, for a sequence of times).

For the parabolic analog of wave maps (the harmonic map heat flow) thesequestions have been studied earlier (see e.g. [77], [64], [65], [81]). In full generality,in the parabolic case, the soliton resolution is only known for a sequence of times.An example of Topping [81] shows that for a general target manifold, two sequencesof times may lead to different decompositions. In the case when this target is thetwo-sphere, it is conjectured that the analog of the soliton resolution holds, butthis is still open. For a recent work in the parabolic analog of (1.9), in dimensiongreater than or equal to 7, without radial symmetry, near the ground state, see [6].

All the results in non-integrable cases and in the absence of size constraints,mentioned in the last two paragraphs, are proven using monotonicity laws after timeaveraging, and hence hold only for well-chosen sequences of times. The example ofTopping just mentioned shows the necessity of this. Thus, in order to obtain resultsthat hold for all times, new methods that go beyond monotonic time averages haveto be created, which address the notoriously difficult issue of time oscillations. This,for the case of (1.1) and the radial case of (1.7), is a key contribution of this work.

For the case of the energy-critical nonlinear wave equation (1.9), classificationresults for solutions “below the ground state” were obtained in [48], [27]. (Corre-sponding classification results for co-rotational wave maps are in [11] and in [8], [9]).These results are inspired by earlier results for the nonlinear Schrodinger equationgoing back to [59]. (See also the results for gKdV in [58]).

Decomposition results for (1.9) in the 3D radial case, near the ground state, arein [17], and corresponding results in the non-radial case, near the ground state arein [19]. Constructions near the ground state of center stable manifold, in the radialcase and non-radial cases, are in [50], [51]. The soliton resolution for sequencesof time in the radial case, for solutions which are bounded in the energy norm,was proved by [18] in 3 dimensions, [68] in all other odd dimensions, in [10] in 4dimensions and in [45] in 6 dimensions. In [15], the first, second and fourth authors,with Hao Jia, proved the decomposition for sequences of time, in the nonradial case,for solutions which are bounded in the energy norm, in dimensions 3, 4 and 5. Inthe radial case, W , defined above, is the only soliton up to sign change and scaling.The full resolution (for all times), for solutions bounded in the energy norm, forN = 3, in the radial case, was proved in [20], by the first, second and fourth authors.The key fact in the proof is a rigidity theorem giving a dynamical characterizationof the static solutions, in terms of outer energy lower bounds. The proof of thisfact used techniques that are very specific to radial solutions in 3 space dimensions.Recently, combining the papers [22], [23] and [25], the first, second and fourthauthors were able to prove the soliton resolution for solutions of (1.9) that remainbounded in energy norm, in the radial case, for all odd dimensions. An importanttool in this proof is the odd dimensional linear outer energy inequalities proved in[46], that hold up to certain finite dimensional subspaces with increasing dimensionas a function of N . When N = 3, the dimension is 1 and one can use scaling to dealwith this exceptional subspace. For higher odd dimensions, the dimension is largerthan 1 and this was a stumbling block for many years. The proofs mentioned aboveinvolve modulation analysis to analyse the collisions of solitons, showing that theyare inelastic and hence pure multisolitons in both time directions are ruled out.


1.2. Background on rigidity theorems for dispersive equations. As seen inmany recent works, the proof of rigidity (also called Liouville) theorems, classifyingsolutions that are non-dispersive (in a sense to be specified) is crucial in the un-derstanding of the asymptotic dynamics of semilinear dispersive equations as (1.6)and (1.9). A typical statement is that the only non-dispersive solutions are thestationary solutions (or more generally the solitons) of the equation.

A first notion of non-dispersive solutions is given by solutions with the compact-ness property, that are solutions whose trajectory is precompact up the modulationsof the equation. These solutions are also sometimes called quasi-periodic or almostperiodic. This concept goes back to (at least) the work of the third and fourth au-thors [56], in the context of the KdV equation. See also [47] for the energy-criticalNLS, [49], [80] for mass-critical NLS. We refer to [79] and [26] for works highlightingthe importance of solutions with the compactness property in the asymptotic studyof bounded solutions of nonlinear dispersive equations.

For equation (1.9), solutions with the compactness property were first consideredin [48], where a rigidity theorem with a size constraint is proved. A general rigiditytheorem, without a size constraint (but with an additional nondegeneracy assump-tion for solutions without symmetry) is proved in [21] (see also [17, Theorem 2] forthe radial, 3D case).

Nevertheless, this notion is not useful to understand the collision of solitons. In-deed a pure multisoliton in both time directions, that is a solution that is, asymp-totically as t→ +∞ and as t→ −∞, a sum of decoupled solitons, is not a solutionwith the compactness property, but it should be definitively considered as a non-dispersive solution.

It is believed that for completely integrable equations, the collision of solitons isalways elastic, but that, for non-integrable equations such as (1.6) and (1.9), it isnot elastic and should always generate some radiation (see e.g. [57] in the contextof generalized Korteweg-de Vries equations).

To deal with this problem, the first, second and fourth authors have introducedthe concept of non-radiative solutions of (1.9). By definition, these are solutions of(1.9), defined for |x| > |t|, and such that

(1.10)∑

±lim

t→±∞

∫

|x|>|t|

(|∇u(t, x)|2 + (∂tu(t, x))

2)dx = 0.

(see [20] where this concept is used without formal definition, and [23] where theterm “non-radiative” is introduced). This definition can be easily adapted to (1.6).

Note that solitary waves (which always travel at speed ℓ < 1 [26]) are non-radiative. As shown in [20] using the profile decomposition of [2], a rigidity theoremstating that all non-radiative solutions are solitons (or in a radial context, stationarysolutions) essentially implies the soliton resolution.

The usefulness of this concept is that, using finite speed of propagation, it can beapplied by first studying solutions in the exterior of a wave cone |x| > R+ |t|, forlarge R, thus restricting to small solutions, that are close to solutions of the linearwave equation. In connection with this, the study of lower bounds of the form

(1.11) C∑

±lim

t→±∞

∫

|x|>R+|t||∇t,xuL(t, x)|2dx ≥

∫

|x|>R(u1(x))

2 + |∇u0(x)|2dx


for radial solutions of the linear wave equation

(1.12) ∂2t uL −∆uL = 0, (t, x) ∈ R× RN , |x| > R + |t|

with initial data ~ut=0 = (u0, u1) is crucial to prove the rigidity theorem for theequations (1.9) and (1.6). The linear estimate (1.11) depends strongly on thedimension N . In the case R = 0 it was proved that for N odd, (1.11) holds for any(u0, u1) ∈ H (see [19]), whereas when N is even, it is not valid in full generality,but it holds (at least in the radial case) for initial data of the form (u0, 0) or (0, u1)depending on the congruence of N modulo 4 (see [12]). In the case R > 0, explicitcounterexamples to (1.11) exist and one can hope to prove (1.12) only on a strictsubspace of the energy space H. For N = 3, (1.11) is valid for all radial initial data(u0, u1) ∈ H(r > R), that are orthogonal to

(1r , 0). This single nondegenerate

direction can be handled with the scaling of the equation, and corresponds to the

stationary solutionW =(1 + |x|2

3

)−1/2

. This leads to the proof of a strong rigidity

theorem, as was mentioned earlier, and the soliton resolution for all radial solutionsof (1.9) with N = 3 that are bounded in the energy space H [20].

For N odd, N ≥ 5, (1.11) holds in the radial case, for all radial data in an N−12

co-dimensional subspace of H(r > R), which does not seem sufficient to deduce astrong rigidity result for (1.9) as in space dimension 3, using the scaling invarianceof the equation. However, as mentioned earlier, combining asymptotic estimateson non-radiative solutions of (1.7) deduced from (1.11) with a careful study of themodulation equations close to a multisoliton, the soliton resolution for all radialsolutions of (1.7) that are bounded in the energy space, for all times, was provedby three of the authors in [23, 25, 22].

In even space dimension, up to now, no lower bound of the form (1.11) has beenknown. Note that this is relevant for equivariant wave maps (verifying (1.6)), forwhich the underlying space dimension, after linearization, is 2k + 2.

1.3. Main results and ideas of proofs. In this article, we consider co-rotationalwave maps (equation (1.1)), for which the underlying space dimension is 4, and(1.7). Besides the difficulty arising from linear estimates in even space dimensions,these are also special issues at the nonlinear level, since the modulation equationsclose to a multisoliton degenerate, and this makes the method used in [22] for largeodd dimensions ineffective.

Our first step is to prove that (1.11) is valid for R > 0, when N = 4, for radial

solutions of (1.12), with initial data of the form (u0, 0), where u0 ∈ H1(r > R)is radial and orthogonal to 1

r2 . The proof of this is based on the result of [12] andinvolved (but elementary) explicit computations, using a different method of proofthan in the corresponding results in [46].

When N = 4, the inequality (1.11) is not valid for radial solutions of (1.12) withdata of the form (0, u1), u1 ∈ L2(R4), even if u1 is taken in a finite co-dimensionalsubspace of L2. This is essentially due to the resonant solution t

r2 , whose initialdata barely fails to be in the energy space. In Subsection 5.2 of this paper, we willconstruct an approximate solution for each of the equations (1.1) and (1.7), whichis odd in time and whose time derivative is non-radiative. The initial data for thissolution again barely fail to be in the energy space, but the solution itself satisfiesthe same global LpLq estimates in the exterior of wave cones, as the global, finiteenergy solutions of (1.1) or (1.7). Although this solution has similar properties as


the linear solution tr2 , it is not a perturbation of this solution, but a truly nonlinear

object, which comes from a cancellation between the linear and nonlinear parts anddepends on the specific form of the nonlinearity in equations (1.1) and (1.7).

Using the new estimate (1.11) for the linear problem, and the approximate non-radiative solution, we obtain in this paper strong rigidity theorems for the equations(1.1) and (radial) (1.7), which are identical to the one proved in [20] for radialsolutions of (1.9) in dimension N = 3. The proofs, which are much deeper than theone in [20], are first based on the decoupling of a non-radiative solution of (1.1) or(radial) (1.7) into its odd and even parts (in the time variable), which satisfy similarapproximate equations, with a cubic interaction term. Using this decoupling, weconsider two cases.

• If the non-radiative solution has constant sign, one easily sees that the evenpart is dominant. Using that (1.11) holds for initial data of the form (u0, 0),

where u0 is taken on a codimension 1 subspace of H1(r > R), we provethat the solution is equal, up to the invariances of the equation, to one ofthe nonzero explicit stationary solutions ±Q(λ ·)+pπ for (1.1) or ±λW (λ ·)for (1.7) for large r. We then extend the result for all r > 0. This strategycan be seen as a dispersive analog of Alexandrov’s method of moving planefor elliptic equations.

• If, on the other hand, u vanishes at some point (t0, r0), we first prove using(1.11) and some kind of maximum principle for non-radiative solutions,that u(t0, r) = 0 for r > r0. Thus u is after time translation odd in timein the exterior of the wave cone r > |t| + r0. In this case we prove bycontradiction that ∂tu(t0, r) = 0 for r > r0. If not, using (1.11) on thetime derivative of the solution, we prove that u is asymptotically close tothe non-radiative nonlinear solution mentioned above, which does not havefinite energy, giving a contradiction. Extending the result to r > 0 as inthe previous step, we conclude that u(t, r) is the zero solution.

Precisely, we obtain the following two rigidity theorems for equations (1.1) and(radial) (1.7).

Theorem 1 (Rigidity for co-rotational wave maps). Let (ψ0, ψ1) ∈ E (and thusto Hℓ,m for some (ℓ,m) ∈ Z

2). Assume that (ψ0, ψ1) is not the initial data of astationary solution of (1.1), i.e. if ℓ = m, (ψ0, ψ1) 6= (mπ, 0) and otherwise, forall λ > 0, (ψ0 −mπ,ψ1) is not equal to ±(π −Q(λ ·), 0). Then there exists η > 0such that the following holds for all t > 0 or for all t < 0:

(1.13)

∫ ∞

|t|

((∂tψ(t, r))

2 + (∂rψ(t, r))2 +

1

r2sin2(ψ(t, r))

)rdr ≥ η.

We refer to §2.3 for the definition of solutions outside a wave cone. As provedthere, for any initial data (ψ0, ψ1) ∈ E, the solution ψ with initial data (ψ0, ψ1) isalways well-defined for r > |t|.

Theorem 2 (Rigidity for 4D radial critical wave). Let (u0, u1) ∈ H, radial. Assumethat (u0, u1) is not the initial data of a radial stationary solution of (1.7), i.e. that∀λ ∈ R, (u0, u1) is not equal to (λW (λ ·), 0). Then there exists R > 0 such that thesolution u with initial data (u0, u1) is defined on |x| > R + |t| and there exists


η > 0 such that the following holds for all t > 0 or for all t < 0:

(1.14)

∫

|x|>R+|t|(∂tu(t, x))

2 + |∇xu(t, x)|2dx ≥ η.

Note that Theorems 1 and 2 imply the fact that the collision of two or moresolitons yields dispersion, and thus that there is no pure multisoliton solution ofequation (1.1) and of equation (1.7) in the radial case. This was previously knownonly in the case of equation (1.1), for two-solitons (see [69], and also [41] for two-soliton solutions of equation (1.6) with k ≥ 2).

As a consequence of the two rigidity theorems, we obtain the soliton resolutionfor both equations. We first state the result for co-rotational wave maps.

If f(t) and g(t) are positive functions defined in a neighborhood of a ∈ R∪±∞,we will write f(t) ≪ g(t) as t→ a when limt→a f(t)/g(t) = 0.

Theorem 3 (Soliton resolution for co-rotational wave maps). Consider any solu-tion ψ in the energy space of (1.1) with maximal time of existence T+. Then oneof the following holds:

Blow-up. T+ < ∞, and there exist (φ0, φ1) ∈ E, an integer J ∈ N \ 0, andfor each j ∈ 1, . . . , J, a sign ιj ∈ ±1, and a positive function λj(t) defined fort close to T+ such that

λ1(t) ≪ λ2(t) ≪ . . .≪ λJ (t) ≪ T+ − t as t→ T+(1.15)∥∥∥∥∥∥(ψ(t), ∂tψ(t))−

φ0 +

J∑

j=1

ιjQ

( ·λj(t)

), φ1

∥∥∥∥∥∥H

−→t→T+

0.(1.16)

Global solution. T+ = +∞ and there exist ℓ ∈ Z, a solution ψL of

∂2t ψL − ∂2rψL − 1

r∂rψL − 1

r2ψL = 0,

with initial data in H, an integer J ∈ N, and for each j ∈ 1, . . . , J, a signιj ∈ ±1, and a positive function λj(t) defined for large t such that

λ1(t) ≪ λ2(t) ≪ . . .≪ λJ (t) ≪ t as t→ +∞(1.17)∥∥∥∥∥(ψ(t), ∂tψ(t))−

ψL(t) + ℓπ +

J∑

j=1

ιjQ

( ·λj(t)

), ∂tψL(t)

∥∥∥∥∥H

−→t→+∞

0.(1.18)

Remark 1.1. In the finite time-blow-up case, if (ψ0, ψ1) ∈ Hℓ,m, (φ0, φ1) ∈ Hℓ′,m′ ,the expansion (1.16) yields the following constraints:

ℓ = ℓ′, m = m′ +J∑

j=1

ιj .

Similarly, in the global case, (ψ0, ψ1) ∈ Hℓ,m, where ℓ is as in (1.18) and m =

ℓ+∑Jj=1 ιj .

Remark 1.2. We highlight here the fact that Theorem 3 is a soliton resolution forany finite energy initial data of equation (1.1), which should be compared withthe results for radial solutions of (1.7), or (1.9), where we must assume that thesolution is bounded in energy norm or global in time, or sometimes both.


The strategy of the proof of the soliton resolution from the rigidity theorem isthe same as in [20]. However, since the exterior energy bound is not valid for dataof the form (0, u1), we need the additional fact, proved in [7], that there existsa sequence tnn → T+ such that ∂tψ(tn) − φ1 in the finite time blow-up case(respectively ∂tψ(tn) − ∂tψL(tn) in the global case) goes to 0 in L2. Moreover,the proof uses a “nonlinear profile decomposition” outside wave cones which wasnot known earlier for equivariant wave maps (see Proposition 7.8). We prove thisnonlinear profile decomposition using a new space-time bound outside wave cones,involving the nonlinearity (see Lemma 7.5). These results should be of independentinterest for further study of equivariant wave maps.

Remark 1.3. Solutions of (1.1) satisfying (1.16) with J = 1 are known: see [52],[66] and [34]. See also [67] for solutions of (1.6) with J = 1, k ≥ 4. We do not knowany construction of solutions of (1.1) satisfying (1.20) or (1.21) with J ≥ 2, knownexamples of solutions satisfying (1.21) with J = 1 are the stationary solutions andthe recently constructed solutions in [62], [63]. For construction of solutions of (1.6)with k ≥ 3, satisfying (1.21) with J = 2, see [40].

Theorem 4 (Soliton resolution for radial 4D critical wave). Let u be a radialsolution of (1.7) and T+ its maximal time of existence. Then one of the followingholds:Type I blow-up. T+ <∞ and

(1.19) limt→T+

‖(u(t), ∂tu(t)‖H = +∞.

Type II blow-up. T+ < ∞, and there exist (v0, v1) ∈ H, an integer J ∈ N \ 0,and for each j ∈ 1, . . . , J, a sign ιj ∈ ±1, and a positive function λj(t) definedfor t close to T+ such that (1.15) holds and

∥∥∥∥∥∥(u(t), ∂tu(t))−

v0 +

J∑

j=1

ιjλj(t)

W

(x

λj(t)

), v1

∥∥∥∥∥∥H

−→t→T+

0.(1.20)

Global solution. T+ = +∞ and there exist a solution vL of the linear waveequation, an integer J ∈ N, and for each j ∈ 1, . . . , J, a sign ιj ∈ ±1, and apositive function λj(t) defined for large t such that (1.17) holds and

(1.21)

∥∥∥∥∥(u(t), ∂tu(t))−

vL(t) +

J∑

j=1

ιjλj(t)

W

(x

λj(t)

), ∂tvL(t)

∥∥∥∥∥H

−→t→+∞

0.

As in the wave maps case, Theorem 4 follows from the rigidity theorem, Theorem2, using the strategy of [20] and the extra-fact, proved in [10], that there exists asequence tnn → T+ such that ∂tu(tn) − v1 (in the finite time blow-up case) or∂tu(tn)− ∂tvL((tn) (in the global case) goes to 0 with n going to infinity.

Remark 1.4. Except for the scattering solutions ((1.21) with J = 0) and the sta-tionary solutions ±λW (λ ·), the only constructed radial solutions of (1.7), whichare bounded in H, are the blow-up solutions constructed in [38], satisfying (1.20)with J = 1. Note however that the construction in [51] of center-stable manifoldsfor (1.9), with N = 3 or N = 5, should be adaptable to the case N = 4 in theradial case, using Theorem 4. This scarcity of contructions seems to be the result ofthe very strong interaction between solitons, due to their slow decay, and the fact


that the modulation equations for (1.9) degenerate when N = 4, as was mentionedearlier.

Remark 1.5. The results in Theorems 3 and 4 naturally suggest the possibility ofproving analogous results for solutions of (1.6) with k ≥ 2, for the radial Yang-Millsequation, and for radial solutions of (1.9) in the even dimensional case, N ≥ 6. Weexpect that the ideas in this paper, together with the work in [22], [23] and [25]will set the path to solve these problems.

1.4. Outline of the paper. We conclude the introduction with a very brief outlineof the paper. Section 2 introduces notations, reviews the well-posedness theory andrelevant space-times estimates (Strichartz estimates) and reviews solutions outsidea wave cone. Section 3 is devoted to the proof of our new linear estimate (1.11).Sections 4 and 5 prove the rigidity theorems, with Section 4 treating constant signsolutions and Section 5 treating odd solutions in time. In Section 6 we prove thesoliton resolution for radial solutions of the non-linear wave equation (1.7), whilein Section 7 we prove the necessary results on co-rotational wave maps, and thenprove the soliton resolution for them. The Appendices A, B, C collect technicalresults that are needed in the proofs.

2. Preliminaries

2.1. Notations. We will reduce co-rotational wave maps to a radial wave equationin space dimension 4, and work in most of the article with functions that are definedon R

4 or Rt × R4x and that are radial in the space variable, i.e. depending only on

r = |x|, x ∈ R4. Unless explicitly mentioned, the notation Lp will always denote

the space Lp(R4). We normalize the Lp norm as follows:

‖f‖pLp =

∫ ∞

0

|f(r)|pr3dr, f ∈ Lp, radial.

We will denote by LpLq the space Lp(Rt, Lqx(R

4)). If A is a space of functionson R

4, we will denote by A(R) the space of restrictions of radial elements of A to[R,∞). If B is a space of functions on Rt × R

4x, we denote by B(R) the space of

restriction of radial elements of B in the space variable x to (t, x) |x| > R + |t|.We will endow this space with the usual norms, e.g.

‖ϕ‖pLp(R) =

∫ +∞

R

|ϕ(r)|pr3dr, ‖ϕ‖2H1(R)

=

∫ ∞

R

∣∣∣∣dϕ

dr(r)

∣∣∣∣2

r3dr

‖f‖p(LpLq)(R) =

∫

R

(∫ +∞

R+|t||f(r)|qr3dr

) pq

dt.

If f is a function of space and time, we write ∇t,xf = (∂tf,∇xf), and (if f isradial), ∂t,rf = (∂tf, ∂rf). The notation H denotes the energy space for the wave

equation H1(R4) × L2(R4). The notation E stands for the space of finite energydata for the co-rotational wave maps equation (1.1). We recall that E is the disjointunion

E =⋃

ℓ,m

Hℓ,m

where the affine spaces Hℓ,m parallel to the vector space H := H0,0, are defined inthe introduction. Note that H = H × (L2((0,∞), rdr)), where H is the space of


radial L1loc functions ψ0 such that:

(2.1) ‖ψ0‖2H :=

∫ ∞

0

(∂rψ0)2rdr +

∫ ∞

0

1

r2ψ20rdr <∞.

As above, we will use the notations H(R) to denote the space of restriction to r > Rof pair of radial functions in H and H(R) for the space of restrictions to r > R ofpair of functions in H, endowed with the following norm

‖(ψ0, ψ1)‖2H(R) =

∫ ∞

R

(ψ1)2rdr +

∫ ∞

R

(∂rψ0)2rdr +

∫ ∞

R

1

r2ψ20rdr.

If u is a function of space and time, ~u denotes the pair (u, ∂tu).

In all the article (except in Appendix A), ∆ = ∂2

∂r2 +3r2

∂∂r will denote the radial

Laplace operator in dimension 4.

2.2. Strichartz estimates and well-posedness. We first recall some results onStrichartz estimates and local well-posedness. We refer to [35], [36], [55] and [48]for the details.

Let I be an interval, t0 ∈ I, (u0, u1) ∈ H, f ∈ L1(I, L2(R4))). By definition, thesolution of the wave equation

(2.2) ∂2t u−∆u = f

with initial data

(2.3) ~ut=t0 = (u0, u1)

is given by the Duhamel formulation:

u(t, x) = cos((t− t0)√−∆)u0+

sin((t− t0)√−∆)√

−∆u1+

∫ t

t0

sin((t− s)

√−∆

)√−∆

f(s)ds.

Note that ~u ∈ C0(I,H). Furthermore, u satisfies the following Strichartz estimates(see [76], [36]):

(2.4) ‖u‖L2(I,L8) + ‖u‖L3(I,L6) + supt∈I

‖~u(t)‖H . ‖(u0, u1)‖H + ‖f‖L1(I,L2).

Radial solutions of (2.2) in the case where I = R and f = 0 also satisfy the followingdispersive property:

(2.5) limt→±∞

‖ru(t)‖L∞ = 0.

This can be proved for smooth compactly supported initial data using the explicitformula for the solution. The general case can be deduced by a density argumentand the radial Sobolev inequality:

‖rU‖L∞ . ‖U‖H1 , U ∈ H1rad(R

4).

If (u0, u1) ∈ H, a solution u of the nonlinear wave equation (1.7) on I with initialdata (2.3) is a function u ∈ L3

loc(I, L6) which is a solution of (1.7) in the preceding

Duhamel sense, on every compact subinterval of I. Using Strichartz estimates and astandard fixed point argument, one can prove that for any initial data (u0, u1) ∈ H,there exists a unique maximal solution u such that ~u ∈ C0 ((T−(u), T+(u)),H)satisfying the blow-up criterion

T+ <∞ ( respectively |T−| <∞)

=⇒ ‖u‖L3((t0,T+),L6) = +∞ (respectively ‖u‖L3((T−,t0),L6) = +∞),


where we have denoted T± = T±(u).We say that a solution u of (1.7) scatters forward in time if and only if T+(u) = ∞

and there exists a solution uL of the free wave equation ∂2t uL = ∆uL such that

limt→∞

‖~u(t)− ~uL(t)‖H = 0,

or equivalently, u ∈ L3((t0, T+), L6).

2.3. Solutions outside a wave cone. We will now restrict to radial functionsand define solutions of wave equations outside wave cones. Let us mention that itis possible to construct a local well-posedness theory in this context (see Subsection2.3 of [22]) and also the space-time maximal domain of influence of a given solution(in the spirit of [1]). We will not need these notions here.

To simplify notations, we will restrict to initial data at t0 = 0. Recall from §2.1above the notations H(R) = H1(R) × L2(R) and (LpLq)(R). If f ∈ (L1L2)(R)and (u0, u1) ∈ H(R), we define the solution of (2.2), (1.8) on r > R + |t| as therestriction to r > R+ |t| of a solution u of the equation

∂2t u−∆u = f , ~ut=0 = (u0, u1),

where (u0, u1) ∈ H, f ∈ L1L2, (u0, u1)(r) = (u0, u1)(r) for r > R, and f(t, r) =f(t, r) for r > |t| + R. By finite speed of propagation, u does not depend on

the choice of the extensions u0, u1 and f . Furthermore, (2.4) and finite speed ofpropagation implies the Strichartz estimates outside the wave cone:

(2.6) ‖u‖(L2L8)(R) + ‖u‖(L3L6)(R) + supt∈R

‖~u(t)‖H(R+|t|)

. ‖(u0, u1)‖H(R) + ‖f‖(L1L2)(R).

Let (u0, u1) ∈ H(R). We say that u is a solution of (1.7) in r > R + |t| ifu ∈ (L3L6)(R) and u is a solution of (2.2), (1.8) with f = u3 on r > R + |t|.Using the usual small data theory and finite speed of propagation, one sees thatthere exists δ0 (independent of R) such that if ‖(u0, u1)‖H(R) < δ0, then there existsa (unique) solution u on |x| > R+ |t| with initial data (u0, u1). As a consequence,for any (u0, u1) ∈ H, there exists R ≥ 0 and a solution of (1.7) in |x| > R + |t|with initial data (u0, u1). One also has the following uniqueness properties:

• If (u0, u1) ∈ H, R ≥ 0, and there exists a solution of (1.7) on |x| > R+ |t|with initial data (u0, u1), then this solution coincides with the solution of(1.7) defined in §2.2 in t ∈ (T−(u0, u1), T+(u0, u1) ∩ |x| > R + |t|.We will consider u as a solution of (1.7) with initial data on the domaint ∈ (T−(u0, u1), T+(u0, u1) ∪ |x| > R + |t|, with obvious modificationsof the definitions above.

• If (u0, u1) ∈ H and 0 ≤ R ≤ R′, and there exists a solution of (1.7) withinitial data (u0, u1) in |x| > R + |t|, then the restriction of this solutionto |x| > R′ + |t| is the solution of (1.7) with initial data (u0, u1)r>R′in |x| > R′ + |t|.

We can define the same way solutions of the co-rotational wave maps equation (1.1)outside wave cones. However in this case the solution is always global in time:

Claim 2.1. For any initial data (ψ0, ψ1) ∈ E, the solution ψ of (1.1) is well-

defined on r > |t|, and satisfies ~ψ(t) ∈ H(|t|) + (mπ, 0) for all t, where mπ =limr→∞ ψ0(r).


Proof. We assume (ψ0, ψ1) ∈ Hℓ,m. By the standard well-posedness theory for (1.1)(see [72]), there exist t0 > 0 and a solution of (1.1) defined on [−t0, t0]×(0,∞) which

satisfies ~ψ(t) ∈ Hℓ,m for all t ∈ [−t0, t0]. We are thus reduced to prove the existence

of a solution of (1.1) with initial data ~ψ(t0) at t = t0, defined for r > t > t0, and

a solution of (1.1) with initial data ~ψ(−t0) at t = −t0, defined for r > −t > t0.Translating in time, we see that the conclusion of the claim will follow from thefact that for all R > 0 and m ∈ Z, for all initial data (ψ0, ψ1) ∈ H(R) + (mπ, 0),there exists a solution ψ of (1.1) defined for r > R + |t|, with initial data (ψ0, ψ1)at t = 0.

Let T be the positive maximal time of existence of ψ as a solution of (1.1) in r >R+ |t|, and assume that T <∞. Let R1 > T +R. We have 11|T |+R<r<R1

sin(2ψ)2r2 ∈

L1((0, T ), L2). Thus by standard energy estimates, ~ψ(t) has a limit, as t → T , in(H1 × L2)(R + T < r < R1). Taking R1 large, and combining with small datatheory and finite speed of propagation to obtain convergence for r > R1, we obtain

that ~ψ(t) converges in H(R+ T ) + (mπ, 0) as t→ T , a contradiction.

3. Exterior energy estimates

This section concerns lower bounds on the exterior energy for radial solutions ofthe free wave equation

(3.1)

∂2t u−∆u = 0, (t, x) ∈ R× R

4

~ut=0 = (u0, u1) ∈ Hand of its inhomogeneous counter part

(3.2)

∂2t u−∆u = f, (t, x) ∈ R× R

4

~ut=0 = (u0, u1) ∈ H.Exterior energy estimates were first etablished for radial solutions of the 3D linearwave equation in [17] where the following typical result was obtained (see [17,

Lemma 4.2 and proof of Corollary 4.3]): for any (u0, v0) ∈ H1(R3) × L2(R3) withradial symmetry, the solution of the 3D linear wave equation uL satisfies, either forall t ≥ 0 or for all t ≤ 0,

(3.3) ‖∇uL(t)‖2L2(|x|>t) + ‖∂tuL(t)‖2L2(|x|>t) ≥1

2

(‖∇u0‖2L2 + ‖u1‖2L2

).

This result was later extended to any odd dimension d ≥ 3, see [19] and referencestherein. Moreover, [17, 46] proved suitable variants of such estimates over channels,i.e. in space time regions of the form |x| > t+R, for any R > 0: see [46, Corollary 1]and Proposition 3.2 below.

The possiblity of extending estimates of the form (3.3) to even dimensions d ≥ 2is thoroughly discussed in [12]. While estimate (3.3) is proved to fail in any evendimension for general data (u0, u1) (even after replacing 1

2 by any constant C > 0),it is proved to hold for any data of the form (u0, 0), if d = 0 mod 4 and for any dataof the form (0, u1), if d = 2 mod 4. More specifically, in dimension 4, [12, Corollary2] first proved the existence of a constant C > 0 such that any radial solution ofthe free wave equation (3.1) satisfies either for all t ≥ 0 or for all t ≤ 0,

(3.4) ‖∇uL(t)‖2L2(|x|>t) + ‖∂tuL(t)‖2L2(|x|>t) ≥1

C‖∇u0‖2L2 .


Note that by decomposing the solution uL in even and odd parts (in the timevariable), it is equivalent to state an estimate of the type (3.3) for initial data ofthe form (u0, 0) for both t ≥ 0 and t ≤ 0, or to state (3.4) for any general initial data(u0, u1), either for all t ≥ 0 or for all t ≤ 0. We also recall that the exterior energybeing a nonincreasing function of |t|, it is also equivalent to formulate an asymptoticestimate, i.e. in terms of the limits lim±∞

‖∇uL(t)‖2L2(|x|>t)+‖∂tuL(t)‖2L2(|x|>t)

.

In the sequel, we will restrict ourselves to initial data of the form (u0, 0) and wewill focus on limits of the exterior energy as t→ ∞.

The main goal of this section is to obtain lower bounds on the exterior energyover channels |x| > t + R in space dimension 4, in the same spirit as in [17, 46]for odd dimensions. We will also deduce, by a simple trick, an analogous estimatein space dimension 6 (see Subsection 3.3). This will leave open the same questionfor dimensions d = 4n, n ≥ 2 integer, and the analogue estimate for initial data ofthe form (0, u1) for dimension d = 4n+ 2, n ≥ 2 integer.

3.1. Exterior energy estimates for the linear equation. Here, we discuss aslightly more precise version of (3.4) which follows from combining [12] and [24].

Proposition 3.1 ([12],[24]). Let u0 ∈ H1(R4) have radial symmetry. Let uL bethe solution of (3.1) with initial data (u0, 0). Then

(3.5) limt→∞

‖∇uL(t)‖2L2(|x|>t) = limt→∞

‖∂tuL(t)‖2L2(|x|>t) ≥1

4‖∇u0‖2L2 .

For the reader’s convenience, we provide a simplified proof of (3.5) compared to[12], using computations from [12, 46] and the notion of radiation profile (see [24,Theorem 2.1] and references therein).

Proof. Let u0 ∈ H1(R4) with radial symmetry and let uL be the solution of (3.1)with initial data (u0, 0). From Proposition A.1 in the case f = 0, there exists afunction G ∈ L2(R), called the radiation profile, such that

(3.6)

limt→∞

∫ ∞

0

∣∣∣r 32 ∂ruL(t, r)−G(r − t)

∣∣∣2

dr = 0,

limt→∞

∫ ∞

0

∣∣∣r 32 ∂tuL(t, r) +G(r − t)

∣∣∣2

dr = 0,

with c3‖G‖2L2 = 12

∫|∇u0|2, where c3 is the measure of the unit sphere of R4. We

observe that the equality of the two limits in (3.5) is a direct consequence of (3.6).This being known, the result of Proposition 3.1 follows from [12, Corollary 2],since looking carefully at the proof of this result, one sees that the constant c(d)appearing there is exactly c(d) = 1

2 . As in [12], the proof extends to any spacedimension d = 4n, where n ≥ 1 is integer.

Notation. We use the following notation for the Fourier transform of a Schwartzfunction f

f(ξ) =

∫

R4

e−ixξf(x)dx, f(x) = (2π)−4

∫

R4

eixξ f(ξ)dξ.

In particular,

‖f‖2L2(R4) = (2π)4‖f‖2L2(R4),(Plancherel)∫

R4

f(ξ)g(ξ)dξ = (2π)4∫

R4

f(x)g(x)dx.(Parseval)


Recall that if f has the radial symmetry, then f also has the radial symmetry andthe Plancherel identity yields

∫ ∞

0

|f(ρ)|2ρ3dρ = (2π)4∫ ∞

0

|f(r)|2r3dr.

Moreover,

(3.7) f(r) = (2π)−2

∫ ∞

0

f(ρ)J1(rρ)(rρ)−1ρ3dρ

where J1 is the Bessel function of first type of order one.Reduction of the proof. We assume without loss of generality that u0 : R4 → R

is a Schwartz function with radial symmetry such that for some 0 < ρ∗ < ρ∗, itholds

supp u0 ⊂ Bρ∗(0) \Bρ∗(0).Using the formulas (12), (14) of [12] with f = u0 and g ≡ 0, we have

(3.8) (2π)4 limt→∞

∫ ∞

t

|∂tuL(t, r)|2r3dr =2

πlimt→∞

limε↓0

Aε(t)

where

Aε(t) =

∫ ∞

t

∫ ∞

0

∫ ∞

0

u0(ρ1)u0(ρ2)k(t, ρ1, ρ2)ρ52

1 ρ52

2 e−εrdρ1 dρ2 dr

the function k being defined by

k(t, ρ1, ρ2) = sin(tρ1) sin(tρ2) cos

(rρ1 −

3π

4

)cos

(rρ2 −

3π

4

).

Recall that (3.8) follows from the Fourier inversion formula for radial functions (3.7),and the asymptotic behavior of the Bessel functions, see [12, (10)]. Now, we usestandard trigonometry formulas to obtain

2 cos

(rρ1 −

3π

4

)cos

(rρ2 −

3π

4

)= cos r(ρ1 − ρ2) + cos

(r(ρ1 + ρ2)−

3π

2

)

= cos r(ρ1 − ρ2)− sin r(ρ1 + ρ2).

Thus,

k(t, ρ1, ρ2) =1

2[cos r(ρ1 − ρ2)− sin r(ρ1 + ρ2)] sin(tρ1) sin(tρ2)

Now, applying formulas (15) and (16) from [12] with

φ(ρ1) = u0(ρ1) sin(tρ1)ρ52

1 , ψ(ρ2) = u0(ρ2) sin(tρ2)ρ52

2 ,

we obtain

limε↓0

Aε(t) = A1(t) +A2(t) +A3(t)

where

A1(t) =π

2

∫ ∞

0

|u0(ρ)|2 sin2(tρ)ρ5dρ,

A2(t) = −1

2

∫ ∞

0

∫ ∞

0

u0(ρ1)u0(ρ2)k2(t, ρ1, ρ2)ρ52

1 ρ52

2 dρ1 dρ2,

A3(t) = −1

2

∫ ∞

0

∫ ∞

0

u0(ρ1)u0(ρ2)k3(t, ρ1, ρ2)ρ52

1 ρ52

2 dρ1 dρ2,


the functions k2 and k3 being defined by

k2(t, ρ1, ρ2) =sin t(ρ1 − ρ2)

ρ1 − ρ2sin(tρ1) sin(tρ2),

k3(t, ρ1, ρ2) =cos t(ρ1 + ρ2)

ρ1 + ρ2sin(tρ1) sin(tρ2).

Calculation of lim∞ A1. We use sin2(tρ) = 12 (1 − cos(2tρ)) and we remark that

limt→∞

∫ ∞

0

|u0(ρ)|2 cos(2tρ)ρ5dρ = 0,

by integration by parts. Thus, we obtain

(3.9) limt→∞

A1(t) =π

4

∫ ∞

0

|u0(ρ)|2ρ5dρ.

Calculation of lim∞ A2. Applying the formula

2 sin(tρ1) sin(tρ2) = cos t(ρ1 − ρ2)− cos t(ρ1 + ρ2),

and

sin t(ρ1 − ρ2) cos t(ρ1 − ρ2) =1

2sin 2t(ρ1 − ρ2),

we find

k2(t, ρ1, ρ2) =1

4(ρ1 − ρ2)

(sin 2t(ρ1 − ρ2)− 2 sin t(ρ1 − ρ2) cos t(ρ1 + ρ2)

).

For the first term in the above expression of k2, we use the computation after (19)in [46], which yields

limt→∞

∫ ∞

0

∫ ∞

0

u0(ρ1)u0(ρ2)sin 2t(ρ1 − ρ2)

ρ1 − ρ2ρ

52

1 ρ52

2 dρ1 dρ2 = π

∫ ∞

0

|u0(ρ)|2ρ5dρ.

For the second term in the above expression of k2, we rewrite

∫ ∞

0

∫ ∞

0

u0(ρ1)u0(ρ2)sin t(ρ1 − ρ2)

ρ1 − ρ2cos t(ρ1 + ρ2) ρ

52

1 ρ52

2 dρ1 dρ2

=

∫ ∞

0

∫ ∞

0

t sinc t(ρ1 − ρ2) cos t(ρ1 + ρ2)φ(ρ1 + ρ2)dρ1dρ2,

where sinc θ = sin θθ , φ(ρ1, ρ2) = u0(ρ1)u0(ρ2)ρ

52

1 ρ52

2 . By the assumptions on u0 and

u0, the function φ is smooth and compactly supported on (0,∞)2. By integrationby parts, the above expression is equal to

− 1

4t

∫ ∞

0

∫ ∞

0

sinc t(ρ1 − ρ2)

(d

dρ1+

d

dρ2

)2 (cos t(ρ1 + ρ2)

)φ(ρ1, ρ2)dρ1dρ2

= − 1

4t

∫ ∞

0

∫ ∞

0

sinc t(ρ1 − ρ2) cos t(ρ1 + ρ2)

(d

dρ1+

d

dρ2

)2 (φ(ρ1, ρ2)

)dρ1dρ2,

which goes to 0 as t→ ∞. Therefore, we obtain

(3.10) limt→∞

A2(t) = −π8

∫ ∞

0

|u0(ρ)|2ρ5dρ.


Estimate for lim∞A3. Using trigonometry, we compute

−k3(t, ρ1, ρ2) =1

2(ρ1 + ρ2)cos t(ρ1 + ρ2) (cos t(ρ1 + ρ2)− cos t(ρ1 − ρ2))

=1

4(ρ1 + ρ2)(1 + cos 2t(ρ1 + ρ2)− cos 2tρ1 − cos 2tρ2) .

By integration by parts, the contribution of the last three terms to the limit of A3

is zero. Therefore, we obtain

limt→∞

A3(t) =1

8

∫ ∞

0

∫ ∞

0

1

ρ1 + ρ2u0(ρ1)u0(ρ2)ρ

52

1 ρ52

2 dρ1 dρ2

=1

8

∫ ∞

0

u0(ρ1)ρ52

1H(ρ

52 u0)(ρ1)dρ1,

where H denotes the Hankel operator, defined by

(Hu)(ρ1) =

∫ ∞

0

u(ρ)

ρ+ ρ1dρ.

Since H is a positive operator (see references in the Introduction of [12]), and sinceu0 is real-valued (u0 has radial symmetry), we obtain

(3.11) limt→∞

A3(t) ≥ 0.

Conclusion. Gathering (3.8), (3.9), (3.10), (3.11) and using∫ ∞

0

|u0(ρ)|2ρ5dρ = (2π)4∫ ∞

0

|∂ru0(r)|2r3dr,

we have proved

limt→∞

∫ ∞

t

|∂tuL(t, r)|2r3dr ≥1

4

∫ ∞

0

|∂ru0(r)|2r3dr.

Since

limt→∞

∫ ∞

t

|∂ruL(t, r)|2r3dr = limt→∞

∫ ∞

t

|∂tuL(t, r)|2r3dr

as a direct consequence of (3.6), the proof of (3.5) is complete.

3.2. Exterior energy estimates over channels. We prove a lower bound of theasymptotic energy over channels of the form |x| > |t| + R for any R > 0. Werecall that 1/r2 is a solution of (3.1) on the wave cone |x| > R+ |t|, with initial

data (1/r2, 0) ∈ H1(R) × L2(R). We denote by spanR(1/r2) the one dimensional

subspace of H1(R) spanned by the function 1/r2 and by πR the orthogonal projec-

tion on spanR(1/r2) for the H1(R) scalar product, explicitly given for any function

f ∈ H1(R) by

πR(f)(r) =R2f(R)

r2, r > R.

The projection onto the orthogonal complement of spanR(1/r2) is given by

π⊥R(f)(r) = f(r) − R2f(R)

r2, r > R.

We have the following result.


Proposition 3.2. Let R > 0 and let u0 ∈ H1(R) have radial symmetry. Let uLbe the solution of (3.1) with initial data (u0, 0) in the region (t, x) ∈ [0,∞)×R

4 :|x| > R+ |t|. Then

limt→∞

‖∇uL(t)‖2L2(|x|>t+R) = limt→∞

‖∂tuL(t)‖2L2(|x|>t+R) ≥3

20‖∇π⊥

Ru0‖2L2(|x|>R).

To prove Proposition 3.2, we will need the following estimate.

Lemma 3.3. Let v0 : R4 → R be a smooth function with radial symmetry such thatfor some A > 1,

(3.12) supp(v0) ⊂ BA(0) \B1(0).

Let vL be the solution of (3.1) with initial data (v0, 0). Then,

(3.13) lim supt→∞

‖∂rvL(t)‖2L2(t<|x|<t+1) ≤1

10‖∂rv0‖2L2 .

Proof of Proposition 3.2 assuming Lemma 3.3. We start with the case R = 1. Wedefine the space-time region Σ1 = (t, x) ∈ [0,∞)×R

4 : |x| > t+1. First, considera smooth initial data (w0, 0) with radial symmetry such that

(3.14) supp(w0) ⊂ BA(0) \B1(0),

for some A > 0, and denote by wL the corresponding solution of (3.1) on Σ1.Applying Lemma 3.3 to wL, we have

limt→∞

‖∇wL(t)‖2L2(t<|x|<t+1) ≤1

10‖∇w0‖2L2 .

Thus, by Proposition 3.1 and ‖∇w0‖L2 = ‖∇w0‖L2(|x|>1), we obtain

(3.15) limt→∞

‖∇wL(t)‖2L2(|x|>t+1) ≥3

20‖∇w0‖2L2(|x|>1).

Second, consider a radial function w0 ∈ H1(R4) that vanishes at r = 1. By finitespeed of propagation, we can assume that w0(r) is zero on [0, 1] without changingthe solution wL on Σ1. Moreover, such a function w0 can be approximated inH1(R4) by a sequence of smooth functions (w0,n)n satisfying (3.14), with someAn → ∞. By this density argument, (3.15) also holds for any such w0.

Third, consider a general initial data u0 ∈ H1(R4) with radial symmetry. Let

w0(x) = π⊥1 (u0)(x) = u0(x)−

u0(1)

|x|2 .

Since w0 vanishes at r = 1, if we extend it by 0 for r ∈ [0, 1], the property (3.15)

holds for wL, and wL = uL− u0(1)|x|2 on Σ1 since 1

|x|2 is a solution of (3.1) on Σ1, and

uniqueness holds on Σ1. Moreover,

limt→∞

‖∇uL(t)‖2L2(|x|>t+1) = limt→∞

‖∇wL(t)‖2L2(|x|>t+1),

which implies that

limt→∞

‖∇uL(t)‖2L2(|x|>t+1) ≥3

20‖∇π⊥

1 u0‖2L2(|x|>1).

By (3.6), we have

limt→∞

‖∂tuL(t)‖2L2(|x|>t+1) = limt→∞

‖∇uL(t)‖2L2(|x|>t+1),

which proves the result for R = 1.


We complete the proof by a scaling argument. LetR0 > 0 and let uL be a solutionof (3.1). Applying the result for R = 1 to the solution vL(t, x) = uL(R0t, R0x)implies the result for the solution uL for R = R0.

Proof of Lemma 3.3. Recall that vL is given explicitly by the representation for-mula

vL(t, x′) =

1

4π2

(∂

∂t

)(1

t

∂

∂t

)(∫

|x−x′|<t

v0(x)

(t2 − |x− x′|2) 12

dx

)

(see [30, (38) of §2.4] for n = 4, and then use the fact that the volume of the four

dimensional unit ball is π2

2 ).

Denote by e1 the first vector of the canonical basis of R4. For any t, s > 0, set

V (t, s) =1

4π2t3

∫

|x−se1|<t

v0(x)

(t2 − |x− se1|2)12

dx,

so that

vL(t, se1) =

(∂

∂t

)(1

t

∂

∂t

)(t3V

)(t, s)

= 3V (t, s) + 5t∂tV (t, s) + t2∂2t V (t, s)

= (V1 + V2 + V3)(t, s).

We focus on V3 = t2∂2t V . By the change of variable

(3.16) y =x− se1

t, x = se1 + ty

we rewrite V as

(3.17) V (t, s) =1

4π2

∫

|y|<1

v0(se1 + ty)

(1− |y|2) 12

dy.

Thus,

V3(t, s) =t2

4π2∂2t

(∫

|y|<1

v0(se1 + ty)

(1− |y|2) 12

dy

)

=t2

4π2

∫

|y|<1

∑4j,k=1 yjyk∂xj∂xk

v0(se1 + ty)

(1− |y|2) 12

dy.

Turning back to the variable x using

y = −e1 +1

t(x− (s− t)e1),

we obtain

V3 = V4 + V5,


Lemma 3.4. Let f : R4 → R be a continuous, compactly supported function withradial symmetry. For any η > 0,

∫

x1>η

f(x)

(x1 − η)12

dx = η−12

∫ ∞

η

f(ρ)h

(ρ

η

)ρ3dρ

where

(3.19) h(r) =4π

r2

∫ r

1

√r2 − z2

z − 1dz.

Proof. We use the hyperspherical coordinates in dimension four:

x1 = ρ cosψ

x2 = ρ sinψ cos θ

x3 = ρ sinψ sin θ cosϕ

x4 = ρ sinψ sin θ sinϕ

with ρ > 0, ψ ∈ (0, π), θ ∈ (0, π) and ϕ ∈ (0, 2π). Using

dx = ρ3 sin2 ψ sin θ dρ dψ dθ dϕ,

the change of variable yields∫

x1>η

f(x)

(x1 − η)12

dx = 4π

∫ ∞

η

f(ρ)

(∫ arccos η/ρ

0

sin2 ψ

(ρ cosψ − η)12

dψ

)ρ3dρ

(the constant 4π comes from the integral of (θ, ϕ) 7→ sin θ over (0, π) × (0, 2π)).Next, changing variable z = (ρ/η) cosψ,

4π

∫ arccosη/ρ

0

sin2 ψ

(ρ cosψ − η)12

dψ = 4πη

32

ρ2

∫ ρ/η

1

√(ρ/η)2 − z2

z − 1dz = η−

12h

(ρ

η

),

where the function h is defined in (3.19).

By integration by parts, it holds for j = 2, 3, 4,∫

x1>η

∂2xjv0(x)

(x1 − η)12

dx = 0.

This implies that the function G rewrites as

G(η) =1

4√2π2

∫

x1>η

∆v0(x)

(x1 − η)12

dx.

The function ∆v0 has radial symmetry and is equal to r−3(r3v′0

)′. Thus, using

Lemma 3.4,

G(η) =1

4√2π2

η−12

∫ ∞

η

(ρ3v′0

)′(ρ)h

(ρ

η

)dρ.

Thus, integrating by parts twice and using (3.12),

G(η) =1

4√2π2

∫ ∞

1

ρ12 v0(ρ)H1

(ρ

η

)dρ

where we have defined

H1(r) = r−12

(r3h′

)′.


Differentiating in η, we find

G′(η) = − 1

4√2π2

1

η

∫ ∞

1

ρ12 v0(ρ)H

(ρ

η

)dρ

where

(3.20) H(r) = rH ′1(r) = r

(r−

12

(r3h′

)′)′.

The Cauchy-Schwarz inequality and then a change a variable ρ = rη yield

(G′)2(η) ≤ 1

32π4

(1

η2

∫ ∞

1

H2

(ρ

η

)dρ

)(∫ ∞

1

v20(r)rdr

)

≤ 1

32π4

(1

η

∫ ∞

1/η

H2(r)dr

)(∫ ∞

1

v20(r)rdr

).

Integrating on [0, 1] in the variable η and then using Tonelli’s theorem,

∫ 1

0

(G′)2(η)dη ≤ 1

32π4

(∫ 1

0

1

η

∫ ∞

1/η

H2(r)dr dη

)(∫ ∞

1

v20(r)rdr

)

≤ 1

32π4

(∫ ∞

1

H2(r) log rdr

)(∫ ∞

1

v20(r)rdr

).

Lemma 3.5 (Estimate on I). Let

I =1

32π4

∫ ∞

1

H2(r) log r dr.

It holds

I ≤ 1

10.

Proof. We change variable in the expression of h in (3.19)

z = 1 + ζ(r − 1), z − 1 = ζ(r − 1),

r2 − z2 = (r − z)(r + z) = (1 − ζ)(r − 1)(r + 1 + ζ(r − 1)),

and we obtain

h(r) = 4π

(1

r− 1

r2

)∫ 1

0

(1 − ζ)12 ζ−

12K(ζ, r)dζ,

K(ζ, r) = (r(1 + ζ) + 1− ζ)12 .

Denote

K ′ =∂K

∂r=

1

2(1 + ζ)K−1,(3.21)

K ′′ =∂2K

∂r2= −1

4(1 + ζ)2K−3,(3.22)

K ′′′ =∂3K

∂r3=

3

8(1 + ζ)3K−5.(3.23)

We compute H(r). First,

h′ = 4π

(− 1

r2+

2

r3

)∫ 1

0

(1 − ζ)12 ζ−

12Kdζ + 4π

(1

r− 1

r2

)∫ 1

0

(1− ζ)12 ζ−

12K ′dζ


and thus

r3h′ = 4π (−r + 2)

∫ 1

0

(1− ζ)12 ζ−

12Kdζ + 4π

(r2 − r

) ∫ 1

0

(1− ζ)12 ζ−

12K ′dζ.

Second,

(r3h′)′ = −4π

∫ 1

0

(1− ζ)12 ζ−

12Kdζ + 4π (r + 1)

∫ 1

0

(1− ζ)12 ζ−

12K ′dζ

+ 4π(r2 − r

) ∫ 1

0

(1− ζ)12 ζ−

12K ′′dζ

and so

r−12

(r3h′

)′= −4πr−

12

∫ 1

0

(1− ζ)12 ζ−

12Kdζ + 4π

(r

12 + r−

12

)∫ 1

0

(1− ζ)12 ζ−

12K ′dζ

+ 4π(r

32 − r

12

)∫ 1

0

(1 − ζ)12 ζ−

12K ′′dζ.

Third,

(r−

12

(r3h′

)′)′= 2πr−

32

∫ 1

0

(1 − ζ)12 ζ−

12Kdζ

− 2π(r−

12 + r−

32

)∫ 1

0

(1− ζ)12 ζ−

12K ′dζ

+ 2π(5r

12 + r−

12

)∫ 1

0

(1 − ζ)12 ζ−

12K ′′dζ

+ 4π(r

32 − r

12

) ∫ 1

0

(1− ζ)12 ζ−

12K ′′′dζ.

Thus, we have obtained

H(r) = 2πr−12

∫ 1

0

(1− ζ)12 ζ−

12Z(ζ, r)dζ

where

Z(ζ, r) = K − (r + 1)K ′ + (5r2 + r)K ′′ + 2(r3 − r2)K ′′′.

We replace K ′, K ′′ and K ′′′ by their expressions in (3.21), (3.22) and (3.23),

Z(ζ, r) = K−5

[K6 − 1

2(r + 1)(1 + ζ)K4

− 1

4(5r2 + r)(1 + ζ)2K2 +

3

4(1 + ζ)3(r3 − r2)

].

Next, we insert

K2(ζ, r) = (r(1 + ζ) + 1− ζ)


and we expand in powers of r

Z(ζ, r) = K−5

[r3(1 + ζ)3 − 1

2(1 + ζ)3 − 5

4(1 + ζ)3 +

3

4(1 + ζ)3

+ r23(1 + ζ)2(1− ζ) − (1 + ζ)2(1− ζ)− 1

2(1 + ζ)3

− 5

4(1 + ζ)2(1− ζ)− 1

4(1 + ζ)3 − 3

4(1 + ζ)3

+ r

3(1 + ζ)(1 − ζ)2 − 1

2(1 + ζ)(1 − ζ)2

− (1 + ζ)2(1 − ζ)− 1

4(1 + ζ)2(1− ζ)

+

(1 − ζ)3 − 1

2(1 + ζ)(1 − ζ)2

].

Note that in the first line, the terms in r3 vanish. This expression simplifies into

Z(ζ, r) =1

4K−5

[−3r2(1+ζ)2(1+3ζ)+5r(1+ζ)(1−ζ)(1−3ζ)+2(1−ζ)2(1−3ζ)

].

We observe that for all ζ ∈ [0, 1], |1− 3ζ| ≤ 1 + ζ. Indeed, for ζ ∈ [0, 13 ], |1− 3ζ| =1 − 3ζ ≤ 1, while for ζ ∈ [ 13 , 1], |1 − 3ζ| = 3ζ − 1 = 1 + ζ − 2(1 − ζ) ≤ 1 + ζ. Itfollows that

|Z(ζ, r)| ≤ 1

4K−5

[3r2(1 + ζ)2(1 + 3ζ) + 5r(1 + ζ)2(1− ζ) + 2(1− ζ)2(1 + ζ)

].

Therefore,

|H(r)| ≤ 3π

2r

32

∫ 1

0

(1− ζ)12 ζ−

12 (1 + ζ)2(1 + 3ζ)K−5(ζ, r)dζ

+5π

2r

12

∫ 1

0

(1− ζ)32 ζ−

12 (1 + ζ)2K−5(ζ, r)dζ

+ πr−12

∫ 1

0

(1 − ζ)52 ζ−

12 (1 + ζ)K−5(ζ, r)dζ

Now, we estimate K−5 to reduce to simple integrals. Since

K−5(ζ, r) = r−52 (1 + ζ)−

52

(1 +

1− ζ

r(1 + ζ)

)− 52

≤ r−52 (1 + ζ)−

52 ,

we have

|H(r)| ≤ ar−1 + br−2 + cr−3

where

a =3π

2

∫ 1

0

(1− ζ)12 ζ−

12 (1 + ζ)−

12 (1 + 3ζ)dζ

b =5π

2

∫ 1

0

(1− ζ)32 ζ−

12 (1 + ζ)−

12 dζ

c = π

∫ 1

0

(1− ζ)52 ζ−

12 (1 + ζ)−

32 dζ.


We estimate a, b and c. For a, using the change of variable ζ = ζ12 and then the

Cauchy-Schwarz inequality,

a = 3π

∫ 1

0

(1− ζ2)12 (1 + ζ2)−

12 (1 + 3ζ2)dζ

≤ 3π

√∫ 1

0

dζ

1 + ζ2

√∫ 1

0

(1− ζ2)(1 + 3ζ2)2dζ = 3π

√π

4

(1 +

5

3+

3

5− 9

7

)≤ 12.

We proceed similarly for b and c, which yields

b = 5π

∫ 1

0

(1 − ζ2)32 (1 + ζ2)−

12 dζ

≤ 5π

√∫ 1

0

dζ

1 + ζ2

√∫ 1

0

(1 − ζ2)3dζ = 5π

√π

7≤ 11,

c = 2π

∫ 1

0

(1 − ζ2)52 (1 + ζ2)−

32 dζ ≤ 2π.

For the sake of simplicity, we bound b ≤ 12 and c ≤ 12, so that we have proved

|H(r)| ≤ 12(r−1 + r−2 + r−3

).

Using the expression of I and the estimate on H , it holds∫ ∞

1

H2(r) log rdr ≤ 144

∫ ∞

1

(r−1 + r−2 + r−3)2 log rdr

≤ 144

∫ ∞

1

(r−2 + 2r−3 + 3r−4 + 2r−5 + r−6

)log rdr.

Using∫∞1r−k log rdr = (k − 1)−2, we obtain

I ≤ 144

32π4

(1× 1 + 2× 1

4+ 3× 1

9+ 2× 1

16+ 1× 1

25

)=

3597

400

1

π4≤ 1

10

where the last estimate, checked numerically, is given for the sake of simplicity.

We continue the proof of Lemma 3.3. By Lemma 3.5 and then the Hardy in-equality (see e.g. [73, Appendix A.4])

∫ ∞

0

v20rdr ≤∫ ∞

0

(v′0)2r3dr,

we obtain∫ 1

0

(G′)2(η)dη ≤ 1

10

∫ ∞

1

v20(r)rdr =1

10

∫ ∞

0

v20(r)rdr ≤1

10

∫ ∞

0

(v′0)2(r)r3dr.

For the term V7, we conclude∫ t+1

t

(∂sV7)2(t, s)s3ds = t−3

∫ t+1

t

(G′)2(s− t)s3ds

= t−3

∫ 1

0

(G′)2(η)(t + η)3dη ≤(1 +

1

t

)3 ∫ 1

0

(G′)2(η)dη,

and so

(3.24) lim supt→∞

∫

t≤|x|≤t+1

|∇V7|2(t, x)dx ≤ 1

10

∫|∇v0(x)|2dx.


Now, we claim that for k = 1, 2, 4, 6, for t large and for any η ∈ [0, 1], it holds

(3.25) |∂sVk(t, t+ η)| . t−52 .

Observe that setting s = t+ η, estimate (3.25) implies that

∫ t+1

t

(∂sVk)2(t, s)s3ds =

∫ 1

0

(∂sVk)2(t, t+ η)(t + η)3dη

≤ (t+ 1)3∫ 1

0

(∂sVk)2(t, t+ η) dη . t−2.

Thus, for k = 1, 2, 4, 6,

(3.26) limt→∞

∫ t+1

t

(∂sVk)2(t, s)s3ds = 0.

Since estimates (3.24) and (3.26) imply Lemma 3.3, we only have to prove (3.25).We start with the following technical result.

Lemma 3.6. Let A > 0 and let f : R4 → R be a smooth function such that

supp f ⊂ BA(0). Define

Ω[f ](t) =

∫

|x−te1|<t

f(x)

(t2 − |x− te1|2)12

dx,

For t large enough,

(3.27)

∣∣∣∣∣(2t)12Ω[f ](t)−

∫

x1>0

f(x)

x12

1

dx

∣∣∣∣∣ ≤C

t

and

(3.28) |Ω[f ](t)| ≤ Ct−12

where the constant C depends on A, ‖f‖L∞ and ‖∇f‖L∞.

Remark 3.7. We do not assume that f has radial symmetry.

Proof. Observe that (3.28) is a direct consequence of (3.27). We prove (3.27). Sincet2 − |x− te1|2 = 2tx1 − |x|2, we rewrite

(2t)12Ω[f ](t) =

∫

x1>|x|2

2t

f(x)(x1 − |x|2

2t

) 12

dx.

We change variables, setting

y = Φ(x) =

(x1 −

|x|22t

, x2, x3, x4

).

For t large enough (depending on A), the map Φ is a diffeomorphism from B2A(0)to its image Φ(B2A(0)), and BA(0) ⊂ Φ(B2A(0)) ⊂ B4A(0). We compute Φ−1.Using the notation x = (x2, x3, x4), y = (y2, y3, y4), we have for y = Φ(x),

|y|2 = |x|2, x21 − 2tx1 + 2ty1 + |y|2 = 0,

and thus

Φ−1(y) =(t−√t2 − 2ty1 − |y|2, y2, y3, y4

).


By the change of variable y = xt − e1, we rewrite

V2(t, t+ η) =5

4π2t2

∫

|x−te1|<t

(x1

t − 1)∂x1

v0(x+ ηe1)

(t2 − |x− te1|2) 12

dx

+5

4π2t3

∫

|x−te1|<t

∑4j=2 xj∂xjv0(x+ ηe1)

(t2 − |x− te1|2) 12

dx.

Arguing as for V1, using (3.28), for any η ∈ [0, 1], we find |∂sV2(t, t+ η)| . t−52 .

Estimate for V4. The first term in the expression of V4 in (3.18) is rewritten as

V4,1(t, t+ η) = − 1

2π2t2

∫

|x−te1|<t

x1∂2x1v0(x + ηe1)

(t2 − |x− te1|2) 12

dx.

Arguing as for V1, using (3.28), for η ∈ [0, 1], we find |∂sV4,1(t, t+ η)| . t−52 . The

other terms in the expression of V4 are estimated similarly.Estimate for V6. Recall that

V6(t, t+ η) =t−

32

4√2π2

((2t)

12 Ω[∂2x1

v0(·+ ηe1)](t)−∫

x1>0

∂2x1v0(x+ ηe1)

x12

1

dx

).

Thus, by (3.27) of Lemma 3.6, applied to the function x 7→ ∂3x1v0(x+ηe1), it follows

that for any η ∈ (0, 1), |∂sV6(t, t+η)| . t−52 and the proof of (3.25) is complete.

3.3. Exterior energy in space dimension 6. In this subsection, we consider thelinear wave equation in space dimension 6:

(3.29) ∂2t uL −∆uL = 0, (t, x) ∈ R× R6, |x| > R+ |t|.

With initial data of the form:

(3.30) ~ut=0 = (0, u1), u1 ∈ L2(x ∈ R6, |x| > R).

We note that t/r4 is a radial solution of (3.29),(3.30), with initial data (0, 1/r4).We will denote by ΠR the orthogonal projection, on span(1/r4) in the space ofradial functions in L2(x ∈ R

6, |x| > R), and by Π⊥R = Id−ΠR. We will prove:

Proposition 3.8. Let u be a radial solution of (3.29) with initial data of the form(3.30). Then

(3.31) limt→∞

∫ ∞

t+R

|∇uL(t, r)|2r5dr = limt→∞

∫ ∞

t+R

|∂tuL(t, r)|2r5dr

≥ 3

20

∫ ∞

R

∣∣Π⊥Ru1(r)

∣∣2 r5dr.

Proof. This is a simple consequence of Proposition 3.2. By using the radiationprofile (see Proposition (A.1)), it suffices to prove the inequality in (3.31). Byscaling, we can assume R = 1. Consider

v(t, r) =

∫ ∞

r

ρ∂tuL(t, ρ)dρ.

A calculation shows that v(t, r) solves the linear wave equation in R × R4 for

|x| > 1 + |t|. Moreover, ∂rv(0, r) ∈ L2(x ∈ R4, |x| > 1) and ∂tv(0, r) = 0 since

uL(0, r) = 0. By Proposition 3.2, we have that

limt→∞

∫ ∞

1+t

(∂tuL(t, r))2r5dr = lim

t→∞

∫ ∞

1+t

(∂rv(t, r))2r3dr ≥ 3

20

∫ ∞

1+t

∣∣∇π⊥1 v(0)

∣∣2 r3dr.


Now ∂rv(0, r) = −ru1(r), and

∂r(π⊥1 v(0, r)) =

∂

∂r

(v(0, r) − v(0, 1)

r2

)

= −ru1(r) +2

r3

∫ ∞

1

ρu1(ρ)dρ = −r(u1(r) −

2

r4

∫ ∞

1

ρu1(ρ)dρ

),

which by a simple calculation equals −rΠ⊥1 (u1), and the proposition follows.

3.4. Exterior energy for the linear inhomogeneous wave equation. Wededuce from Propositions 3.1 and 3.2 similar lower bounds on the exterior energyfor the inhomogeneous problem (3.2).

Lemma 3.9. Let u0 ∈ H1(R4) and f ∈ L1((0,∞), L2(R4)) have radial symmetry.Let u be the solution of (3.2). Then

(3.32) ‖u0‖H1 ≤ 2 limt→+∞

‖∇u(t)‖L2(|x|>t) + 2∥∥11|x|>|t|f

∥∥L1((0,∞),L2)

,

and for R > 0,(3.33)

‖π⊥Ru0‖H1(R) ≤

√20

3

(lim

t→+∞‖∇u(t)‖L2(|x|>t+R) +

∥∥11|x|>|t|+Rf∥∥L1((0,∞),L2)

).

Proof. Let v be the solution of∂2t v −∆v = 11|x|>|t|f

~vt=0 = (0, 0).

On the one hand, by standard energy computations and the Cauchy-Schwarz in-equality, one sees that ∣∣∣∣

d

dt‖∇t,xv(t)‖L2

∣∣∣∣ ≤ ‖11|x|>|t|f(t)‖L2

and so by integration on (0,∞),

‖∇t,xv(t)‖L2 ≤ ‖11|x|>|t|f‖L1L2 ,

where in this proof, we denote ‖ · ‖L1L2 = ‖ · ‖L1((0,∞),L2(R4)).On the other hand, we observe that the function u− v satisfies

∂2t (u− v)−∆(u− v) = 11|x|≤|t|f

(~u− ~v)t=0 = (u0, 0)

and so u−v coincides in the region |x| > |t| with the solution uL of the free waveequation (3.1) with initial data (u0, 0). As a consequence, by Proposition 3.1,

‖u0‖H1 ≤ 2 limt→+∞

‖∇uL‖L2(|x|>t)

≤ 2 limt→+∞

‖∇u‖L2(|x|>t) + 2 limt→+∞

‖∇v‖L2(|x|>t)

≤ 2 limt→+∞

‖∇u‖L2(|x|>t) + 2 ‖11|x|>|t|f‖L1L2 ,

which completes the proof of (3.32).The proof of (3.33) is similar, using the solution v of

∂2t v −∆v = 11|x|>|t|+Rf

~vt=0 = (0, 0).


and Proposition 3.2 instead of Proposition 3.1.

4. Rigidity theorem part I: compactly supported pertubations and

constant sign solutions

In this section, we will start the proof of Theorems 2 and 1. We will prove thetwo theorems at the same time, reducing the co-rotational wave maps equation toa critical wave equation in four space dimension (see §4.1). In §4.2, we considersolutions with initial data that are equal to a stationary solution for large r. In§4.3 we will treat the case of solutions that have a constant sign outside a wavecone. Section 5 concerns solutions that are odd in time.

4.1. From co-rotational wave maps to four dimensional waves. Let R > 0,(ψ0, ψ1) ∈ H(R). Let ψ be the solution of (1.1) on |x| > R+ |t|. Let

u(t, r) =

√2

3

ψ

r, r > R+ |t| and (u0, u1)(r) = ~ut=0(r), r > R.

Then (u0, u1) ∈ H(R) and

∂2t u− ∂2ru− 3

r∂ru− 1

r2u+

sin(√6ru)√

6ru= 0, r > R+ |t|.

We write this equation as

(4.1) ∂2t u− ∂2ru− 3

r∂ru =

1

r3Λ(ru),

where

(4.2) Λ(U) = U − sin(√6U)√6

.

We note that Λ is odd, smooth, and that it satisfies:

(4.3) ∀k ∈ 0, 1, 2,∣∣∣∣∣

(d

dU

)k (Λ(U)− U3

)∣∣∣∣∣ . min

(U5−k, U3−k) .

As a consequence, we have the following Lipschitz bound:

(4.4) |Λ(U)− Λ(V )| . |U − V |(U2 + V 2).

Note that (4.1) with Λ(U) = U3 is exactly the 4D radial wave equation (and thatin this case, (4.3) is trivially satisfied). We have the following small data statement:

Proposition 4.1. Let Λ ∈ C∞(R), odd, such that (4.3) holds. Then there existsε > 0 with the following property. For all (u0, u1) ∈ H(R) such that ‖uL‖(L3L6)(R) ≤ε, the solution u of (4.1) with initial data (u0, u1) is defined for r > R+ |t| and

supt∈R

‖~u(t)− ~uL(t)‖H(R+|t|) + ‖u− uL‖(L3L6)(R) . ‖uL‖3(L3L6)(R),

where uL is the solution of the linear wave equation (3.1) with initial data (u0, u1).

Sketch of proof. Using equation (4.1), the Strichartz estimate outside the cone (2.6)and a straightforward bootstrap argument, one first obtains:

‖u‖(L3L6)(R) . ‖uL‖(L3L6)(R).

The statement then follows from (4.1) and (2.6).


Recall that Q(r) = 2 arctan r is a stationary solution of (1.1). By the above

reduction, we obtain a solution W =√

231r (π − Q) of (4.1) (with Λ is defined by

(4.2)), such that for large r,∣∣∣∣∣W (r) − 2

√2√

3r2

∣∣∣∣∣ .1

r4, r ≥ 1.

Rescaling, we obtain a solution W (r) = λW (λr) of (4.1) such that

(4.5)

∣∣∣∣W (r) − 8

r2

∣∣∣∣ ≤C

r4,

that is, with exactly the same asymptotic behaviour as the stationary solutionW of(1.7) defined in the introduction. We denote by H(0+) the set of functions (u0, u1)defined on (0,∞) such that

∀R > 0, (u0, u1)(R,∞) ∈ H(R),

and define similarly H1(0+). Theorems 2 and 1 reduce to the following rigiditytheorem:

Theorem 4.2. Let Λ ∈ C∞(R) odd, such that (4.3) holds. Assume that the wave

equation (4.1) admits a stationary solution W ∈ H1(0+) such that (4.5) holds. Let(u0, u1) ∈ H(0+), and assume that ∀λ ∈ R, (u0, u1) is not equal to (λW (λ·), 0).Then there exists R > 0 such that the solution u with initial data (u0, u1) is definedon |x| > R+ |t| and

(4.6)∑

±lim

t→±∞

∫

|x|>R+|t||∇t,xu(t, x)|2dx > 0.

Claim 4.3. Theorem 4.2 implies Theorems 2 and 1.

Proof. The wave equation (1.7) is (4.1) with Λ(U) = U3. The ground stateW (x) =(1 + |x|2/8)−1 is a stationary solution of (1.7) that satisfies (4.5). Thus Theorem4.2 implies that for any solution u of (1.7) that is not a stationary solution, thereexists R > 0 such that u is defined for |x| > R+ |t| and

(4.7)∑

±lim

t→±∞

∫

|x|>R+|t|(∂tu(t, x))

2 + |∇xu(t, x)|2dx > 0.

Next, observe that from finite speed of propagation and small data theory, if

∑

±inf

t→±∞

∫

|x|>R+|t|(∂tu(t, x))

2 + |∇xu(t, x)|2dx = 0,

then (4.7) cannot hold, concluding the proof of Theorem 2.Let ψ as in Theorem 4.2. Thus (ψ0, ψ1) ∈ Hℓ,m, which implies (ψ0 − πm,ψ1) ∈

Hℓ−m,0. Let

u(t, r) =

√2

3

ψ(t, r) − πm

r, (u0, u1)(r) = ~u(0, r).

Since (ψ0−πm,ψ1) ∈ H(R) for all R, we have (u0, u1) ∈ H(0+). By the assumptionon (ψ0, ψ1), we have that (u0, u1) is not identically 0, and that it is not equal to


±( √

2√3r(π −Q(λ·)), 0

)for all λ > 0. By the definition ofW , we deduce that (u0, u1)

satisfies the assumptions of Theorem 4.2. Furthermore,

∫ +∞

R+|t|

((∂tu)

2 + (∂ru)2)r3dr =

2

3

∫ +∞

R+|t|

((∂tψ)

2 +(∂rψ − r−1(ψ − πm)

)2)rdr

.

∫

R+|t|

(∂tψ)

2 + (∂rψ)2 +

1

r2(ψ − πm)2

)rdr,

which shows that the conclusion (4.6) of Theorem 4.2 implies

(4.8)∑

±lim

t→±∞

∫ ∞

R+|t|

((∂tψ)

2 + (∂rψ(t, r))2 +

1

r2(ψ(t, r) − πm)2

)rdr > 0.

It remains to prove that (4.8) implies the conclusion (1.13) of Theorem 1. This iselementary, but requires some preliminaries on equation (1.1) and we postpone theproof to Section 7, after the proof of Claim 7.1.

The remainder of this section and the next section are dedicated to the proofof Theorem 4.2. Until the end of Section 5, Λ is an odd, smooth function on R

satisfying (4.3) and such that there exists a stationary solution W ∈ H1 of (4.1)satisfying (4.5).

4.2. Compactly supported perturbation of a stationary solution. We firstprove:

Proposition 4.4. Let (u0, u1) ∈ H(0+), radial. Assume that there exists R0 > 0and λ ∈ R such that

∀r > R0, (u0, u1)(r) = (λW (λr), 0) ,

and that the preceding equality does not hold for all r > 0. Then there exists R > 0such that the solution u of (4.1) with initial data (u0, u1) is defined for r > R+ |t|and ∑

±lim

t→±∞

∫

|x|>R+|t||∇t,xu(t, x)|2dx > 0.

Proof. Rescaling, we may assume that λ = 1. Let

h = u−W, (h0, h1) = (u0 −W,u1) = ~ht=0.

Then the equation (4.1) satisfied by u is equivalent to

(4.9) ∂2t h−∆h =1

r3(Λ(r(W + h))− Λ(rW )) .

Furthermore by (4.3)

(4.10)1

r3|Λ(r(W + h))− Λ(rW )| . |h|(h2 +W 2).

We define

ρ = inf

σ > 0,

∫

|x|>σ|∇h0|2 + h21 = 0

By the assumptions of the proposition, 0 < ρ <∞. We fix a small ε > 0 such that

(4.11) 0 < ‖(h0, h1)‖H(ρ−ε) ≪ 1.


Using the Strichartz estimates (2.6), and standard small data theory, we obtainthat (4.9) has a solution h with initial data (h0, h1) defined for |x| > ρ− ε+ |t|,and that

supt∈R

‖~h‖H(ρ−ε+|t|) . ε0.

By the radial Sobolev embedding and the property (4.5) ofW , we obtain that thereexists a constant C > 0 such that

∀t ∈ R, ∀r > ρ− ε+ |t|, h2 +W 2 ≤ C

r2.

Combined with (4.10), and taking a smaller ε if necessary, we see that Proposition4.7 in [23] implies that the following holds for all t > 0 or for all t < 0:

(4.12)

∫

|x|≥ρ−ε+|t||∇t,xh|2dx ≥ 1

8

∫

|x|≥ρ−ε

(|∇h0|2 + h21

)dx > 0,

which concludes the proof, since W ∈ H1(0+).

Proposition 4.4 reduces the proof of Theorem 4.2 to

Proposition 4.5. Let R0 > 0 and u be a radial solution of (4.1) for |x| > |t|+R0,with initial data in H(R0). Assume

∑

±lim

t→±∞

∫

|x|>R0+|t||∇t,xu(t, x)|2dx = 0.

Then there exists R ≥ R0 such that one of the following holds:

(4.13) ∀r > R, (u0, u1)(r) = (0, 0) ,

or

(4.14) ∃λ > 0, ι ∈ ±1, ∀r > R, (u0, u1)(r) = ι (λW (λr), 0) .

The proof of Proposition 4.5 is divided into §4.3, and Section 5.

4.3. Constant sign solutions. In this section, we prove

Proposition 4.6. Let R0 > 0 and u be a radial solution of (4.1) defined for|x| > |t|+R0 with

~ut=0 = (u0, u1) ∈ H(R0).

Assume

(4.15)∑

±lim

t→±∞

∫

|x|>|t|+R0

|∇t,xu(t, x)|2dx = 0.

Then one of the following holds:

∃t0 ∈ R, ∃R > R0, ∀r > |t0|+R, u(t0, r) = 0(4.16)

∃µ ∈ R \ 0, ∃R > R0, ∀r > R, (u0, u1)(r) = (µW (µr), 0) .(4.17)

Proposition 4.6 reduces the proof of Proposition 4.5 to the classification of solu-tions satisfying (4.15) and (4.16), which we will carry out in the next section. Notethat (4.16) is not sufficient to conclude immediately that u is identically 0 outsidea wave cone, since ∂tu(t0, r) does not have to vanish for r > |t0|+R.

We split the proof of Proposition 4.6 in a few lemmas.


Lemma 4.7. The exists a small ε0 > 0 with the following property. Let u be aradial solution of (4.1) on |x| > R+ |t|, where R ≥ 0. Assume

∫

|x|>R|∇t,xu(0, x)|2dx ≤ ε20(4.18)

∑

±lim

t→±∞

∫

|x|>R+|t||∇t,xu(t, x)|2dx = 0.(4.19)

Then

∀ρ ≥ R, |u0(ρ)| ≈1

ρ‖u‖L4(ρ) ≈

1

ρ‖u0‖H1(ρ)(4.20)

∀r ≥ ρ ≥ R, |u0(r)| ≤ 2(ρr

) 116 |u0(ρ)|.(4.21)

Proof. Let

u+(t) =u(t) + u(−t)

2, u−(t) =

u(t)− u(−t)2

.

Then

(4.22)

∣∣∂2t u+ −∆u+∣∣ . u+(u

2+ + u2−)

~u+t=0 = (u0, 0),

where we have used that, Λ being odd,

|Λ(ru(t)) + Λ(ru(−t))| = |Λ(ru(t))− Λ(−ru(−t))|. r3|u(t) + u(−t)|

(u2(t) + u2(−t)

),

by (4.4). By the small data theory for (4.1) (see Proposition 4.1), the condition(4.18) implies ‖u‖(L3L6)(R) . ε20. Combining with the equation (4.22) and Strichartzestimates, we deduce that for all ρ ≥ R,

(4.23) ‖u+‖(L3L6)(ρ) + supt∈R

‖~u+(t)‖H(ρ+|t|) . ‖u0‖H1(ρ).

Using the lower bound for the exterior energy (Lemma 3.9) and equation (4.22)again, we obtain

(4.24)∥∥π⊥

ρ u0∥∥H1(ρ)

≤ C(‖u3+‖(L1L2)(ρ) + ‖u2−u+‖(L1L2)(ρ)

)≤ Cε20‖u0‖H1(ρ).

Since

π⊥ρ u0(r) = u0(r) −

ρ2

r2u0(ρ),

∥∥∥∥1

r2

∥∥∥∥H1(ρ)

=1√2ρ,

we deduce from (4.24), taking ε0 small enough,

(4.25) ‖u0‖H1(ρ) ≤ Cρ|u0(ρ)|.Next, we deduce from (4.24), (4.25) and Sobolev embedding that for ρ ≥ R,

(4.26)

∥∥∥∥u0 −ρ2

r2u0(ρ)

∥∥∥∥L4(ρ)

≤ Cε20ρ|u0(ρ)|.

This implies, noting that∥∥ 1r2

∥∥L4(ρ)

= 1√2ρ,

(4.27)

∣∣∣∣‖u0‖L4(ρ) −ρ|u0(ρ)|√

2

∣∣∣∣ ≤ Cε20ρ|u0(ρ)|.


Combining with (4.25) and Sobolev embedding, we obtain (4.20). We also obtain,from (4.27),

(4.28) ‖u0‖4L4(ρ) ≤ρ4

4−|u0(ρ)|4,

where 4− is a positive constant, smaller than 4, that can be chosen arbitrarily closeto 4 provided ε0 is small enough. Letting

f(ρ) =

∫ +∞

ρ

|u0(r)|4r3dr = ‖u0‖4L4(ρ),

we can write (4.28) as

(4.29) f(ρ) ≤ − ρ

4−f ′(ρ), ρ ≥ R,

that is

(4.30)f ′(ρ)

f(ρ)≤ −4−

ρ, ρ ≥ R.

Integrating, we deduce that for r ≥ ρ ≥ R,

(4.31)f(r)

f(ρ)≤ ρ4

−

r4−,

that is

‖u0‖4L4(r) ≤(ρr

)4−‖u0‖4L4(ρ).

Combining with (4.27), we deduce (4.21).

Lemma 4.8. Let u, R and ε0 be as in Lemma 4.7 and assume that for all t, u(t)is not identically 0 on [R + |t|,+∞). Then u never vanishes and has a constantsign on |x| > R+ |t|. Furthermore, for all t ∈ R, there exists ℓ(t) 6= 0 such that

(4.32) ∀r > R+ |t|,∣∣r2u(t, r)− ℓ(t)

∣∣ ≤ Cr4|u(t, r)|3.Remark 4.9. Recall that by the radial Sobolev inequality, for all t, limr→∞ ru(t, r) =0. Thus (4.32), dividing by r2u(t, r), implies

(4.33) limr→∞

r2u(t, r) = ℓ(t).

Proof. Taking a smaller ε0 > 0 if necessary, we see by finite speed of propagationand the small data theory that for every time t0, the function (t, r) 7→ u(t +t0, r) satisfies the assumptions of Lemma 4.7 with R replaced by R + |t0|. By theconclusion of this lemma and the assumption that for all t, u(t, r) is not identically0 for r > R + |t|, we obtain that u does not vanish for r > R + |t|. Since u is acontinuous function, we deduce that it has a constant sign for r > R+ |t|. We willassume to fix ideas u(t, r) > 0 for all r > |t|+R. By (4.21),

(4.34) ∀r > ρ ≥ R, 0 < u0(r) ≤ 2(ρr

) 116

u0(ρ).

Next, we observe that for all (t, r) with r > t+R,

|u−(t, r)| =1

2|u(t)− u(−t)| ≤ u(t) + u(−t)

2= u+(t),


where we have used that u is positive. The first inequality in (4.24), together with(4.23) implies, for ρ ≥ R,

(4.35)∥∥π⊥

ρ u0∥∥H1(ρ)

≤ C‖u3+‖(L1L2)(ρ) ≤ C‖u0‖3H1(ρ).

Using (4.20) and Sobolev inequalities as in the proof of Lemma 4.7, we deduce

(4.36)

∣∣∣∣‖u0‖L4(ρ) −ρ√2u0(ρ)

∣∣∣∣ ≤ Cρ3u0(ρ)3, ρ ≥ R.

Letting as before f(ρ) = ‖u0‖4L4(ρ), we deduce, for r ≥ ρ,

∣∣∣f(r) + r

4f ′(r)

∣∣∣ ≤ Cr6|u0(r)|6 ≤ C

r5ρ11u0(ρ)

6,

that is ∣∣∣∣d

dr

(r4f(r)

)∣∣∣∣ ≤C

r2ρ11u0(ρ)

6.

As a consequence, r4f(r) has a limit ℓ′ ≥ 0 as r → ∞ and∣∣ℓ′ − ρ4f(ρ)

∣∣ ≤ Cρ10u0(ρ)6, ρ ≥ R.

Recall that by (4.20), ρu0(ρ) ≈ f(ρ)1/4 goes to 0 as ρ→ ∞. By (4.36),∣∣∣∣f(ρ)−

ρ4

4u0(ρ)

4

∣∣∣∣ ≤ Cρ6u0(ρ)6,

and thus ∣∣∣∣4ℓ′

ρ8− u0(ρ)

4

∣∣∣∣ ≤ Cρ2|u0(ρ)|6.

This implies ℓ′ > 0 since u0(ρ) 6= 0 and ρu0(ρ) is small for ρ large. Using the bound

|A−B| ≤ |A4−B4|A3 , A,B > 0, we also deduce

∣∣∣∣∣

√2ℓ′

14

ρ2− u0(ρ)

∣∣∣∣∣ ≤ Cρ2|u0(ρ)|3, ρ ≥ R.

This yields (4.32) for t = 0. Applying this to the solution (t, r) 7→ u(t+ t0, r) whichsatisfies the same assumptions as u with R replaced by R + |t|, we deduce (4.32)for all t ∈ R.

Lemma 4.10. Let u, R, ε0 be as in Lemmas 4.7 and 4.8. Then ℓ is independentof t and

(4.37) ∀t ∈ R, ∀r ≥ R+ |t|, |u(t, r)| + 1

r‖u(t)‖H(R) .

|ℓ|r2.

Proof. We prove that ℓ is constant by contradiction. Let (τ, t) ∈ R2, and assume

ℓ(t) > ℓ(τ). Since ℓ does not vanish, and u has a constant sign, we can also assumewithout loss of generality (using Remark 4.9), ℓ(τ) > 0.

Fix a small δ > 0, to be specified. By (4.33), we have that for large r

0 < (1− δ)r2u(t, r) ≤ ℓ(t) ≤ (1 + δ)r2u(t, r),

and similarly0 < (1− δ)r2u(τ, r) ≤ ℓ(τ) ≤ (1 + δ)r2u(τ, r).

As a consequence,

r2(u(t, r)− u(τ, r)) ≥ 1

1 + δℓ(t)− 1

1− δℓ(τ) = c > 0


for δ > 0 small enough. Thus for large r,

(4.38) u(t, r)− u(τ, r) ≥ c

r2.

Since u is a finite energy solution of (4.1), on r > R+ |t|, we have

u(t)− u(τ) =

∫ t

τ

∂tu(s)ds ∈ L2(ρ)

for large ρ, contradicting (4.38). Thus ℓ is independent of t.Going back to (4.32), we obtain

∀ρ > R+ |t|, |ρ2u(t, ρ)| − Cρ4|u(t, ρ)|3 ≤ ℓ.

Since by (4.20)

ρ|u(t, ρ)| ≈ ‖~u(t)‖H(ρ) . ε0, ρ ≥ R+ |t|,we deduce (4.37) using that ε0 is small.

Proof of Proposition 4.6. We let u be as in the proposition, and choose R large, sothat

(4.39)

∫

|x|>R|∇u0|2 + |u1|2dx ≤ ε0,

where ε0 is small as in the preceding lemmas. We assume that (4.16) does not holdso that (taking a larger R if necessary), u satisfies the assumptions of Lemmas 4.8and 4.10. We assume to fix ideas that u is positive. Let ℓ be as in the two Lemmas,so that by (4.32), (4.37),

(4.40) ∀t ∈ R, ∀r ≥ R+ |t|,∣∣∣∣u(t, r)−

ℓ

r2

∣∣∣∣ ≤ Cℓ3

r4,

for some absolute constant C > 0. Rescaling, we may assume that ℓ = 8, so that,by the assumption (4.5) on W ,

(4.41) ∀t ∈ R, ∀r ≥ R+ |t|, |u(t, r)−W (r)| ≤ K

r4,

for some absolute constant K > 0. Let, for ρ ≥ R,

M(ρ) = supt0∈R

σ≥ρ+|t0|

σ2|u(t0, σ)−W (σ)|.

By (4.41),

(4.42) ∀ρ ≥ R, M(ρ) ≤ K

ρ2.

We will show that for R′ ≥ R large enough,

(4.43) ∀ρ ≥ R′, M(2ρ) ≥ 1

2M(ρ).

Note that (4.42) and (4.43) would imply, for ρ > R′ and k ∈ N,

M(ρ) ≤ 2kM(2kρ) ≤ K1

2kρ2

and thus M(ρ) = 0. Thus u(t, r) =W (r) for r ≥ R′ + |t|, as desired. It remains toprove (4.43).


We fix t0 ∈ R and ρ ≥ R′. We let

u(t) =W + h(t), h±(t) =h(t)± h(2t0 − t)

2.

Using the equation −∆W = 1r3Λ(rW ), and the Taylor expansion

Λ(w + ε) = Λ(w) + Λ′(w)ε + Λ2(w, ε)ε2,

where

Λ2(w, ε) =

∫ 1

0

Λ′′(w + θε)(1 − θ)dθ, |Λ2(w, ε)| . |w|+ |ε|,

by (4.3). As a consequence, denoting ht0(t) = h(2t0 − t),

(4.44) ∂2t h+ −∆h+

=1

2r3

Λ′(rW )(rh+) + Λ2(rW, rh)(rh)

2 + Λ2(rW, rht0 )(rht0)2,

with the initial data ~h+t=0 = (u(t0)−W, 0).We have

(4.45) |Λ′(rW )| . r2W 2, |Λ2(rW, rh)| . r|W | + r|h|.Let σ ≥ ρ+ |t0|. We have

r ≥ σ + |t− t0| =⇒ r ≥ max(ρ+ |t|, ρ+ |2t0 − t|),and thus

(4.46) r ≥ σ + |t− t0| =⇒ |h+(t, r)|+ |ht0(t, r)| + |h(t, r)| . M(ρ)

r2.

Combining (4.44), (4.45), (4.46) and the lower bound for the exterior energy (Lemma3.9), we obtain

(4.47)∥∥π⊥

σ h+(t0)∥∥H1(σ)

.∑

k=1,2,3

‖W‖3−k(L3L6)(σ+|t−t0|)

∫

R

(∫ ∞

σ+|t−t0|

M(ρ)6

r9dr

) 12

dt

k3

.

Let ε(σ) = ‖W‖2(L3L6)(σ+|t|) = ‖W‖2(L3L6)(σ+|t−t0|). Note that ε(σ) is finite, inde-

pendent of t0, and goes to 0 as σ goes to infinity. The inequality (4.47) implies

(4.48)

∥∥∥∥h+(t0, r)−σ2

r2h+(t0, σ)

∥∥∥∥H1

r (σ)

≤ Cε(σ)

σM(ρ) +

CM(ρ)3

σ3, σ ≥ ρ+ |t0|.

By the radial Sobolev inequality,∣∣∣∣h+(t0, 2σ)−

σ2

(2σ)2h+(t0, σ)

∣∣∣∣ ≤Cε(σ)

σ2M(ρ) +

CM(ρ)3

σ4.

As a consequence

1

4|h+(t0, σ)| ≤

Cε(σ)

σ2M(ρ) +

CM(ρ)3

σ4+ |h+(t0, 2σ)|

≤ Cε(σ)

σ2M(ρ) +

CM(ρ)3

σ4+M(2σ)

4σ2,

where we have used (4.46) and the fact that 2σ ≥ 2ρ + 2|t0| ≥ 2ρ + |t0|, so that|h+(t0, 2σ)| ≤ 1

(2σ)2M(2σ).


Going back to the definition of h+, we deduce

σ2|u(t0, σ)−W (σ)| ≤M(2ρ) + Cε(ρ)M(ρ),

where ε(ρ) = ε(ρ)+ M(ρ)2

ρ2 tends to 0 as ρ tends to infinity. This holds for all t0 ∈ R

and σ ≥ ρ+ |t0|. HenceM(ρ) ≤M(2ρ) + Cε(ρ)M(ρ).

Taking ρ large enough, we obtain (4.43), concluding the proof.

5. Rigidity theorem, part II: odd solutions

In this section, we conclude the proof of Proposition 4.5, and thus of the rigiditytheorem (Theorem 4.2) by the following proposition:

Proposition 5.1. Let R > 0 and u a radial solution of (4.1) on r ≥ R + |t|,such that

(u, ∂tu)t=0 = (0, u1), u1 ∈ L2(R)(5.1)∑

±lim

t→±∞

∫

|x|>R+|t||∇t,xu(t, x)|2dx = 0.(5.2)

Then

∀t ∈ R, ∀r > R+ |t|, u(t, r) = 0.

We first check that Proposition 4.5 (and thus Theorem 4.2) follows from Propo-sitions 4.6 and 5.1. Let u be as in Proposition 4.5. Then by Proposition 4.6, oneof the properties (4.16) or (4.17) holds. We must prove that on of the conclu-sions (4.13) or (4.14) of Proposition 4.5 hold. Property (4.17) implies immediately(4.14). If (4.16) holds, we can assume, translating in time and taking a largerR, that u0(0, r) = 0 for all r > R. Thus u satisfies (after a time translation),the assumptions of Proposition 5.1, and the conclusion of this proposition implies(4.13).

We will prove Proposition 5.1 in §5.3 after giving two preliminary results. In §5.1,we prove a gain of regularity for a solution u of (4.1) satisfying the assumptions ofthis proposition. More precisely, we prove that for such a solution, (∂tu, ∂

2t u)(t) is

in H(R′+ |t|) for all t (where R′ is large) and that ∂tu is non-radiative, in the sensethat

(5.3)∑

±lim

t→±∞

∫

|x|>R′+|t||∇t,x∂tu(t, x)|2dx = 0.

In §5.2, we give an explicit, radial, smooth approximate solution a of (4.1) on|x| > |t|, which satisfies the non-radiative property (5.3) and whose initial data isof the form(0, a1(r)), where a1 /∈ L2(R) for R > 0 (indeed a1(r) equals 1

r2(log r)1/2

up to a multiplicative constant).The proof of Proposition 5.1 in §5.3 consists in showing that the initial data

u1(r) of a solution as in Proposition 5.1 is of the same order as or greater thana1(r) for large r, contradicting the fact that u1 ∈ L2(R). This is done by carefullycomparing u1 with an appropriate rescaling of a1, using again the improved lowerenergy bound of Lemma 3.9.


5.1. Gain of regularity. In this subsection we prove:

Lemma 5.2. Let u be as in Proposition 5.1. Then there exists a large constantR′ ≥ R such that

ru1 ∈ H1(R′)(5.4)

∀ρ ≥ R′,1

ρ‖u1‖H1(ρ) ≈ |u1(ρ)|(5.5)

Furthermore, (∂tu, ∂2t u) is the restriction to r > R′ + |t| of a C0(R, H1 × L2)

function and

(5.6)∑

±lim

t→±∞

∫

|x|>R′+|t||∇t,x∂tu|2dx = 0.

Proof of Lemma 5.2. Step 1: gain of decay. We let, for ρ ≥ R+ |T |, T ∈ R,

(5.7) η(T, ρ) = max0≤t≤T

‖∂tu(t)‖L2(ρ).

In this step, we prove that if R′ is large enough and ρ ≥ R′ + |T |, then r∇u(T ) ∈L2(2ρ) and

(5.8) ‖r∇u(T )‖L2(2ρ) .

√∫ ∞

ρ

|u(T, r)|2r3dr . Tη(T, ρ).

Indeed, using that u(t, r) =∫ t0 ∂tu(τ, r)dτ , we have, for all t ≥ 0 and ρ ≥ R+ |t|,

(5.9) ‖u(t)‖L2(ρ) ≤∫ t

0

‖∂tu(τ)‖L2(ρ)dτ ≤ |t|η(t, ρ).

By Lemma 4.7, for ρ > R′ + |T |, R′ large,

(5.10) |u(T, ρ)| ≈ 1

ρ‖∇u(T )‖L2(ρ).

Integrating, we obtain, for σ > R′ + |T |,∫ +∞

σ

|u(T, ρ)|2ρ3dρ ≈∫ +∞

σ

∫ +∞

ρ

|∂ru(T, r)|2r3drρ dρ,

and thus, by Tonelli’s Theorem,∫ ∞

σ

|u(T, ρ)|2ρ3dρ ≈∫ +∞

σ

(∂ru(T, r)|2(r5 − σ2r3)dr.

Combining with (5.9), we obtain, for σ > R′ + |t|,∫ ∞

2σ

|∂ru(T, r)|2r5dr .∫ ∞

σ

|u(T, r)|2r3dr . T 2η(T, σ)2,

which is exactly (5.8).

Step 2. Gain of regularity. In this step, we prove (5.4) and (5.5). By (5.8), forT ∈ [0, 1], ρ > R′ + |T |,

(5.11)

∫ ∞

2ρ

r5∣∣∣∣1

T∂ru(T, r)

∣∣∣∣2

dr . η(T, ρ)2 . η(1, R′)2.


As a consequence, there exist v1 ∈ L2([2R′,∞), r5dr) and a sequence Tnn goingto 0 as n goes to infinity such that

(5.12)1

Tn∂ru(Tn) −−−−

n→∞v1 weakly in L2

([2R′,∞), r5dr

).

We have1

Tnu(Tn, 2ρ) = − 1

Tn

∫ ∞

2ρ

∂ru(Tn, r)dr,

and thus, using the weak convergence of ∂ru(Tn),

(5.13) limn→∞

1

Tnu(Tn, 2ρ) = −

∫ ∞

2ρ

v1(r)dr,

for all ρ ≥ R′. Next, we observe that by (5.11) and a radial Sobolev inequality,

1

T 2n

|u(Tn, 2ρ)|2 .1

ρ4η(1, R′)2

for large n. This yields by the dominated convergence theorem that the convergence(5.13) also holds locally in L2(R′,∞). Since

limt→0+

1

tu(t) → u1 in L2(2R),

we deduce, by uniqueness of the limit that for ρ > R′

(5.14) u1(2ρ) = −∫ ∞

2ρ

v1(r)dr,

where∫∞R′ (v1(r))

2r5dr <∞. Hence (5.4) (taking a larger R′).The estimate (5.5) follows from (5.10), dividing by T and letting T go to 0.

Step 3. Proof that ∂tu is non-radiative. In this step we prove (5.6). We fix a smallconstant ε0 > 0 to be specified. Taking a larger R′ if necessary, we can assume‖r∂ru1‖L2 + ‖u1‖H1(R′) ≤ ε0. We define u1 by

u1(r) = u1(R′) if r < R′, u1(r) = u1(r) if r ≥ R′.

Using that by the radial Sobolev embedding

|u1(R′)| . 1

(R′)2‖r∂ru1‖L2(R′) .

ε0(R′)2

,

we obtain

(5.15) ‖u1‖H1 . ε0.

Let u be the solution of (4.1) with initial data (0, u1) at t = 0. Since (0, u1) ∈(H2 ∩ H1)×H1), a standard persistence of regularity argument yields

−→∂tu ∈ C0

(R, (H2 ∩ H1)×H1

).

We observe that ∂tu satisfies the equation

(5.16) ∂2t (∂tu)−∆∂tu =1

r2Λ′(ru)∂tu,

with initial data

(5.17)−→∂tut=0 = (u1, 0) ∈ H1 × L2.


Noting that

1

r2|Λ′(ru)| . u2,

∥∥u2∥∥(L3/2L3)

= ‖u‖2(L3L6) . ε0,

we deduce by Strichartz estimates, and taking ε0 small enough,

(5.18) ‖∂tu‖(L3L6) . ‖u1‖H1 .

As a consequence, 1r2Λ

′(ru)∂tu ∈ (L1L2). By finite speed of propagation,

(5.19) r > R+ |t| =⇒ u(t, r) = u(t, r).

Thus the assumption (5.2) implies

∑

±lim

t→±∞

∫

|x|>R+|t||∇t,xu(t, x)|2dx = 0.

Combining with Proposition A.1 in the appendix, we obtain (5.6) for u and thus,by (5.19), for u.

5.2. Approximate non-radiative solution.

Lemma 5.3. There exists a C∞ function a(t, r) defined for t ∈ R, r > 2 such that∂ta ∈ C0(R,H(1)) and

∀t ∈ R, ∀r > 2, a(t, r) = −a(−t, r)(5.20)

∀r > 2, a1(r) := ∂ta(0, r) =2√

3r2(log r)1/2(5.21)

∀t ∈ R, ∀r > max(|t|, 2), |a(t, r)|+ |t∂ta(t, r)| .|t|

r2(log r)1/2(5.22)

∑

±lim

t→±∞

∫

|x|>|t||∇t,x∂ta(t, x)|2dx = 0(5.23)

and b = ∂2t a−∆a− 1r3Λ(ra) satisfies

(5.24) ∀t ∈ R, ∀r > max(|t|, 2), |b(t, r)| + |t∂tb(t, r)| .|t|

r4(log r)5/2.

Proof. Let

a(t, r) =2t√

3r2(log r)1/2− t3

3√3r4(log r)3/2

.

It is easy to see that a satisfies (5.20), (5.21), (5.22) and (5.23). Using the formula

(5.25) ∆

(1

rα(log r)β

)=

1

rα+2

(α(α − 2)

(log r)β+

(2α− 2)β

(log r)β+1+

β(β + 1)

(log r)β+2

)

we can prove that b satisfies (5.24). Indeed by (4.3),∣∣∣∣a

3 − 1

r3Λ(ra)

∣∣∣∣ . r2|a|5 .|t|

r4(log r)5/2, r > max(|t|, 2),

∣∣∣∣∂t(a3 − 1

r3Λ(ra)

)∣∣∣∣ . r2a4|∂ta| .1

r4(log r)5/2, r > max(|t|, 2),


and ∂2t a−∆a− a3 is a linear combination of:

t

r4(log r)5/2,

t3

r6(log r)5/2,

t3

r6(log r)7/2,

t5

r8(log r)5/2

t7

r10(log r)7/2,

t9

r12(log r)9/2.

The key point is the double (approximate) cancellation given by:

∂2t

(t3

3√3r4(log r)3/2

)= −∆

(2t√

3r2(log r)1/2

)+ l.o.t

and

∆

(t3

3√3r4(log r)3/2

)=

(2t√

3r2(log r)1/2

)3

+ l.o.t.,

where l.o.t. is dominated by t/r4(log r)5/2 for r ≥ |t|. The nonlinearity appears inthe last line: the approximate solution a is not the perturbation of a linear solution,but a fully nonlinear object.

Remark 5.4. An explicit computation shows that

11|x|>1+|t|a ∈ L3L6.

However ∂ta(0) barely fails to be in L2. This fact will be crucial in the proof ofProposition 5.1.

5.3. Proof of the rigidity result. We are now in position to prove Proposition5.1.

Step 1. Preliminaries. We argue by contradiction, assuming that u1 is notidentically 0 for r > R. By (5.5) in Lemma 5.2, u1 does not change sign, and wecan assume that u1(r) > 0 for large r. Furthermore, by (5.4) in Lemma 5.2, andthe radial Sobolev inequality,

(5.26) limr→∞

r2u1(r) = 0.

We fix ρ ≫ 1 and choose λ = λ(ρ) such that u1(ρ) = ∂taλ(0, ρ), where a is thenon-radiative approximate solution given by Lemma 5.3 and

aλ(t, r) = λa(λt, λr).

In other words,

(5.27) u1(ρ) =2√

3ρ2 log(ρλ)1/2.

Note that the equation (5.27) always has a unique solution λ(ρ) > 1/ρ, and thatby (5.26) and (5.27),

(5.28) limρ→∞

ρλ(ρ) = ∞.

We also note that by (5.22),

(5.29) r > max(|t|, 2/λ) =⇒ |aλ(t, r)| + |t∂taλ(t, r)| .|t|

r2 log1/2(rλ).

We have

(5.30) ∂2t aλ −∆aλ −1

r3Λ(raλ) = b[λ],


where b[λ](t, r) = λ3b(λt, λr) satisfies, by (5.24),

(5.31) r > max(|t|, 2/λ) =⇒∣∣b[λ](t, r)

∣∣ +∣∣t∂tb[λ](t, r)

∣∣ . |t|r4 log(λr)5/2

.

Let w = ∂taλ − ∂tu, so that w satisfies the equation

(5.32) ∂2tw −∆w =1

r2Λ′(raλ)∂taλ −

1

r2Λ′(ru)∂tu+ ∂tb[λ]

with initial data w(0) = (∂taλ(0)− u1, 0). By explicit computation, using (5.31),

(5.33) ‖∂tb[λ]‖(L1L2)(ρ) .1

ρ(log(λρ))5/2.

Step 2. Channels of energy. Fix a small ε0 > 0, to be specified later. In this stepwe prove that if ρ is chosen large enough,

(5.34) ‖u1 − ∂taλ(0)‖H1(ρ) .1

ρ(log(λρ))5/2.

By the equation (5.32), Strichartz inequalities and finite speed of propagation,

(5.35) ‖w‖(L2L8)(ρ) + ‖w‖(L3L6)(ρ) + supt∈R

‖~w(t)‖H(ρ+|t|)

. ‖w(0)‖H1(ρ) +

∥∥∥∥1


1


∥∥∥∥(L1L2)(ρ)

.

By the exterior energy lower bound of Lemma 3.9 (which also applies to linearsolutions outside a wave cone by finite speed of propagation)

(5.36) ‖π⊥ρ w(0)‖H1(ρ) .

∥∥∥∥1


1


∥∥∥∥(L1L2)(ρ)

.

Since by the choice of λ, w(0, ρ) = 0, we obtain w(0) = π⊥ρ w(0). Combining with

(5.33), (5.35) and (5.36), we deduce

(5.37) ‖w‖(L2L8)(ρ) + ‖w‖(L3L6)(ρ) + ‖w0‖H1(ρ)

.

∥∥∥∥1


1

r2Λ′(ru)∂tu

∥∥∥∥(L1L2)(ρ)

+1

ρ(log(λρ))5/2,

and we are left with bounding from above the first term in the second line of (5.37).We have

(5.38)1


1

r2Λ′(ru)∂tu =

1

r2Λ′(ru)w +

1

r2(Λ′(raλ)− Λ′(ru)) ∂taλ.

By the bound |Λ′(U)| . U2 and Holder’s inequality,

(5.39)

∥∥∥∥1

r2Λ′(ru)w

∥∥∥∥(L1L2)(ρ)

. ‖w‖(L3L6)(ρ)‖u‖2(L3L6)(ρ) ≤ε03‖w‖(L3L6)(ρ)

for large ρ. Next, we have, by the bound |Λ′′(U)| . |U | which implies |Λ′(U) −Λ′(V )| . |U − V | (|U |+ |V |),

(5.40)1

r2

∣∣∣∣Λ′(raλ)−

1

r2Λ′(ru)

∣∣∣∣ . |u||u− aλ|+ |aλ||u− aλ|.


We will bound separately ‖u(u− aλ)∂taλ‖(L1L2)(ρ) and ‖aλ(u− aλ)∂taλ‖(L1L2)(ρ);

By Holder inequality,

‖u∂taλ(u − aλ)‖(L1L2)(ρ) . ‖u‖(L2L8)(ρ)‖∂taλ(u− aλ)‖(L2L8/3)(ρ).

Furthermore, using the bound (5.29), and the fact that w = ∂t(aλ − u), we obtainfor large ρ

|∂taλ(u− aλ)| .1

r2 log(rλ)1/2

∣∣∣∣∫ t

0

w(τ, r)dτ

∣∣∣∣ .ε0r2

∣∣∣∣∫ t

0

w(τ, r)dτ

∣∣∣∣ ,

where we have used (5.28). By Lemma C.1 in Appendix C,

‖∂taλ(u− aλ)‖(L2L8/3)(ρ) . ε0‖w‖(L2L8)(R),

and thus

(5.41) ‖u∂taλ(u− aλ)‖(L1L2)(ρ) . ε0‖w‖(L2L8)(ρ).

Using again the bound (5.29) on aλ and ∂taλ, we obtain

|aλ∂taλ(u − aλ)| .|t|

r4 log(λr)

∣∣∣∣∫ t

0

w(τ, r)dτ

∣∣∣∣ .

By Lemma C.2 in Appendix C,

(5.42) ‖aλ∂taλ(u− aλ)‖(L1L2)(ρ) .1

(log(λρ))1/3‖w‖(L3L6)(ρ).

Combining (5.37), (5.38), (5.39), (5.41) and (5.42), we deduce that for large ρ,

(5.43) ‖w‖(L2L8)(ρ) + ‖w‖(L3L6)(ρ) + ‖w0‖H1(ρ) .

ε0(‖w‖(L2L8)(ρ) + ‖w‖(L3L6)(ρ)

)+

1

ρ(log(λρ))5/2.

This implies (5.34) if ε0 is small enough.

Step 3. Differential inequality. In this step, we deduce from (5.34) the followinginequality for large ρ:

(5.44)

∣∣∣∣‖u1‖4L4(ρ) −ρ4u1(ρ)

4

4

∣∣∣∣ . ρ8u1(ρ)6.

Recall that ρ2u1(ρ) is small when ρ is large (see (5.26)), so that ρ8u1(ρ)6 ≪

ρ4u1(ρ)4. The inequality (5.44) can be seen as an approximate differential equation

on the L4 norm of u. We will deduce a contradiction from this inequality in thenext step.

By (5.34) and the critical Sobolev inequality, we obtain, for ρ large

(5.45) ‖u1 − ∂taλ(0)‖L4(ρ) .1

ρ(log(λρ))5/2.

Using the explicit value of ∂taλ(0), we have

‖∂taλ(0)‖4L4(ρ) =

(2√3

)4 ∫ ∞

ρ

dr

r5 log2(rλ),

which yields, after an integration by parts,

(5.46) ‖∂taλ‖4L4(ρ) =4

9ρ4 log2(ρλ)+O

(1

ρ4 log3(ρλ)

).


By (5.45),∣∣‖u1‖L4(ρ) − ‖∂taλ(0)‖L4(ρ)

∣∣ . 1ρ log(λρ)5/2

. Thus

‖u1‖L4(ρ) ∼√

2

3

1

ρ (log(ρλ))1/2

and we deduce from (5.45)

(5.47)

∣∣∣∣‖u1‖4L4(ρ) −4

9ρ4 log2(ρλ)

∣∣∣∣ .1

ρ4 log3(ρλ).

Going back to the definition of λ (see (5.27)), we deduce (5.44)).

Step 4. Lower bound on u1 and contradiction.In this step, we deduce from (5.44) and the fact that ρ2u1(ρ) goes to 0 that for

some constant C > 0,

(5.48) u1(ρ) ≥1

Cρ2(log ρ)1/2, ρ→ ∞,

a contradiction since ρ 7→ 1ρ2(log ρ)1/2

is not in L2(r3dr).

Let

f(ρ) = ‖u1‖4L4(ρ).

Since by (5.44), f(ρ) ∼ ρ4u1(ρ)4

4 , we have

(5.49) limρ→∞

ρ4f(ρ) = 0.

Also, we can rewrite (5.44) as

(5.50)∣∣∣f(ρ) + ρ

4f ′(ρ)

∣∣∣ . f(ρ)3/2ρ2.

Next, we let g(ρ) = ρ4f(ρ) and note that by (5.49),

(5.51) limρ→∞

g(ρ) = 0.

Since f ′(ρ) = − 4ρ5 g(ρ) +

1ρ4 g

′(ρ), the estimate (5.50) reads

(5.52)

∣∣∣∣g′(ρ)

g(ρ)3/2

∣∣∣∣ .1

ρ.

Integrating between ρ0 and ρ > ρ0, where ρ0 is large, we deduce that for large ρ,

1

g(ρ)1/2≤ C log ρ,

where C is a large positive constant (depending on g). This implies that for largeρ

g(ρ) ≥ 1

C(log ρ)2.

Since g(ρ) = ρ4f(ρ) and by (5.44), f(ρ) ≈ ρ4u1(ρ)4, we deduce (5.48) as announced.

6. Soliton resolution for the wave equation

In this Section we prove Theorem 4. We start with some preliminaries on profiledecomposition for the wave equations (3.1) and (1.7).


6.1. Profile decomposition. Let(u0,n, u1,n)

nbe a bounded sequence of radial

functions in H. We say that it admits a profile decomposition if for all j ≥ 1, thereexist a solution U jL to the free wave equation with initial data in H and sequencesof parameters λj,nn ∈ (0,∞)N, tj,nn ∈ R

N such that

(6.1) j 6= k =⇒ limn→∞

λj,nλk,n

+λk,nλj,n

+|tj,n − tk,n|

λj,n= +∞,

and, denoting

U jL,n(t, r) =1

λj,nU jL

(t− tj,nλj,n

,r

λj,n

), j ≥ 1(6.2)

wJn(t) = SL(t)(u0,n, u1,n)−J∑

j=1

U jL,n(t),(6.3)

one has

(6.4) limJ→∞

lim supn→∞

‖wJn‖L3L6 = 0.

We recall (see [2], [4]) that any bounded sequence in H has a subsequence thatadmits a profile decomposition. The properties above imply that the followingweak convergences hold:

(6.5) j ≤ J =⇒(λj,nw

Jn (tj,n, λj,n·) , λ2j,n∂twJn (tj,n, λj,n·)

)−−−−n→∞

0 in H.

Furthermore we have the Pythagorean expansions valid, for all J ≥ 1

‖(u0,n, u1,n)‖2H =

J∑

j=1

∥∥∥~U jL(0)∥∥∥2

H+∥∥~wJn(0)

∥∥2H + on(1)(6.6)

E(u0,n, u1,n) =

J∑

j=1

E(~U jL,n(0)) + E(~wJn(0)) + on(1).(6.7)

If (u0,n, u1,n)n admits a profile decomposition, we can assume, extracting sub-sequences and time-translating the profiles if necessary, that the following limitsexist:

limn→∞

−tj,nλj,n

= τj ∈ −∞, 0,∞.

If τj = 0, we will assume without loss of generality tj,n = 0 for all n. If τj = 0, wedefine U j as the unique solution to the nonlinear wave equation (1.7) such that

(6.8) ~U j(0) = ~U jL(0).

If τj ∈ ±∞, we simply let U j = U jL. We denote by U jn the rescaled profile:

U jn(t, r) =1

λj,nU j(t− tj,nλj,n

,r

λj,n

).

We will now state a superposition principle (sometime called nonlinear profiledecomposition) outside wave cones using the profiles U j . We denote by J the setof indices j ≥ 1 such that τj = 0 and the solution U j of (1.7) with initial data

(U j0 , Uj1 ) cannot be extended to a solution with finite L3L6 norm to the wave cone


|x| > |t|. By the small data theory and (6.6), J is finite. If j /∈ J , then thesolution U jn(t, x) is well defined for |x| > |t| and

(6.9) supn→∞

‖U jn‖(L3L6)(|x|>|t|) <∞.

If J is empty, we let Rn = 0 for all n. If not, after extraction, (6.1) implies thatthere is a unique j0 ∈ J such that

∀j ∈ J \ j0, limn→∞

λj,nλj0,n

= 0.

In this case, we fix a R > 0 such that U j0 is defined (with a finite L3L6 norm) in|x| > R + |t|, and we let Rn = Rλj0,n for all n.

Proposition 6.1. Let (u0,n, u1,n)n, U jn, wJn , Rn be a above. Then for largen, there is a solution un of (1.7) defined in r > Rn + |t| with initial data(u0,n, u1,n)n at t = 0. Furthermore, denoting, for J ≥ 1, r > Rn + |t|,

ǫJn(t, r) = un(t, r)−J∑

j=1

U jn(t, r) − wJn(t, r),

we have

limJ→∞

limn→∞

[‖ǫJn‖(L3L6)(Rn) + sup

t≥0

∥∥~ǫJn(t)∥∥H(t+Rn)

]= 0.

Proof. The proof uses

∀j ≥ 1, lim supn→∞

‖U jn‖(L3L6)(Rn) <∞

∀j, k, ℓ ≥ 1, j 6= k =⇒∥∥U jnUknU ℓn

∥∥(L1L2)(Rn)

= 0,

and long-time perturbation theory arguments.We omit the details, since they are similar to the proof when the solution is not

restricted to the exterior of a wave cone (see e.g. the Main Theorem p. 135 in[2] in the defocusing case, and a sketch of proof in the focusing case right afterProposition 2.8 in [17]). The proof is also similar to the one of the correspondingresult for wave maps which is detailed below.

We next state a pseudo-orthogonality property of the profiles that will be neededin the proof of Theorem 4. See Appendix B for the proof.

Lemma 6.2. Let (u0,n, u1,n)n, U jn, wJn , Rn be a above. Let snn be a sequenceof times, ρn, ρ

′n > 0 be such that Rn + |sn| ≤ ρn < ρ′n (ρ′n = ∞ is allowed). Then

j 6= k =⇒ limn→∞

∫

ρn<|x|<ρ′n∇t,xU

jn(sn, x) · ∇t,xU

kn(sn, x)dx = 0(6.10)

j ≤ J =⇒ limn→∞

∫

ρn<|x|<ρ′n∇t,xU

jn(sn, x) · ∇t,xw

Jn(sn, x)dx = 0.(6.11)

6.2. Finite time blow-up case. In this subsection we deduce the finite time blow-up case of Theorem 4 from the rigidity theorem, Theorem 2. The proof is inspiredby the one of the analogous result in space dimension 3 (see [20]). However, unlike inspace dimension 3, we will need the fact (proved in [10]) that the soliton resolutionholds for a sequence of times.


We consider a solution u of (1.7), with initial data (1.8) such that T+ = T+(u)is finite and

(6.12) lim inft

<−→T+

‖~u(t)‖H <∞

6.2.1. Convergence of the solution. We first recall that the solution u convergesoutside the wave cone:

Lemma 6.3. There exists (v0, v1) ∈ H such that

limt→T+

∫

|x|>|T+−t||∇(u(t, x) − v0(x))|2 + (∂tu(t, x)− v1(x))

2dx = 0.

Lemma 6.3 is identical to Lemma 4.1 in [20]. We refer to this article for theproof. Note that [20, Lemma 4.1] is stated in space dimension 3, however the proofis independent of the dimension.

6.2.2. Analysis along a sequence of times. The core of the proof of Theorem 4 inthe finite time blow-up case is the following proposition, that we will deduce fromTheorem 2:

Proposition 6.4. Let tnn be a sequence of times such that

limn→∞

tn = T+, ∀n, 0 ≤ tn < T+

lim supn→∞

‖~u(tn)‖H <∞.

Then there exists a subsequence of tnn (that we still denote by tnn), J ≥ 0,(ιj)j ∈ ±1J, sequences λj,nn, 1 ≤ j ≤ J with

λ1,n ≪ . . .≪ λJ,n ≪ T+ − tn,

such that

limn→∞

∥∥∥∥∥∥u(tn)− v0 −

J∑

j=1

ιjλj,n

W

( ·λj,n

)∥∥∥∥∥∥H1

= 0(6.13)

E(u0, u1) = E(v0, v1) + JE(W, 0) +1

2‖∂tu(tn)− v1‖2L2 + on(1),(6.14)

where (v0, v1) are given by Lemma 6.3.

Proposition 6.4 is the analog in space dimension 4 of Proposition 4.2 of [20].However the conclusion (6.13) of Proposition 6.4 is weaker than the one of Propo-sition 4.2 of [20], which states in addition that ∂tu(tn)− v1 goes to 0 as n goes toinfinity in L2. This weaker conclusion is due to the fact that in space dimension 4,the lower bound for the asymptotic linear energy (3.4) is weaker than its analog inspace dimension 3.

Proof. To simplify notations, we assume without loss of generality in all the proofthat

T+(u) = 1.

Step 1. We first prove that we can assume

(6.15) ‖(v0, v1)‖H ≤ ε,


where ε is a fixed small constant. Let v be the solution of (1.7) such that ~v(1) =(v0, v1). By finite speed of propagation and the definition of (v0, v1), v(t, r) = u(t, r)for r > 1− t, t < 1 close to 1. Thus

∫ 3−3t

1−t

((∂t,ru(t, r))

2 +1

r2(u(t, r))2

)r3dr

=

∫ 3−3t

1−t

((∂t,rv(t, r))

2 +1

r2(v(t, r))2

)r3dr −→

t→10.

Hence, we can choose t0 < 1 close to 1 such that

(6.16) t0 < t < 1 =⇒∫ 3−3t

1−t

((∂t,ru(t, r))

2 +1

r2(u(t, r))2

)r3dr ≤ ε2.

We fix (u0, u1) ∈ H such that

r ≤ 3− 3t0 =⇒ (u0, u1)(r) = (u(t0, r), ∂tu(t0, r))(6.17)∫ ∞

1−t0

((∂ru0(r))

2 + (u1(r))2 +

1

r2(u0(r))

2

)r3dr . ε2.(6.18)

Let u be the solution of (1.7) such that (u(t0), ∂tu(t0)) = (u0, u1). By (6.17) andfinite speed of propagation,

(6.19) u(t, r) = u(t, r), r ≤ 3− 2t0 − t, t0 < t < 1.

By finite speed of propagation, small data theory and (6.18),

(6.20) ‖(u, ∂tu)‖H(1−2t0+t). ε, t0 < t < 1.

Combining (6.16), (6.19) and (6.20), we see that

‖(u, ∂tu)(t)‖H(1−t) . ε, t0 < t < 1.

Replacing u by u, we obtain a solution of (1.7) that blows up at t = 1, has notchanged in r < 1− t and satisfies the additional condition (6.15). Note that thefact that u = u in r < 1 − t implies (defining v in the same way as v with ureplaced by u)

limt→1

∥∥(u(t)− v(t))− (u(t)− v(t)), ∂tu(t)− ∂tv(t)) − (∂tu(t)− ∂tv(t))∥∥H = 0,

so that if (6.13), (6.14) hold for u, then it holds for the original u.

Step 2. Profile decomposition. Extracting subsequences, we assume that ~u(tn) −(v0, v1) admits a profile decomposition as in §6.1. We use the same notations U jL,U j, λj,nn, tj,nn, wJn as in §6.1. Extracting subsequences again and rescalingthe profiles, we can assume that there is a partition N \ 0 = Is ∪ Ic ∪ I+ ∪ I− ofthe set of indices such that

j ∈ Is ⇐⇒ ∀n, tj,n = 0, U jL(0) ∈ W,−W, 0 and ∂tUjL(0) = 0

j ∈ Ic ⇐⇒ ∀n, tj,n = 0 and U j is not a stationary solution of (1.7)

j ∈ I± ⇐⇒ limn→∞

−tj,nλj,n

= ±∞.

In Step 3, we prove that Ic = ∅. In Step 4, we prove that I± = ∅. In Step 5, weconclude the proof, by showing that the first component of the dispersive remainderwJn goes to 0 in H1 as n→ ∞.


Step 3. Compact profiles. In this step we prove by contradiction that Ic is empty.Assume that Ic is not empty. As in Subsection 6.1, we denote by J the set ofindices j ≥ 1 such that ∀n, tj,n = 0 and the solution U j cannot be extended tor > |t| with finite L3L6 norm. We recall that by the small data theory, J isfinite, and note that J ⊂ Ic. If J is not empty, we can extract subsequences suchthat there exists a unique j0 ∈ J such that

∀j ∈ J \ j0, limn→∞

λj,nλj0,n

= 0.

By Theorem 2, there exists R > 0 such that the solution U j0 is defined for |x| >R+ |t| and

(6.21)∑

±∞lim

t→±∞

∫

|x|>R+|t|

∣∣∇t,xUj0(t, x)

∣∣2 dx = η0 > 0.

If J is empty, we let j0 ∈ Ic and R = 0, so that U j0 is defined on |x| > |t| =|x| > R + |t| and, by Theorem 2, (6.21) holds.

In both cases, combining (6.21) with finite speed of propagation and the smalldata theory, we see that the following holds for all t > 0 or for all t < 0:

(6.22)

∫

|x|>R+|t||∇t,xU

j0(t, x)|2dx ≥ η1 > 0.

We let Rn = Rλj0,n for all n. By Proposition 6.1, for large n, the solution u isdefined for r > Rn + |t− tn| and

(6.23) u(tn + τ, r) = v(1 + τ, r) +J∑

j=1

U jn(τ, r) + wJn(τ, r) + εJn(τ, r),

where

(6.24) limJ→∞

lim supn→∞

supτ

∥∥εJn∥∥H(Rn+|τ |) = 0,

and v is as above the solution of (1.7) with initial data (v0, v1) at t = 1, which isglobally defined and scattering by the small data theory. Using the continuity of v,we can rewrite (6.23) as

(6.25) u(tn + τ, r) = v(tn + τ, r) +

J∑

j=1

U jn(τ, r) + wJn(τ, r) + εJn(τ, r),

where the remainder εJn has slightly changed but still satisfies (6.24). We distinguishtwo cases.

Case 1. Channels of energy in the future. We assume that (6.22) holds for all t > 0.Recall that by finite speed of propagation u(t, r) = v(t, r) for r > 1 − t, t close to1. By (6.25) at τ = 1− tn, we obtain, denoting ρn = Rλj0,n + 1− tn,

(6.26) 0 =∑

1≤j≤J

∫

|x|>ρn∇t,xU

jn(1− tn, x) · ∇t,xU

j0n (1− tn, x) dx

+

∫

|x|>ρn∇t,xw

Jn(1− tn, x) · ∇t,xU

j0n (1− tn, x)dx

+

∫

|x|>ρn∇εJn(1− tn, x) · ∇t,xU

j0n (1− tn, x)dx.


By (6.22), at t = (1− tn)/λj0,n,

(6.27)

∫

|x|>ρn|∇t,xU

j0n (1 − tn, x)|2dx

=

∫

|y|>R+ 1−tnλj0,n

∣∣∣∣∇t,xUj0

(1− tnλj0,n

, y

)∣∣∣∣2

dy ≥ η12

for large n. On the other hand, by Lemma 6.2,

j 6= j0 =⇒ limn→∞

∫

|x|>ρn∇t,xU

jn(1− tn, x) · ∇t,xU

j0n (1 − tn, x) dx = 0(6.28)

J ≥ j0 =⇒ limn→∞

∫

|x|>ρn∇t,xw

Jn(1− tn, x) · ∇t,xU

j0n (1− tn, x) dx = 0.(6.29)

Combining (6.26), (6.27), (6.28), (6.29) with the property (6.24) of εJn, we obtain acontradiction.

Case 2. Channels of energy in the past. Next, we assume that (6.22) holds for allt ≤ 0. We fix t0 < 1 in the domain of existence of u. By (6.25), at τ = t0 − tn, wehave, letting τn = t0 − tn, ρn = Rλj0,n + |τn|.

(6.30)

J∑

j=1

∫

|x|>ρn∇U jn(τn, x) · ∇U j0n (τn, x)dx

+

∫

|x|>ρn∇t,xw

Jn(τn, x) · ∇t,xU

j0n (τn, x)dx +

∫

|x|>ρn∇εJn(τn, x) · ∇t,xU

j0n (τn, x)dx

=

∫

|x|>ρn(∇t,xu(t0, x)−∇t,xv(t0, x)) · ∇t,xU

j0n (τn, x)dx −→

n→∞0,

where we have used that lim τn = 1 − t0, and that ~u(t0, x) − ~v(t0, x) = 0 for|x| > 1− t0. Since (6.22) holds for all t ≤ 0, we have

(6.31) ∀n,∫

|x|>ρn|∇U j0n (τn, x)|2dx ≥ η1.

By Lemma 6.2,

j 6= j0 =⇒ limn→∞

∫

|x|>ρn∇t,xU

jn(τn, x) · ∇t,xU

j0n (τn, x) dx = 0(6.32)

J ≥ j0 =⇒ limn→∞

∫

|x|>ρn∇t,xw

Jn(τn, x) · ∇t,xU

j0n (τn, x) dx = 0.(6.33)

Combining (6.24), (6.30), (6.31), (6.32) and (6.33) we obtain a contradiction.

Step 4. Scattering profiles. We next prove that I+ and I− are empty. The proof isclose to the proof of the fact that Ic is empty, and we only sketch it, focusing onthe arguments that are new with respect to Step 3. By Step 3, J ⊂ Ic = ∅, andthus (6.25) and (6.24) hold with Rn = 0.

Assume that I+ 6= ∅, and let j0 ∈ I+. Recall that U j0 is a nonzero solutionof the linear wave equation. By Proposition A.1 and Remark A.2, we obtain thatthere exist A ∈ R, η1 > 0 and T such that

∀t ≥ T,

∫

|x|>t+A|∇t,xU

j0(t, x)|2dx ≥ η1 > 0.


Changing variables, we deduce

∀τ ≥ λj0,nT + tj0,n,

∫

|y|>τ−tj0,n+λj0,nA

|∇τ,yUj0n (τ, y)|2dy ≥ η1.

Wewill use the preceding bound with τ = 1−tn, which is possible since−tj0,n/λj0,n →∞ as n→ ∞. Indeed, λj0,nT + tj0,n < 0, and thus 1− tn ≥ λj0,nT + tj0,n for largen. Denoting ρn = 1− tn − tj0,n + λj0,nA, we obtain that for large n

∫

|y|>ρn|∇t,yU

j0n (1− tn, y)|2dy ≥ η1.

We note that ρn > 1− tn for large n. The end of the proof is the same as the oneof Case 1 of Step 3 above, and we omit it.

The proof of the fact that I− = ∅ is along the same lines (following Case 2 ofStep 3) and is also omitted.

Step 5. We have proved the existence of J ≥ 0, ιj1≤j≤J ∈ ±1J , 0 < λ1,n ≪. . .≪ λJ,n such that

(6.34) ~u(tn) = ~v(tn) +

J∑

j=1

(ιjλj,n

W

( ·λj,n

), 0

)+ (w0,n, w1,n),

where, denoting by wn the solution of the linear wave equation with initial data(w0,n, w1,n),

limn→∞

‖wn‖L3(R,L6) = 0.

Using that the support of u− v is included in r < 1− t, that, by (6.34),

λJ,n(u− v)(tn, λJ,n·) −−−−n→∞

ιJW,

and that W is not compactly supported, we obtain

(6.35) limn→∞

λJ,n1− tn

= 0.

In this step, we prove that

(6.36) limn→∞

‖w0,n‖H1 = 0,

which will conclude the proof of Proposition 6.4. By (3.4) and the time decay ofthe exterior energy for the linear wave equation, the following holds for all t ≥ 0 orfor all t ≤ 0:

(6.37) C

∫

|x|>|t||∇t,xwn(t, x)|2dx ≥ ‖w0,n‖2H1 .

On the other hand, since by (6.34),

limn→∞

∫

|x|≥1−tn

(|∇w0,n(x)|2 + w2

1,n(x))dx = 0,

we see that for any ε > 0, if n is large enough,

∀t,∫

|x|≥1−tn+|t||∇t,xwn(t, x)|2dx ≤ ε.

Thus for large n, the following holds for all t ≥ 0 or for all t ≤ 0.

(6.38) C

∫

|t|<|x|<|t|+(1−tn)|∇t,xwn(t, x)|2dx ≥ ‖w0,n‖2H1 .


Arguing by contradiction along the same lines as in Step 3, we obtain (6.36). Thisand (6.34) give (6.13) and (6.14).

6.2.3. End of the proof. We are now in position to prove Theorem 4 in the finitetime blow-up case. We consider u as above, and let (v0, v1) be given by Lemma6.3.Step 1. Boundedness. We first prove that u remains bounded in H as t→ T+. Forthis, we let tnn → T+ be a sequence of times such that ~u(tn)n is bounded inH. By Proposition 6.4, there exists J such that

‖u(tn)‖2H1 = ‖v0‖2H1 + J‖W‖2H1 + on(1)(6.39)

E(u0, u1) = E(v0, v1) + JE(W, 0) +1

2‖∂tu(tn)− v1‖2L2 + on(1).(6.40)

From (6.40), we deduce

lim supn→∞

‖∂tu(tn)− v1‖2L2 ≤ E(u0, u1)− E(v0, v1)

and

J ≤ E(u0, u1)− E(v0, v1)

E(W, 0).

From the equation −∆W =W 3, we obtain∫|∇W |2 =

∫|W |4 and thus ‖W‖2

H1=

4E(W, 0). Using also (6.39), we deduce

lim supn→∞

‖u(tn)‖2H1 ≤ ‖v0‖2H1 + 4(E(u0, u1)− E(v0, v1)).

Combining, we see that there exists a constant C0, depending only on u (butindependent of the choice of the sequence tnn) such that if ~u(tn)n is boundedin H,

lim supn→∞

‖~u(tn)‖H ≤ C0.

Using thatlim inft→∞

‖~u(t)‖H <∞we deduce immediately, since t 7→ ‖~u(t)‖H is continuous,

lim supt→T+

‖~u(t)‖H ≤ C0.

Step 2. Convergence to 0 of the time derivative inside the wave cone. In this stepwe prove

(6.41) limt→T+

‖∂tu(t)− v1‖L2 = 0.

By Step 1 and the soliton resolution for a sequence of times proved in [10], thereexists a sequence tnn, J ≥ 1, (ιj)j ∈ ±1J , sequences λj,nn, 1 ≤ j ≤ J with

(6.42) λ1,n ≪ . . .≪ λJ,n ≪ T+ − tn,

such that

(6.43) limn→∞

∥∥∥∥∥∥~u(tn)− (v0, v1)−

J∑

j=1

(ιjλj,n

W

( ·λj,n

), 0

)∥∥∥∥∥∥H

= 0.

Note that the conclusion of Step 1 is necessary here, since the main result of [10] isvalid assuming that ~u(t) is bounded inH, a stronger statement than our assumption(6.12).


As an immediate consequence of this, we obtain

(6.44) E(u0, u1) = E(v0, v1) + JE(W, 0).

We prove (6.41) by contradiction. If it does not hold, using (6.43), we can find anarbitrarily small ε0 > 0 and a sequence t′n → T+ such that ‖∂tu(t′n) − v1‖L2 = ε0.By Proposition 6.4, there exists an integer J ′ ≥ 0 such that

E(u0, u1) = E(v0, v1) + J ′E(W, 0) +1

2ε20.

Combining with (6.44), we deduce that

1

2ε20 = (J − J ′)E(W, 0),

a contradiction if ε20 is smaller than E(W, 0).

Step 3. End of the proof. Combining Steps 1 and 2 with Proposition 6.4, we seethat for any sequence tnn → T+, extracting subsequences if necessary, there existJ ≥ 1, (ιj)j ∈ ±1J , sequences λj,nn, 1 ≤ j ≤ J such that (6.42) and (6.43)hold. We note that

J =E(u0, u1)− E(v0, v1)

E(W, 0)

is independent of the choice of tnn. The case J = 0 is excluded since T+ is themaximal time of existence of u.

It remains to construct the scaling parameters λj(t) such that the expansion(1.20) holds. This can be done as in [20], Section 3.5 defining, for j = 1 . . . J andt < 1 close to 1,

Bj := (j − 1)‖∇W‖2L2 +

∫

|x|≤1

|∇W (x)|2 dx

and

λj(t) := inf

λ > 0 s.t.

∫

|x|≤λ|∇(u− v)(t, x)|2 dx ≥ Bj

.

We refer to Section 3.5 of [20] for the proof that the conclusion of Theorem 4 inthe finite time blow-up case holds with this choice of λj(t)

6.3. Comments on the proof in the global case. We recall:

Proposition 6.5. Let u be a solution of (1.7) such that T+(u) = +∞. Then

• lim inft→∞

‖~u(t)‖H <∞.

• There exists a finite energy solution vL of the free wave equation such that

∀A ∈ R, limt→∞

∫

|x|>t+A|∇t,x(u− vL)(t, x)|2dx = 0.

See Subsections 3.2 and 3.3 in [20] for the proofs. The proofs are written inspace dimension 3 there, but are indeed independent of the dimension. See also[10], Subsection 4.1, for the proof of the second point in space dimension 4.

In view of Proposition 6.5, the proof of Theorem 4 in the global case is very closeto the corresponding proof in the finite time blow-up case, replacing (v0, v1) andt = 1 by ~vL(t) and t = +∞, and we omit it.


7. Soliton resolution for corotional wave maps

7.1. Preliminaries on wave maps.

7.1.1. Miscellaneous results on wave maps. We gather here a few standard resultson wave maps.

Claim 7.1. Let (ψ0, ψ1) ∈ E, such that EM (ψ0, ψ1) is small. Then there exists ℓsuch that (ψ0, ψ1) ∈ Hℓ,ℓ and

‖(ψ0 − ℓπ, ψ1)‖2H ≈ EM (ψ0, ψ1).

Similarly, if R > 0, (ψ0 − ℓπ, ψ1) ∈ H(R) for some ℓ ∈ Z, and∫ ∞

R

(ψ21 + (∂rψ0)

2 +1

r2sin2 ψ0(r)

)rdr

is small, then∫ ∞

R

(ψ21 + (∂rψ0)

2 +1

r2sin2 ψ0(r)

)rdr

≈∫ ∞

R

(ψ21 + (∂rψ0)

2 +1

r2(ψ0(r) − ℓπ)2

)rdr.

Proof. (See also [45, Lemma 5.3]). Let (ψ0, ψ1) ∈ E, and (ℓ,m) ∈ Z2 such that

(ψ0, ψ1) ∈ Hℓ,m. Since ψ0(0) = ℓπ, we have, for any r > 0,∣∣∣∣∣

∫ ψ0(r)

ℓπ

| sin(σ)|dσ∣∣∣∣∣ ≤

∫ r

0

| sin(ψ0(σ))| |∂rψ0(σ)|dσ ≤ EM (ψ0, ψ1).

Since EM (ψ0, ψ1) is small, we deduce that |ψ0(r) − ℓπ| is small for all r. Thisimplies in particular ψ0 ∈ Hℓ,ℓ,

12 |ψ0(r)− ℓπ| ≤ | sin(ψ0(r))| ≤ |ψ0(r) − ℓπ| and

the first assertion of the claim follows. The proof of the second assertion is similarand we omit it.

Recalling that the outer energy

Eout(~ψ(t)) =1

2

∫ ∞

|t|

((∂tψ(t, r))

2 + (∂rψ(t, r))2 +

sin2 ψ(t, r)

r2

)rdr

is nonincreasing for t > 0 (and nondecreasing for t < 0), we can now complete theproof of Theorem 1.

End of the proof of the rigidity theorem. Let ψ be a solution of (1.1) which is nota stationary solution, with data (ψ0, ψ1) ∈ Hℓ,m. We have already proved (4.8).Assume to fix ideas that

(7.1) limt→+∞

∫ ∞

|t|

((∂t,rψ(t, r))

2 +1

r2(ψ(t, r) −mπ)2

)rdr > 0.

Let

η = limt→∞

∫ ∞

t

((∂t,rψ(t, r))

2 +1

r2sin2 ψ(t, r)

)rdr = lim

t→∞Eout(~ψ(t)).

Since Eout is nonincreasing for t ≥ 0, we are reduced to prove that η > 0. However

if η = 0, then for large t, Eout(~ψ(t)) is small and the second assertion of Claim 7.1applies. This yields a contradiction with (7.1), concluding the proof of (1.13).


We will need the following estimates on the sinus functions:

Claim 7.2. Let J ≥ 2, (aj)1≤j≤J ∈ RJ . Then

∣∣∣∣∣∣sin(2

J∑

j=1

aj

)−

J∑

j=1

sin(2aj)

∣∣∣∣∣∣.∑

j 6=k| sin(2aj)| sin2(ak) .

∑

j 6=k| sin(aj)| sin2(ak)

(7.2)

∣∣∣∣∣∣sin2

( J∑

j=1

aj

)−

J∑

j=1

sin2(aj)

∣∣∣∣∣∣.∑

j 6=k|sinaj sin ak|(7.3)

Proof. Proof of (7.2). The inequality (7.2) with J = 2 follows from

sin(2(a1 + a2))− sin(2a1)− sin(2a2) = −2 sin(2a1) sin2 a2 − 2 sin(2a2) sin

2 a1.

The general case follows from a straightforward induction and the bound | sin(a+b)| ≤ | sin a|+ [sin b|.Proof of (7.3). In the case J = 2, we have∣∣sin2(a1 + a2)− sin2 a1 − sin2 a2

∣∣ = 2 |sin a1 sin a2 cos(a1 − a2)| ≤ 2| sina1 sin a2|.The general case follows from an elementary induction.

In the following proposition, we gather some important facts about finite timeblow-up solutions of (1.1)

Proposition 7.3. Let ψ be a finite energy solution of (1.1) with maximal time ofexistence T+ <∞. Then there exists (ϕ0, ϕ1) ∈ E such that for all R > 0,

limt→T+

∫ ∞

R

((∂tψ(t)− ϕ1)

2 + (∂rψ(t)− ϕ0)2 +

1

r2(ψ(t)− ϕ0)

2

)r dr = 0.

The solution of (1.1), with initial data (ϕ0, ϕ1) at t = T+ satisfies

ψ(t, r) = ϕ(t, r), 0 < r < T+ − t, t < T+ close to T+.

Furthermore,

(7.4) limt→T+

1

2

∫ T+−t

0

((∂tψ(t, r))

2 + (∂rψ(t, r))2 +

sin2(r)

r2

)rdr ≥ EM (Q, 0).

Finally, there exists tn → T+ such that

(7.5) limn→∞

∫ T+−tn

0

(∂tψ(tn, r)2rdr = 0.

Note that the energy inside the wave cone

Ein(~ψ(t)) =1

2

∫ T+−t

0

((∂tψ(t, r))

2 + (∂rψ(t, r))2 +

sin2(r)

r2

)rdr

is a nonincreasing function of t ∈ [0, T+), so that the limit appearing in (7.4) exists.The existence of (ϕ0, ϕ1) follows from small data well-posedness arguments and

finite speed of propagation. See [8, Lemma 5.2], [7, Proposition 5.2]. The lowerbound (7.4) is an immediate consequence of the work of Struwe [78]. Finally, (7.5)can be seen as a consequence of the main result of [7], but is indeed a step of theproof of this main result (see [7, Corollary 2.7]).


Recall the linearized equation for (1.1) around a constant solution:

(7.6)

∂2t ψL − ∂2rψL − 1

r∂rψL +

1

r2ψL = 0

~ψLt=0 = (ψ0, ψ1) ∈ H.

Proposition 7.4. Let ψ be a solution of (1.1) with maximal time of existenceT+ = +∞ and initial data (ψ0, ψ1) ∈ Hℓ,m. Then there exists a solution ψL of(7.6) with initial data in H and an increasing positive function α(t) = o(t) ast→ ∞ such that

(7.7) limt→∞

‖(ψ(t)−mπ − ψL(t), ∂tψ(t)− ∂tψL(t))‖H(α(t)) = 0.

Furthermore, there exists tn → +∞ such that

(7.8) limn→∞

∫ ∞

0

(∂tψ(tn, r)− ∂tψL(tn, r))2rdr = 0.

See [7, Proposition 5.1] for (7.7) and [7, Corollary 2.3] for (7.8).

7.1.2. Rigidity theorem.

7.1.3. A space-time bound outside the wave cone. We will denote by S the space ofmeasurable functions ψ of (t, r) ∈ R× (0,∞) such that the following norm is finite.

(7.9) ‖ψ‖S =

(∫

R

(∫ ∞

0

|ψ(t, r)|6 drr3

) 12

dt

) 13

,

and by S(r > |t| the space of restrictions of S to r > |t|, with the norm

(7.10) ‖ψ‖S(r>[t[) =

∫

R

(∫ ∞

|t||ψ(t, r)|6 dr

r3

) 12

dt

13

,

By the change of function u = 1rψ and the Strichartz estimate for the wave equation

in dimension 1 + 4, all solutions of (7.6) are in S. This is of course not the casefor general finite energy solution of the wave map equation. However we have thefollowing space time bound, which will be useful to construct a nonlinear profiledecomposition for (1.1) outside wave cones.

Lemma 7.5. Let (ψ0, ψ1) ∈ Hℓ,m, (ℓ,m) ∈ Z2, and let ψ be the solution of (1.1)

with initial data (ψ0, ψ1). Then sinψ ∈ S(r > |t|), i.e.∫ ∞

−∞

(∫ ∞

|t|

sin6 ψ(t, r)

r3dr

)1/2

dt <∞.

Remark 7.6. The solution ψ is well-defined for r > |t|, see Claim 2.1.

Proof. Let u = ψ−ℓπr , u0 = (ψ0−ℓπ)

r , u1 = ψ1

r . Since (ϕ0, ϕ1) ∈ Hℓ,m, (u0, u1)

can be considered as a radial function in (H1 × L2)loc(R4). By the local well-

posedness for the equation (4.1) and finite speed of propagation, u is defined on−t0 ≤ t ≤ t0, 0 < r ≤ 2 for some small t0 > 0, and

∫ t0

−t0

(∫ 2

0

u6(t, r)r3dr

)1/2

dt <∞.


Hence

(7.11)

∫ t0

−t0

(∫ 2

|t|

(sinψ)6

r3dr

)1/2

dt ≤∫ t0

−t0

(∫ 2

|t|

(ψ − ℓπ)6(t, r)

r3dr

)1/2

dt <∞.

Similarly, for all R > 0, u = ψ−mπr is a solution of (4.1) defined for r > R+ |t| with

initial data in H(R). Thus for all T > 0,

∫ T

−T

(∫ ∞

R+|t|u6(t, r)r3dr

)1/2

dt <∞,

which yields∫ T

−T

(∫ ∞

R+|t|

(sinψ)6

r3dr

)1/2

dt <∞.

Combining with a space translation in time and (7.11) we obtain

(7.12) ∀T > 0,

∫ T

−T

(∫ ∞

|t|

(sinψ)6

r3dr

)1/2

dt <∞

Using that Eout is nonincreasing for t > 0, ¡e define

(7.13) Eout(∞) = limt→+∞

Eout

(~ψ(t)

).

Let T such that

(7.14) Eout

(~ψ(T )

)≤ Eout(∞) +

1

4EM (Q, 0).

We extend ~ψ(T, r), which is defined for r > T , as follows. We define

(φ0, φ1)(r) = (ψ(T, r), ∂tψ(T, r)), r ≥ T,

and ϕ1(r) = 0 if r < T . Let p ∈ Z such that pπ ≤ ψ(T, T ) < (p+ 1)π. For r < T ,we let

φ0(r) = pπ if ψ(T, T ) = pπ

φ0(r) = Q(λr) + pπ if pπ < ψ(T, T ) ≤ pπ + π/2

φ0(r) = (p+ 1)π −Q(λr) if pπ + π/2 < ψ(T, T ) < (p+ 1)π,

where in the second and third case, λ is chosen so that φ0 is continuous at r = T .We note that (φ0, φ1) ∈ E. We claim

(7.15) EM (φ0, φ1) ≤ Eout(∞) +3

4EM (Q, 0).

In view of (7.14), it is sufficient to check that

(7.16)

∫ T

0

(∂rφ0)2rdr +

∫ T

0

sin2 φ0dr

r≤ EM (Q, 0).

In the case where φ0(r) = pπ for r < T , this is trivial. In the two other cases, itfollows from the fact that Q(1) = 2 arctan1 = π/2 and that

∫ 1

0

(∂rQ)2rdr +

∫ 1

0

sin2Qdr

r= EM (Q, 0)

which can be proved by an explicit computation.


Let φ be the solution of (1.1) with initial data ~φ(T ) = (φ0, φ1). By finite speed of

propagation, ψ(t, r) = φ(t, r), r > t ≥ T . As a consequence, for t ≥ T , Eout(~φ(t)) ≥Eout(∞). Combining with the conservation of the energy and (7.15), we deduce

∀t ≥ T, (EM − Eout)(~φ(t)) ≤3

4EM (Q, 0).

In view of the energy concentration for blow-up solutions (see (7.4)), the solutionφ is global in the future. Let φL be the solution of (7.6) given by Proposition 7.4,so that

(7.17) limt→∞

‖(φ(t) −mπ − φL(t), ∂tφ(t)− ∂tφL(t))‖H(α(t)) = 0,

where α(t)/t→ 0 as t→ ∞. Let T ≫ 1, so that

∥∥∥(φ(T)−mπ − φL

(T), ∂tφ

(T)− ∂tφL

(T))∥∥∥

H(T )+

∫ ∞

T

(∫ ∞

|t|

φ6L(t, r)

r3dr

)1/2

dt ≪ 1.

Recall that 1rφL is a solution of the linear wave equation in R × R

4, and that√2√3r(φ−mπ) is solution of the nonlinear equation (4.1). Then by Proposition 4.1,

∫ ∞

T

(∫ ∞

|t|

(φ(t)−mπ)6

r3dr

)1/2

dt <∞,

which implies

∫ ∞

T

(∫ ∞

|t|

sin6 ψ(t, r)

r3dr

)1/2

dt =

∫ ∞

T

(∫ ∞

|t|

sin6 φ(t, r)

r3dr

)1/2

dt <∞.

Using the same argument for negative times, we obtain the desired conclusion.

7.1.4. Profile decomposition. We next state a profile decomposition which is adaptedto the wave maps equation (1.1).

Proposition 7.7 (Profile decomposition for wave maps). Let (ℓ,m) ∈ Z2. Con-

sider a sequence (ψ0,n, ψ1,n)n of elements of Hℓ,m such that

(7.18) supnEM (ψ0,n, ψ1,n) <∞.

Then there exists a subsequence of (ψ0,n, ψ1,n)n (that we denote the same) and,

for all j ∈ N \ 0, for all n,(Ψj0,n,Ψ

j1,n

)∈ E, λj,n > 0, tj,n ∈ R such that

(7.19) j 6= k =⇒ limn→∞

λj,nλk,n

+λk,nλj,n

+|tj,n − tk,n|

λj,n= +∞,

and, for all j ≥ 1, one the following holds:

Compact profile: ∀n, tj,n = 0 and

∃(Ψj0,Ψ

j1

)∈ E, (Ψj0,n(r),Ψ

j1,n(r)) =

(Ψj0

(r

λj,n

),

1

λj,nΨj1

(r

λj,n

)).

Linear wave profile: limn→∞−tj,nλj,n

∈ ±∞ and there exists a solution ΨjL

of (7.6) with initial data (Ψj0,Ψj1) ∈ H such that

(Ψj0,n(r),Ψj1,n(r)) =

(ΨjL

(−tj,nλj,n

,r

λj,n

),

1

λj,nΨjL

(−tj,nλj,n

,r

λj,n

)).


Furthermore, denoting by

(7.20)(ωJ0,n, ω

J1,n

)= (ψ0,n, ψ1,n)−

J∑

j=1

(Ψj0,n,Ψ

j1,n

),

then for large J , (ωJ0,n, ωJ1,n) ∈ H for all n,

(7.21) EM (ψ0,n, ψ1,n) =J∑

j=1

EM(ΨJ0,n,Ψ

J1,n

)+ EM

(ωJ0,n, ω

J1,n

)+ on(1),

and, letting ωJn be the solution of (7.6) with initial data (ωJ0,n, ωJ1,n), one has

(7.22) limJ→∞

lim supn→∞

∥∥ωJn∥∥S+ ‖ωJn‖L∞

t,r= 0.

Finally, for large J ,(ωJ0,n, ω

J1,n)nis bounded in H and

(7.23) 1 ≤ j ≤ J =⇒(ωJn(tj,n, λj,n·), λj,n∂tωJn(tj,n, λj,n·)

)−−−−n→∞

0 in H.

We will denote by JC the indices corresponding to compact profiles, and by JLthe indices corresponding to nonzero linear wave profiles. We can assume that thereexists at most one nonzero constant profile (which is a compact profile). We will

consider the profiles such that (Ψj0,Ψj1) ≡ 0, as compact profiles. Of course, if there

is an infinite number of nonzero profiles, we can assume (rearranging the indices ifnecessary) that there are no such profiles.

By (7.21), ∑

j∈JC

EM (Ψj0,Ψj1) <∞.

By Claim 7.1, for large j, (Ψj0,Ψj1) ∈ Hℓj ,ℓj for some ℓj ∈ Z, and, since (ωJ0,n, ω

J1,n) ∈

H for large J , (Ψj0,Ψj1) ∈ H for large j.

Proposition 7.7 is [45, Lemma 5.5], with some changes in the notations:

• The constants denoted by φl0(0) that appear in the decomposition (5.16) of[45] can be summed up all together and considered (if this sum is not zero)as a constant profile (Ψ1

0,Ψ11) = (pπ, 0), p ∈ Z \ 0 with 1 ∈ JC .

• In [45, Lemma 5.5] the solutions of (7.6) are denoted by ruL where uL is aradial solution of the 1 + 4-dimensional wave equation.

• In [45], the profiles corresponding to indices j ∈ JC are divided into twotypes of profiles, the ones defined as rescaled (ψl0, 0) ∈ E, and the ones

defined as rescaled U jL, with tj,n = 0 for all n (see (5.16) in this article).

The only part of Proposition 7.7 which is not included in [45] is the Pythagoreanexpansion (7.21) of the energy for wave maps. The proof is given in Appendix B.2.We refer to [8, Lemma 2.16] for a proof of this Pythagorean expansion in a specialcase.

We now state the analog of Proposition 6.1 (a nonlinear profile decompositionoutside a wave cone) for wave maps. Since the solutions of (1.1) are always definedon the entire exterior of a wave cone r > |t| and satisfy the space-time boundgiven by Lemma 7.5 there, this decomposition will be valid on r > |t|.

Let (ψ0,n, ψ1,n), and, for j ∈ N \ 0,(Ψj0,Ψ

j1

), tj,nn, λj,nn, ωjn be as in

Proposition 7.7. If j ∈ JC , we denote by Ψj the solution of (1.1) with initial data


(Ψj0,Ψ

j1

)at t = 0, which is well-defined for r > |t| (see Claim 2.1). If j ∈ JL, we

will denote Ψj = ΨjL. Let

(7.24) Ψjn(t, r) = Ψj(t− tj,nλj,n

,r

λj,n

).

Proposition 7.8. Let ψn be the solution of (1.1) with initial data (ψ0,n, ψ1,n).Then, denoting,

(7.25) ΘJn(t, r) = ψn(t, r)−J∑

j=1

Ψjn(t, r) − ωJn(t, r), r > |t|,

one has for large J that ~ΘJn(t) ∈ H(|t|) for all t and

limJ→∞

lim supn→∞

(supt

∥∥∥~ΘJn(t)∥∥∥H(|t|)

+∥∥ΘJn

∥∥S(r>|t|)

)= 0.

Proof. If we assume that the sequence (ψ0,n, ψ1,n)n is bounded in H, then theconclusion of the proposition can be obtained by considering the sequence of radialfinite-energy solutions

1rψn

nof the 1 + 4−dimensional nonlinear wave equation

(4.1) (see e.g. [7, Proposition 2.15]). To work in the more general setting of Propo-sition 7.8, we will need the space-time bound for general finite-energy solutions of(1.1) given in Lemma 7.5.

We denote

(7.26) ψJn =

J∑

j=1

ΨJn + ωJn , ΘJn = ψn − ψJn .

Then

(7.27) ∂2tΘJn − ∂2rΘ

Jn − 1

r∂rΘ

Jn +

1

r2ΘJn

= − sin(2ψn)− 2ψn2r2

+∑

1≤j≤Jj∈JC

sin(2Ψjn)− 2Ψjn2r2

,

with initial data (ΘJn, ∂tΘjn)t=0 = (0, 0). Expanding the second line of (7.27) and

letting θjn = 1rΘ

jn, we rewrite it as a radial 1 + 4-dimensional wave equation:

(7.28) ∂2t θJn − ∂2rθ

Jn − 3

r∂rθ

Jn

= − 1

2r3

sin

(2ωJn + 2ΘJn +

J∑

j=1

2Ψjn

)− sin

(2ωJn

)− sin

(2ΘJn

)−

J∑

j=1

sin(2Ψjn

)

+1

r3

∑

1≤j≤Jj∈JL

(Ψjn − sin(2Ψjn)

2

)+

1

r3

(ΘJn − sin(2ΘJn)

2

)+

1

r3

(ωJn − sin(2ωJn)

2

).


Let ΓT := (t, r) : 0 < |t| ≤ min(r, T ). We will prove

(7.29) ∀T > 0,

∥∥∥∥(∂2t θ

Jn − ∂2rθ

Jn − 3

r∂rθ

Jn

)11ΓT

∥∥∥∥L1L2

≤∫ T

0

∥∥θJn(t)∥∥L6(|t|) Fn(t)dt + C

∫ T

0

‖θJn(t)‖3L6(|t|)dt+ εJn,

where εJn is independent of T and is such that

(7.30) limJ→∞

lim supn→∞

εJn = 0,

and Fn satisfies

(7.31) Fn ∈ L3/2(R), supn

‖Fn‖L3/2(R) <∞.

In (7.29) the notation L6(|t|) means as above

‖f‖L6(|t|) =

(∫ ∞

|t|(f(r))

6r3dr

)1/6

.

We first assume (7.29), (7.30) and (7.31) and prove the proposition. In all thesequel we assume that J and n are taken large enough, so that εJn is small. By(7.29), Strichartz estimates and Holder’s inequality,

(∫ T

0

‖θJn(t)‖3L6(|t|)dt

)1/3

≤ εJn +

∫ T

0

‖θJn(t)‖L6(|t|)(Fn(t) + C‖θJn(t)‖2L6(|t|)

)dt.

Using (7.31) and a Gronwall-type inequality (see the appendix of [31]), we obtainthat for all T > 0,

(∫ T

0


)1/3

≤ εJnΓ

(C + C

(∫ T

0

‖θJn‖3L6(|t|)dt)2/3

),

where Γ is the standard Gamma function.Assuming εJnΓ(C + 1) ≤ 1/C, we obtain that

(∫ T

0

‖θJn‖3L6(|t|)dt)2/3

≤ 1/C

=⇒(∫ T

0


)1/3

≤ εJnΓ (C + 1) ≤ 1/C,

and a standard bootstrap argument yields( ∫ T

0 ‖θJn‖3L6(|t|)dt)1/3

. εJn for all T .

Using (7.29) and again Strichartz estimates we see that it implies

supt∈R

∥∥∥~θJn(t)∥∥∥H(|t|)

. εJn.

Going back to the function Θjn(t) = rθjn, we deduce the conclusion of the proposi-tion. We are thus left with proving (7.29). In view of (7.28), it is sufficient to prove


the following inequalities

limn→∞

∥∥∥∥11r>|t|1

r3


2

)∥∥∥∥L1L2

= 0, ∀j ∈ JL(7.32)

limJ→∞

lim supn→∞

∥∥∥∥11r>|t|1

r3

(ωjn − sin(2ωjn)

2

)∥∥∥∥L1L2

= 0(7.33)

∥∥∥∥11ΓT

1

r3

(ΘJn − sin(2ΘJn)

2

)∥∥∥∥L1L2

.∥∥11ΓT θ

Jn

∥∥3L3L6 , ∀J ≫ 1(7.34)

∥∥∥∥∥∥11ΓT

2r3

sin

(2ωJn + 2ΘJn +

J∑

j=1

2Ψjn

)− sin

(2ωJn

)− sin

(2ΘJn

)−

J∑

j=1

sin(2Ψjn

)∥∥∥∥∥∥L1L2

(7.35)

.

∫ T

0

∥∥θJn(t)∥∥L6(|t|) Fn(t)dt+ εJn +

∥∥11ΓT θJn

∥∥3L3L6 , ∀J ≫ 1,

where εJn satisfies (7.30), Fn satisfies (7.31), and the space-time LpLq norms arealways with respect to r3dr and dt.

Proofs of (7.32), (7.33) and (7.34). We use the bound

(7.36) ∀σ ∈ R,

∣∣∣∣σ − sin(2σ)

2

∣∣∣∣ . σ3.

By (7.36) and the property (7.22) of ωJn , we obtain (7.33). The inequality (7.34)follows directly from (7.36) and Holder’s inequality.

Let j ∈ JL. Then by (7.36) and Holder’s inequality,

(7.37)

∥∥∥∥1

r3


2

)11r>|t|

∥∥∥∥L1L2

.

∫

R

(∫ ∞

|t|

1

r3

(Ψj(t− tj,nλj,n

,r

λj,n

))6

dr

)1/2

dt

=

∫

R

(∫ ∞∣∣∣tj,nλj,n

+τ∣∣∣

1

ρ3(Ψj(τ, ρ)

)6dρ

)1/2

dτ,

which tends to 0 as n→ ∞ by dominated convergence, and since limn→∞tj,nλj,n

+τ ∈±∞ for every τ ∈ R if j ∈ JL. Hence (7.32).

Proof of (7.35). By the Claim 7.2, we have∣∣∣∣∣∣sin

(2ωJn + 2ΘJn +

J∑

j=1

2Ψjn

)− sin

(2ωJn

)− sin

(2ΘJn

)−

J∑

j=1

sin(2Ψjn

)∣∣∣∣∣∣

.∣∣sinωJn

∣∣3+∣∣sinΘJn

∣∣3+(∣∣sinωJn

∣∣+∣∣sinΘJn

∣∣)J∑

j=1

sin2 Ψjn+∑

1≤j,k≤Jj 6=k

sin2 Ψjn∣∣sinΨkn

∣∣ ,

Since | sinωJn |3 . |ωJn |3, we have by (7.22)

(7.38) limJ→∞

lim supn→∞

∥∥∥∥1

r3sin3 ωJn

∥∥∥∥L1L2

= 0.


Furthermore,

(7.39)

∥∥∥∥11ΓT

1

r3sin3 ΘJn

∥∥∥∥L1L2

.∥∥11ΓT θ

Jn

∥∥3L3L6 .

We will next prove

(7.40)

∥∥∥∥∥∥1

r311ΓT sinΘJn

J∑

j=1

(sinΨjn)2

∥∥∥∥∥∥L1L2

.

∫ T

0

∥∥θJn(t)∥∥L6(|t|) Fn(t)dt,

(where Fn satisfies (7.31)), and

limJ→∞

lim supn→∞

∥∥∥∥∥∥1

r311r>|t|(sinω

Jn)

J∑

j=1

sin2 Ψjn

∥∥∥∥∥∥L1L2

= 0(7.41)

j 6= k =⇒ limn→∞

∥∥∥∥1

r311r>|t| sin

2 Ψjn sinΨkn

∥∥∥∥L1L2

= 0,(7.42)

which will complete the proof of (7.35) and thus of the proposition. We first provethat there is a constant C > 0 independent of n such that

(7.43)∞∑

j=1

∥∥∥∥11r>|t|1

rsin(Ψjn)

∥∥∥∥2

L3L6

≤ C.

Let J0 ≥ 1 such that for all j ≥ J0, (Ψj0,Ψ

j1) ∈ H and EM (Ψj0,Ψ

j1) is small. By the

Claim 7.1, for j ≥ J0, EM (Ψj0,Ψj1) ≈ ‖(Ψj0,Ψj1)‖2H. By Strichartz estimates (and

the small data theory for (4.1) if j ∈ JC),∥∥11r>|t|

1rΨ

jn

∥∥L3L6 .

∥∥∥(Ψj0,Ψj1)∥∥∥H

for

large j. Since | sinΨjn| ≤ |Ψjn| we deduce, by the Pythagorean expansion (7.21) ofthe energy,

(7.44)

∞∑

j=J0

∥∥∥∥11r>|t|sin(Ψjn)

r

∥∥∥∥2

L3L6

≤ C.

Let j ∈ 1, . . . , J0 − 1. If j ∈ JC ,

(7.45)

∥∥∥∥11r>|t|sin(Ψjn)

r

∥∥∥∥L3L6

=

∥∥∥∥11r>|t|sin(Ψj)

r

∥∥∥∥L3L6

,

which is finite by Lemma 7.5, and if j ∈ JL,

(7.46)

∥∥∥∥11r>|t|sinΨjnr

∥∥∥∥L3L6

≤∥∥∥∥∥11r>|t|

ΨjLr

∥∥∥∥∥L3L6

,

which is finite by Strichartz estimates. Thus (7.43) holds.

Proof of (7.40). We have

(7.47)∥∥∥∥∥∥1

r311ΓT sinΘJn

J∑

j=1

sin2 Ψjn

∥∥∥∥∥∥L1L2

=

∫ T

0

∥∥∥∥1

r3sinΘJn(t)

J∑

j=1

sin2 Ψjn(t)

∥∥∥∥L2(|t|)

dt

≤∫ T

0

∥∥∥1rΘJn(t)

∥∥∥L6(|t|)

∥∥∥∥1

r2

J∑

j=1

sin2 Ψjn(t)

∥∥∥∥L3(|t|)

dt.


Furthermore,(7.48)∥∥∥∥

1

r2

J∑

j=1

sin2 Ψjn(t)

∥∥∥∥L3(|t|)

≤J∑

j=1

∥∥∥∥sin2 Ψjn(t)

r2

∥∥∥∥L3(|t|)

=

J∑

j=1

∥∥∥∥sinΨjn(t)

r

∥∥∥∥2

L6(|t|).

Let

Fn(t) =∞∑

j=1

∥∥∥∥sinΨjn(t)

r

∥∥∥∥2

L6(|t|).

Then

(7.49) ‖Fn‖L3/2 ≤∞∑

j=1

∥∥∥∥11r>|t|sinΨJnr

∥∥∥∥2

L3L6

≤ C

by (7.44). Combining (7.47), (7.48) and (7.49) we obtain (7.40).

Proof of (7.41). We have∥∥∥∥∥∥11r>|t|r3

(sinωJn)

J∑

j=1

sin2 Ψjn

∥∥∥∥∥∥L1L2

≤∥∥∥∥11r>|t|

rsinωJn

∥∥∥∥L3L6

∥∥∥∥∥∥11r>|t|r2

J∑

j=1

sin2 Ψjn

∥∥∥∥∥∥L3/2L3

≤∥∥∥∥11r>|t|

rsinωJn

∥∥∥∥L3L6

∞∑

j=1

∥∥∥∥11r>|t|sinΨJnr

∥∥∥∥2

L3L6

,

and (7.41) follows from (7.22) and (7.43).

Proof of (7.42). If j ∈ JL, we have by dominated convergence, as in (7.37),

limn→∞

∥∥∥∥11r>|t|1

rsinΨjn

∥∥∥∥L3L6

= 0.

Thus (7.42) holds if j ∈ JL or k ∈ JL. If j ∈ JC and k ∈ JC , (7.42) follows from

limn→∞

λj,nλk,n

+λk,nλj,n

= ∞,

and Lemma 7.5.

7.2. Proof of the soliton resolution. In this subsection, we deduce Theorem 3from the rigidity theorem (Theorem 4.2). The proof is close to the proof of Theorem4 and we will not give all the details. As in the proof of Theorem 4, we will focuson the finite time blow-up case, the proof for global solutions being very similar.

7.2.1. Analysis along a sequence of times. We let ψ be a solution of (1.1) suchthat T+ = T+(ψ) < ∞, and (ϕ0, ϕ1) as in Proposition 7.3. The core of the proofof Theorem 3 is the following proposition, which is the analog of Proposition 6.4.Recall from (2.1) the definition of the norm ‖ · ‖H .

Proposition 7.9. Let tnn be a sequence of times such that

limn→∞

tn = T+, ∀n, 0 ≤ tn < T+.


Then there exists a subsequence of tnn (that we still denote by tnn), J ≥ 1,(ιj)j ∈ ±1J, sequences λj,nn, 1 ≤ j ≤ J with

λ1,n ≪ . . .≪ λJ,n ≪ T+ − tn,

such that

limn→∞

∥∥∥∥∥∥ψ(tn)− ϕ0 −

J∑

j=1

ιjλj,n

Q

( ·λj,n

)∥∥∥∥∥∥H

= 0(7.50)

EM (ψ0, ψ1) = EM (ψ0, ψ1) + JEM (Q, 0) +1

2‖∂tψ(tn)− ψ1‖2L2 + on(1).(7.51)

Proof. The proof of Proposition 7.9 is close to the proof of Proposition 6.4, butsomehow easier, since solutions of the co-rotational maps equation (1.1) with initialdata at t = 0 are always defined for r > |t| and enjoy a global space-time boundthere (see Lemma 7.5).

To simplify notations, we assume without loss of generality in all the proof that

T+(ψ) = 1.

Step 1. Profile decomposition. We let ϕ be the solution of (1.1) with initial data(ϕ0, ϕ1) at t = 1. By Proposition 7.3,

supp(ϕ− ψ) ⊂ r ≤ 1− t.

Extracting subsequences, we assume that~ψ(tn)− (ϕ0, ϕ1)

nhas a profile decom-

position as in §7.1.4. We will use the notations of §7.1.4: Ψjn, λj,nn, tj,nn etc...for this decomposition. Recall the partition N \ 0 = JC ∪JL of the set of indicesdefined after Proposition 7.7. Rescaling the profiles, we can assume that there isanother partition N \ 0 = Is ∪ Ic ∪ I+ ∪ I− defined by

j ∈ Is ⇐⇒ j ∈ JC , Ψj0 ∈ Q,−Q ∪ mπ, m ∈ Z and Ψj1 = 0,

j ∈ Ic ⇐⇒ j ∈ JC and Ψj is not a stationary solution of (1.7),

j ∈ I± ⇐⇒ j ∈ JL and limn→∞

−tj,nλj,n

= ±∞.

In Step 2, we will prove that Ic = ∅, in Step 3, that I± = ∅. In Step 4, we excludeself-similar behaviour for the scaling parameters. In Step 5, we conclude the proof,showing that the first component of the dispersive remainder ωJ0,n goes to 0 in Has n goes to infinity.

Step 2. Compact profiles. We prove that Ic is empty by contradiction. Assumethat it is not empty and let j0 ∈ Jc. Assume (Ψj00 ,Ψ

j01 ) ∈ Hℓ,m.

By Theorem 1, there exists η1 > 0 such that the following holds for all t ≥ 0 orfor all t ≤ 0:

(7.52)

∫ ∞

|t|

((∂t,rΨ

j0(t, r))2 +1

r2sin2 Ψj0(t, r)

)rdr ≥ η1.

The sequence~ψ(tn)

n

has the same profile decomposition as the sequence~ψ(tn)− (ϕ0, ϕ1)

n, with an additional profile

(Ψ00,Ψ

01) = (ϕ0, ϕ1), ∀n, λj0,n = 1, tj0,n = 0.


By Proposition 7.8 and arguing as in Step 3 of the proof of Proposition 6.4, weobtain, for all r > |τ |,

(7.53) ψ(tn + τ) = ϕ(tn + τ) +

J∑

j=1

Ψjn(τ, r) + ωJn(τ, r) + ΘJn(τ, r),

where

(7.54) limJ→∞

lim supn→∞

supτ

∥∥∥~ΘJn(τ)∥∥∥H(τ)

= 0.

We distinguish between two cases.

Case 1. Channels of energy in the future. Assume that (7.52) holds for all t > 0.Then by (7.53) at τ = 1− tn,

J∑

j=1

Ψjn(1 − tn, r) + ωJn(1− tn, r) + ΘJn(1− tn, r) = 0, r > 1− tn,

which yields

(7.55)

∫ ∞

1−tn

(∂t,r

( J∑

j=1

Ψjn(1− tn, r) + ωJn(1− tn, r) + ΘJn(1 − tn, r)))2

rdr

+

∫ ∞

1−tnsin2

( J∑

j=1

Ψjn(1− tn, r) + ωJn(1 − tn, r) + ΘJn(1− tn, r))drr

= 0.

Fixing J large, and using that (7.52) holds for t ≥ 0, we claim that the left-handside of (7.55) is larger than

1

2

∫ ∞

1−tn

(∂t,rΨ

j0n (1 − tn, r)

)2rdr +

1

2

∫ ∞

1−tn

sin2 Ψj0n (1− tn, r)

rdr ≥ η1

2

for large n, a contradiction. In view of (7.54), and Claim 7.2, it is sufficient toprove

j 6= j0 =⇒ limn→∞

∫ ∞

1−tn∂t,rΨ

jn(1 − tn, r) · ∂t,rΨj0n (1− tn, r)rdr = 0(7.56)

j 6= j0 =⇒ limn→∞

∫ ∞

1−tn

∣∣sinΨjn(1 − tn, r) sinΨj0n (1− tn, r)

∣∣ drr

= 0(7.57)

1 ≤ j ≤ J =⇒ limn→∞

∫ ∞

1−tn∂t,rΨ

jn(1− tn, r) · ∂t,rωJn(1− tn, r)rdr = 0(7.58)

1 ≤ j ≤ J =⇒ limn→∞

∫ ∞

1−tn

∣∣sinΨjn(1− tn, r) sinωJn(1− tn, r)

∣∣ drr

= 0.(7.59)

The proof of (7.56),. . . ,(7.58) is very close to the proofs of the corresponding prop-erties for the linear wave profiles in Appendix B.1 and of the Pythagorean expansion(7.21) of the wave maps energy in Appendix B.2 and we omit it.


Case 2. Channels of energy in the past. We assume that (7.52) holds for all t ≤ 0.By (7.53) at τ = −tn,

(7.60)

∫ ∞

tn

∣∣∣∂t,r( J∑

j=1

Ψjn(−tn, r) + ωJn(−tn, r) + ΘJn(−tn, r))∣∣∣

2

rdr

+

∫ ∞

tn

sin2( J∑

j=1

Ψjn(−tn, r) + ωJn(1 − tn, r) + ΘJn(−tn, r))drr

=

∫ ∞

tn

|∂t,rψ(0, r)− ∂t,rϕ(0, r)|2 rdr +∫ ∞

tn

sin2(ψ(0, r) − ϕ(0, r))dr

r−→n→∞

0.

By the same argument as in Case 1, we obtain that the left-hand side of (7.60) islarger than

1

2

∫ ∞

tn

(∂t,rΨ

j0n (−tn, r)

)2rdr +

1

2

∫ ∞

tn

sin2 Ψj0n (−tn, r)r

dr ≥ η12,

a contradiction.

Step 3. Scattering profiles. In this step we prove by contradiction that I+ is empty.This step is very close to Case 1 of the preceding step, and also to Step 4 in theproof of Proposition 6.4, and we only sketch it. The proof that I− is empty is verysimilar and we omit it.

Assume that I+ 6= ∅ and let j0 ∈ I+. Since j0 ∈ JL, we have

Ψj0n (t, r) = Ψj0(t− tj0,nλj0,n

,r

λj0,n

)

where Ψj0 is a solution of the linearized equation (7.6), with nonzero initial datain H. Then U j0 = 1

rΨj0 is a radial solution of the linear wave equation in space

dimension 4, and we deduce as in Step 4 of the proof of Proposition 6.4 that thereexists η1 > 0, such that for large n,

∫ ∞

1−tn

∣∣∂t,rU j0n (1− tn, r)∣∣2 r3dr ≥ 2η1,

where U j0n (t, r) = 1λj0 ,n

U j0(t−tj0,n

λj0,n, rλj0,n

). As a consequence, for large n

∫ ∞

1−tn

∣∣∂t,rΨj0n (1 − tn, r)∣∣2 rdr +

∫ ∞

1−tn

∣∣Ψj0n (1− tn, r)∣∣2 dr

r≥ 2η1.

Using that limn→∞ ‖Ψj0n (1− tn)‖L∞ = 0 (see (2.5)), we deduce that for large n,∫ ∞

1−tn

∣∣∂t,rΨj0n (1− tn, r)∣∣2 rdr +

∫ ∞

1−tnsin2 Ψj0n (1− tn, r)

dr

r≥ η1.

The end of the proof is the same as in Case 1 of Step 2 above, and we omit.

Step 4. Bound on the scaling parameters. According to the preceding steps,(ωJ0,n, ω

J1,n) is, for large J , independent of J , and we will drop the superscript

J . Furthermore, all the profiles are in Is. Using the Pythagorean expansion of

the energy (7.21) for the sequence ~ψ(tn), we see that there exists J0 ≥ 0, m ∈ Z,(ιj)1≤j≤J0

∈ ±1J0, sequences λj,nn with 0 < λ1,n ≪ . . .≪ λJ0,n such that

(7.61) ~ψ(tn) = ~ϕ(tn)+(mπ, 0)+

J0∑

j=1

(ιjQ

( ·λj,n

), 0

)+(ω0,n, ω1,n)+on(1) in H.


Using that (ω0,n, ω1,n) ∈ H, and that ψ(tn, r) = ϕ(tn, r) for r > 1− tn, we see that

m = −∑J0

j=1 ιj . In this step, we prove

(7.62) limn→∞

λJ0,n

1− tn= 0.

Arguing by contradiction and extracting subsequences, we assume that there existsR > 0 such that for large n,

λj0,n ≥ R(1− tn).

By (7.61), and the fact that ψ(tn, r)− ϕ(tn, r) = 0 for r > 1− tn, we obtain that

(7.63) ω0,n(λJ0,nr) +mπ +

J0−1∑

j=1

ιjQ

(λJ0,nr

λj,n

)+Q(r) = on(1) in H(R).

Consider χ ∈ C∞0

((R,∞)

), nonnegative and not identically zero. Since ω0,n(λJ0,n·)

converges to 0 weakly in H , we have

limn→∞

∫ ∞

0

ω0,n(λJ0,nr)χ(r)dr = 0.

Furthermore, for all j ∈ 1, . . . , J0 − 1, we have

limn→∞

ιj

∫ ∞

0

Q

(λJ0,nr

λj,n

)χ(r)dr = ιjπ

∫ ∞

0

χ(r)dr.

Thus (7.63) implies, letting n→ ∞,(m+

J0−1∑

j=1

ιj

)π

∫ ∞

0

χ(r)dr + ιJ0

∫ ∞

0

Q(r)χ(r)dr = 0.

Using that m+∑J0−1

j=1 ιj = −ιJ0, we deduce

ιJ0

∫ ∞

0

(Q− π)(r)χ(r)dr = 0,

a contradiction since Q < π.

Step 5. Dispersive remainder. In this step we prove

(7.64) limn→∞

‖ω0,n‖H = 0

which will conclude the proof of Proposition 7.9. Using that (ω0,n, ω1,n) ∈ H, weobtain as usual that wn = 1

rωn is a radial finite energy solution of the free waveequation (3.1) in space dimension 4. Thus by the result of [12] (see (3.4)),

∑

±lim

t→±∞

∫ ∞

|t||∂t,rwn(t, r)|2 r3dr & ‖w0,n‖2H1 ,

where w0,n = 1rω0,n. Using the decay of the exterior energy for the linear wave

equation, we obtain that the following holds for all t ≥ 0 or for all t ≤ 0∫ ∞

|t||∂t,rwn(t, r)|2 r3dr & ‖w0,n‖2H1 .

Going back to ωn, we deduce that the following holds for all t ≥ 0 or for all t ≤ 0.∫ ∞

|t||∂t,rωn(t, r)|2 rdr +

∫ ∞

|t|

1

r2(ωn(t, r))

2rdr & ‖ω0,n‖2H .


Since by (7.22), limn ‖wn‖L∞ = 0, we deduce that for all t ≥ 0 or for all t ≤ 0∫ ∞

|t||∂t,rωn(t, r)|2 rdr +

∫ ∞

|t|sin2 ωn(t, r)

dr

r& ‖ω0,n‖2H .

The end of the proof of (7.64) is similar to Step 2, and we omit it.

7.2.2. End of the proof of the soliton resolution. The end of the proof of Theorem3 in the finite time blow-up case is close to the end of the proof of Theorem 4.However, the fact that the energy always gives an a priori bound on a solution ψof the wave maps equation (1.1) makes the proof slightly simpler.

As before, we let ψ be a solution of (1.1) such that T+(ψ) = 1. By Proposition7.3, there exists a sequence of times tn → 1 such that

(7.65) limn→∞

‖∂tψ(tn)− ϕ1‖L2 = 0.

(Here and until the end of this proof, L2 denotes the space L2((0,∞)

)with respect

to the measure rdr.) Combining (7.65) with Proposition 7.9, we obtain (extractingsubsequences) that there exists J ≥ 0, (ιj)j ∈ ±1J , scaling parameters 0 <λ1,n ≪ . . .≪ λJ,n such that

(7.66) limn→∞

∥∥∥∥~ψ(tn)− (ϕ0, ϕ1)− (mπ, 0)−J∑

j=1

(ιjQ(λj,n·), 0)∥∥∥∥H

= 0.

Since tn → 1 and 1 is the maximal time of existence of ψ, we must have J ≥ 1.Using the Pythagorean expansion (7.21) of the energy, we see that

(7.67) EM (ψ0, ψ1) = EM (ϕ0, ϕ1) + JEM (Q, 0).

We next prove by contradiction

(7.68) limt→1

‖∂tψ(t)− ϕ1‖L2 = 0.

If (7.68) does not hold, using (7.65) and the intermediate value theorem, thereexists a sequence of times t′n → 1 such that

(7.69) limn→∞

‖∂tψ(t′n)− ϕ1‖L2 = ε0.

Using Proposition 7.9 along the sequence t′nn, we obtain (extracting subsequences)

J ′ ≥ 0, (ι′j)j ∈ ±1J′

, 0 < λ′1,n ≪ . . .≪ λ′J′,n such that

(7.70) limn→∞

∥∥∥∥ψ(t′n)− ϕ0 −mπ −

J′∑

j=1

ι′jQ

( ·λ′j,n

)∥∥∥∥H

= 0.

This yields

(7.71) EM (ψ0, ψ1) = EM (ϕ0, ϕ1) +1

2ε20 + J ′EM (Q, 0).

Combining with (7.67), we see that 12ε

20 = (J − J ′)EM (Q, 0), which yields a con-

tradiction since ε0 is small and not equal to 0, concluding the proof of (7.67).Combining Proposition 7.9, the conservation of the energy and (7.68), we obtain

that J = EM (ψ0,ψ1)−EM (ϕ0,ϕ1)EM (Q,0) is a positive integer, and that for all sequences of

times tn → 1, there exists a subsequence of tnn, (ιj)j ∈ ±1J , and sequencesλj,nn, 1 ≤ j ≤ J with

0 < λ1,n ≪ . . .≪ λJ,n ≪ 1− tn,


such that

(7.72) limn→∞

∥∥∥∥~ψ(tn)− (ϕ0, ϕ1)−mπ −J∑

j=1

(ιjQ

( ·λj,n

), 0

)∥∥∥∥H

= 0,

where m =∑ιj .

The fact that we can choose the ιj (and thus m) independently of the sequencet′nn, and that the preceding statement implies the conclusion of Theorem 3 (withφ0 = ϕ0−mπ, φ1 = ϕ1) is standard and we omit it (see the end of Section 6 above,and Section 3.5 of [20]).

The proof of Theorem 3 in the case where T+(ψ) = +∞ is very close, using thescattering state ψL (defined in Proposition 7.4) instead of (ϕ0, ϕ1). We again omitthe details.

Appendix A. Radiation term for the free wave equation

In this appendix we consider the linear inhomogeneous wave equation in anyspace dimension N ≥ 3:

(A.1)

∂2t u−∆u = f, (t, x) ∈ R× R

N

~ut=0 = (u0, u1),

with radial data.

Proposition A.1. Let

(u0, u1) ∈ H = (H1 × L2)rad(RN ), f ∈ L1(R, L2

rad(RN )),

and u be the corresponding solution of (A.1). Then there exists G ∈ L2(R) suchthat

limt→∞

∫ +∞

0

∣∣∣rN−1

2 ∂ru(t, r)−G(r − t)∣∣∣2

dr = 0(A.2)

limt→∞

∫ +∞

0

∣∣∣rN−1

2 ∂tu(t, r) +G(r − t)∣∣∣2

dr = 0(A.3)

If furthermore (u0, u1) ∈ H2 × H1 and ∂tf ∈ L1L2, then

limt→∞

∫ +∞

0

∣∣∣rN−1

2 ∂r∂tu(t, r) +G′(r − t)∣∣∣2

dr = 0(A.4)

limt→∞

∫ +∞

0

∣∣∣rN−1

2 ∂2t u(t, r)−G′(r − t)∣∣∣2

dr = 0(A.5)

Proof. The fact that (A.2) and (A.3) hold in the case f = 0 is well-known (see [24,Theorem 2.1]) and goes back at least to the work of Friedlander [33].

To prove (A.2) and (A.3) in the general case, one can reduce to the case f = 0,by recalling that if u is a solution of (A.1) with f ∈ L1(R, L2(RN )), there exists afinite energy solution vL of the free wave equation (∂2t −∆)vL = 0 such that

(A.6) limt→∞

‖~u(t)− ~vL(t)‖H = 0.


This follows immediately from the Duhamel formulation of (A.1), which yields that(A.6) holds with

vL(t) = cos(t√−∆)

(u0 −

∫ +∞

0

sin(s√−∆)√

−∆f(s)ds

)

+sin(t

√−∆)√

−∆

(u1 +

∫ +∞

0

cos(s√−∆

)f(s)ds

).

We next assume that u0 ∈ H2, u1 ∈ H1, ∂tf ∈ L1(R, L2) and prove (A.4), (A.5).We first note that ∂tu is solution of

(A.7)

∂2t ∂tu−∆∂tu = ∂tf, (t, x) ∈ R× R

N

~ut=0 = (u1,∆u0 + f(0)) ∈ H,(where f(0) is well defined and in L2(RN ) by an elementary trace lemma). By theconsiderations above, there exists H ∈ L2(R) such that

limt→∞

∫ +∞

0

∣∣∣rN−1

2 ∂r∂tu(t, r) −H(r − t)∣∣∣2

dr = 0(A.8)

limt→∞

∫ +∞

0

∣∣∣rN−1

2 ∂2t u(t, r) +H(r − t)∣∣∣2

dr = 0,(A.9)

and we are reduced to prove that H = −G′.Let χ ∈ C∞(R) such that χ(η) = 0 if η ≤ 1

2 and χ(η) = 1 if η ≥ 1. Since

limt→∞

∫ 1

0

|G(r − t)|2dr = 0,

the equality (A.3) implies

limt→∞

∫

R

∣∣∣χ(r)rN−1

2 ∂tu(t, r) +G(r − t)∣∣∣2

dr = 0,

that is

limτ→∞

∫

R

∣∣∣(η + τ)N−1

2 χ(η + τ)∂tu(τ, η + τ) +G(η)∣∣∣2

dη = 0.

In other words, the family of functions

η 7→ (η + τ)N−1

2 χ(η + τ)(∂tu)(τ, η + τ)

converge to −G in L2(R) as τ goes to infinity. Taking the derivative in η and usingthat

limt→∞

∫ +∞

0

|∂tu(t, r)|2rN−3dr = 0

(which follows from the fact that there exists a finite energy solution wL of ∂2twL−∆wL = 0 such that limt→∞ ‖∂tu(t) − wL(t)‖H1 = 0, and a classical dispersiveestimate, see e.g. [24, Theorem 2.1]) we obtain that the family of functions

η 7→ (η + τ)N−1

2 χ(η + τ)(∂t∂ru)(τ, η + τ)

converge to −G′ in D′(R) as τ → ∞. However by (A.8), we obtain that this familyconverge to H in L2(R) as τ → ∞. By uniqueness of the distributional limit,H = −G′ concluding the proof.

Remark A.2. We recall that for all G ∈ L2(R), there exists a unique finite energysolution u of (A.1) such that (A.2) and (A.3) hold: see again [24, Theorem 2.1].


Appendix B. Pseudo-ortogonality and channels of energy

B.1. Linear wave equation. In this subsection, we prove (6.10) and (6.11)Since

∫ρn<|x|<ρ′n

. . . =∫ρn<|x| . . .−

∫ρ′n<|x|, it is sufficient to treat the case where

ρ′n = ∞ for all n. Rescaling un, we also assume

∀n, λj,n = 1.

Pseudo-orthogonality between two profiles. We first prove (6.10). We will use thepseudo-orthogonality (6.1) of the scaling and time translation parameters. We fixj 6= k. Let

εn :=

∫

|x|>ρn∇t,xU

jn(sn, x) · ∇t,xU

kn(sn, x)dx

=

∫

|x|>ρn∇t,xU

j (sn − tj,n, x) ·1

λ2k,n∇t,xU

k

(sn − tk,nλk,n

,x

λk,n

)dx.

Arguing by contradiction, we can extract subsequences and assume that the follow-ing limits exist in R ∪ ±∞.

limn→∞

sn − tj,n = Tj , limn→∞

sn − tk,nλk,n

= Tk.

Case 1: Tj ∈ R, Tk ∈ R. In this case we have

εn = on(1) +

∫

|x|>ρn∇t,xU

j(Tj , x) ·1

λ2k,n∇t,xU

k

(Tk,

x

λk,n

)dx.

Since (6.1) implies limn λk,n ∈ 0,∞, we obtain right away limn εn = 0.

Case 2. Tj = +∞, Tk ∈ R. We note that in this case, there exists gj ∈ L2(R) suchthat

(B.1) εn = on(1) +

∫ ∞

ρn

gj(r − sn + tj,n)1

λ2k,n∂tU

k

(Tk,

r

λk,n

)r3/2dr

−∫ ∞

ρn


λ2k,n∂rU

k

(Tk,

r

λk,n

)r3/2dr.

Indeed if limn→∞ tj,n ∈ ±∞, U j is a solution of the free wave equation (3.1),and (B.1) follows from the asymptotic behaviour for these solutions, recalled inProposition A.1 above. If tj,n = 0 for all n, then by the definition of Rn and the

assumptions ρn ≥ Rn + |sn|, there exists a solution V jL of (3.1) such that

limn→∞

∫

|x|>ρn

∣∣∣∇t,xUj(sn − tj,n, x)−∇t,xV

jL(sn − tj,n, x)

∣∣∣2

dx = 0,

and the claim follows again from Proposition A.1. Letting hk = r3/2∂tUk(Tk, r) ∈

L2(0,∞), we see that∫ ∞

ρn

gj(r−sn+tj,n)1

λ2k,n∂tU

k

(Tk,

r

λk,n

)r3/2dr =

∫ ∞

ρn

gj(r−sn+tj,n)hk(

r

λk,n

)dr,

which tends to 0 as n tends to infinity because limn sn − tj,n = ∞. Since theterm in the second line of (B.1) can be treated in the exact same way, we obtainlimn→∞ εn = 0.

The proof in the cases Tj = −∞, Tk ∈ R, and Tj ∈ R, Tk = ±∞ are very closeand we omit them.


Case 3. Tj = +∞ and Tk = +∞. Similarly to Case 2, there exist gj ∈ L2(R) andgk ∈ L2(R) such that

εn = on(1) + 2

∫ ∞

ρn


λ2k,ngk(r − sn + tk,n

λk,n

)dr,

and the fact that limn→∞ εn = 0 follows easily from the pseudo-orthogonalityproperty (6.1).

The proof in the case Tj = −∞, Tk = −∞ is the same and we omit it.

Case 4. Tj = +∞ and Tk = −∞. Using Proposition A.1 as in cases 2 and 3, weobtain that the contributions of the integrals of ∂tU

jn∂tU

kn and ∂rU

jn∂rU

kn in the

definition of εn are opposite, and thus

limn→∞

εn = 0.

The proof in the case Tj = −∞, Tk = +∞ is exactly the same.

Pseudo-orthogonality between a profile and the dispersive remainder.We next prove (6.11). We assume again that Tj = limn→∞ sn − tj,n exists in

R ∪ ±∞.Case 1. Tj ∈ R. Then

εn = on(1) +

∫

|x|>ρn∇t,xU

j(Tj, x)∇t,xwJn(sn, x)dx.

If limn→∞ ρn = +∞, we deduce immediately that limn→∞ εn = 0. If not, we canassume limn→∞ ρn = ρ∞ ∈ [0,∞), and we obtain

εn = on(1) +

∫

|x|>ρ∞∇t,xU

j(Tj, x)∇t,xwJn(sn, x)dx.

By (6.10), we can take J arbitrarily large. By the definitions of the profiles, and(6.5) we obtain, for J large enough,

(B.2)(wJn (sn) , ∂tw

Jn (sn)

)−−−−n→∞

0 in H.

As a consequence, limn εn = 0.

Case 2. Tj = +∞. Arguing as in Case 2 in the proof of (6.10), we obtain thatthere exists gj ∈ L2(R) such that

εn = on(1) +

∫ ∞

ρn

gj(r + tj,n − sn)(∂tw

Jn(sn, r)− ∂rw

Jn(sn, r)

)r3/2dr.

If limn ρn + tj,n − sn = +∞, we obtain right away that εn = on(1).If limn ρn + tj,n − sn = −∞, then

∫ ρn

0

∣∣gj(r + tj,n − sn)∣∣2 dr ≤

∫ ρn+tj,n−sn

−∞|g(η)|2 −→

n→∞0,

and thus

εn = on(1) +

∫ ∞

0

gj(r + tj,n − sn)(∂tw

Jn(sn, r)− ∂rw

Jn(sn, r)

)r3/2dr

= on(1) +

∫∇t,xV

jL(sn − tj,n) · ∇t,xw

Jn(sn, x)dx,


where V jL is the radial solution of the linear wave equation such that

limt→∞

∫ ∞

0

|r3/2∂tV jL(t, r) − gj(r − t)|2dr +∫ ∞

0

|r3/2∂rV jL(t, r) + gj(r − t)|2dr = 0.

By conservation of the energy for the free wave equation (3.1),

εn = on(1) +

∫∇t,xV

jL(0) · ∇t,xw

Jn(tj,n, x)dx,

and thus by (6.5), εn = on(1).Finally, if limn ρn + tj,n − sn = c ∈ R, we see that

εn = on(1) +

∫ ∞

0

gjc(r + tj,n − sn)(∂tw

Jn(sn, r)− ∂rw

Jn(sn, r)

)r3/2dr

= on(1) +

∫∇t,xV

jL(sn − tj,n) · ∇t,xw

Jn(sn, x)dx,

where gc(η) = 11η>cg(η), and VjL is the radial solution of the linear wave equation

such that

limt→∞

∫ ∞

0

|r3/2∂tV jL(t, r) − gjc(r − t)|2dr +∫ ∞

0

|r3/2∂rV jL(t, r) + gjc(r − t)|2dr = 0.

The same proof as before yields εn = on(1).Since the case Tj = −∞ can be treated exactly in the same way, the proof of

(6.11) is complete.

B.2. Pythagorean expansion of the energy for wave maps. In this subsec-tion we prove (7.21). In view of Claim 7.2, it is sufficient to prove

j 6= k =⇒ limn→∞

∫ ∞

0

∂rΨj0,n∂rΨ

k0,nrdr +

∫ ∞

0

Ψj1,nΨk1,nrdr = 0(B.3)

j 6= k =⇒ limn→∞

∫ ∞

0

∣∣∣sinΨj0,n sinΨk0,n∣∣∣ drr

= 0(B.4)


∫ ∞

0

∂rΨj0,n∂rω

J0,nrdr +

∫ ∞

0

Ψj1,nωJ1,nrdr = 0(B.5)


∫ ∞

0

∣∣∣sinΨj0,n sinωJ0,n∣∣∣ drr

= 0.(B.6)

Recall that if j ∈ JL, i.e. limn→∞ −tj,n/λj,n ∈ ±∞, then(Ψj0,n(r),Ψ

j1,n(r)

)=

(ΨjL

(−tj,nλj,n

,r

λj,n

),

1

λj,n∂tΨ

jL

(−tj,nλj,n

,r

λj,n

)),

where ΨjL is a solution of the linear equation (7.6) with initial data in H. ThusU j = 1

rΨj is a radial solution of the 1+ 4 dimensional wave equation. By standard

dispersive estimates for linear wave equations,

(B.7) limn→∞

∫ ∞

0

∣∣∣Ψj0,n(r)∣∣∣2 dr

r= lim

n→∞

∫ ∞

0

∣∣∣∣Uj

(−tj,nλj,n

, r

)∣∣∣∣2

rdr = 0.

This yields (B.4) when j ∈ JL or k ∈ Jk, and (B.6) when j ∈ JL.


Next, we assume j ∈ JL and k ∈ JL and see that by (B.7) letting U jn(t, r) =1

λj,nU j(t−tj,nλj,n

, rλj,n

)and defining similarly Ukn , we have

limn→∞

∫ ∞

0

∂rΨj0,n∂rΨ

k0,nrdr +

∫ ∞

0

Ψj1,nΨk1,nrdr

=

∫ ∞

0

∂rUjn(0, r)∂tU

jn(0, r)r

3dr +

∫ ∞

0

∂rUjn(0, r)∂rU

jn(0, r)r

3dr + on(1),

and (B.3) follows from (6.10).It remains to prove (B.3) when j ∈ JC or k ∈ JC , (B.4) when j ∈ JC and

k ∈ JC , and (B.5), (B.6) when j ∈ JC .Proof of (B.3) when j ∈ JC , k ∈ JL. Assume to fix ideas limn→∞ −tj,n/λj,n =+∞. By the asymptotic formulas (A.2) and (A.3), there exists Gk ∈ L2(R) suchthat

limt→+∞

∫ ∞

0

∣∣∣r3/2∂tUk(t, r) +Gk(r − t)∣∣∣ dr +

∫ ∞

0

∣∣∣r3/2∂rUk(t, r) −Gk(r − t)∣∣∣ dr.

Using also that limt→∞∫∞0 |Uk(t, r)|2rdr = 0, we deduce

limt→+∞

∫ ∞

0

∣∣∣r1/2∂tΨk(t, r) +Gk(r − t)∣∣∣ dr +

∫ ∞

0

∣∣∣r1/2∂rΨk(t, r) −Gk(r − t)∣∣∣ dr.

Thus∣∣∣∣∫ ∞

0

∂rΨj0,n(r)∂rΨ

k0,n(r)rdr

∣∣∣∣

=

∣∣∣∣∣

∫ ∞

0

1

λj,n∂rΨ

j0

(r

λj,n

)Gk(tk,n + r

λk,n

)r1/2

λ1/2k,n

dr

∣∣∣∣∣

=

∣∣∣∣∣

∫ ∞

0

∂rΨj0 (s)

λ1/2k,n

λ1/2j,n

Gk(tk,n + λj,ns

λk,n

)s1/2ds

∣∣∣∣∣ ,

which goes to 0 when n goes to infinity (this is immediate when Gk is continuousand compactly supported, and follows by density in the general case). By a verysimilar computation,

limn→∞

∫ ∞

0

Ψj1,n(r)Ψk1,n(r)rdr = 0,

which concludes the proof of (B.3) in this case.

Proof of (B.3) and (B.4) when j, k ∈ JC . We have

∫ ∞

0

∂rΨj0,n∂rΨ

k0,nrdr =

∫ ∞

0

1

λj,n∂rΨ

j0

(r

λj,n

)1

λk,n∂rΨ

k0

(r

λk,n

)rdr

=

∫ µn

0

. . .+

∫ ∞

µn

. . . ,

where µn =√λj,nλk,n. Using Cauchy-Schwarz on each of the integrals, we see that

it goes to 0 when n goes to ∞, since limnλj,n

λk,n+

λk,n

λj,n= ∞. By the same proof for

the term involving Ψj1,n and Ψk1,n, we obtain (B.3).

The proof of (B.4) when j, k ∈ JC is the same.


Proof of (B.6) when j ∈ JC . We have∫ ∞

0

∣∣∣sinΨj0,n(r) sinωJ0,n(r)∣∣∣ drr

=

∫ ∞

0

∣∣∣sinΨj0 (r) sinωJ0,n(λj,nr)∣∣∣ drr.

Let ε > 0. Then

∫ ε−1

ε

∣∣∣sinΨj0(r) sinωJ0,n(λj,nr)∣∣∣ drr

≤(∫ ε−1

ε

∣∣∣sinΨj0(r)∣∣∣3/2 dr

r3/2

)2/3(∫ ε−1

ε

∣∣ωJ0,n(λj,nr)∣∣3 dr

)1/3

.

Let wJn(r) = 1rω

Jn(λj,nr). Then (7.23) implies that λj,nw

Jn(λj,n·), considered as

a radial function on R4, converges weakly to 0 in H1(R4). This implies that

λj,nwJn(λj,n·) converges strongly in L3

loc(R4), and thus

limn→∞

∫ ε−1

ε

∣∣ωJ0,n(λj,nr)∣∣3 dr = 0.

As a consequence, for all ε

limn→∞

∫ ε−1

ε

∣∣∣sinΨj0(r) sinωJ0,n(λj,nr)∣∣∣ drr

= 0,

and (B.6) follows since∫∞0

sin2 Ψj0(r)drr is finite and

∫∞0

sin2 ωJ0,n(λj,nr)drr is uni-

formly bounded.

Appendix C. Boundedness of integral operators on LpLq spaces

In this appendix, we consider, as in the core of the article, functions defined fort ∈ R, x ∈ R

4 that are radial in the space variable. As before we use the notation

‖u‖(LpLq)(R) :=∥∥11|x|>R+|t|u

∥∥Lp(R,Lq(R4))

.

Lemma C.1. Let R ≥ 0. For w ∈ (L2L8)(R), define

(Aw)(t, r) =1

r2

∫ t

0

w(τ, r)dτ.

Then Aw ∈ (L2L8/3)(R) and

‖Aw‖(L2L8/3)(R) . ‖w‖(L2L8)(R),

where the implicit constant is independent of R ≥ 0.

Proof. Step 1. We first claim that if h ∈ (L2L8)(R), then |t|r2h ∈ (L2L8/3)(R).

Indeed(∫ ∞

R+|t||h(t, r)|8/3

( |t|r2

)8/3

r3dr

)3/8

. |t|(∫

R+|t|h(t, r)8r3dr

)1/8(∫ ∞

R+|t|

r3

r8dr

)1/4

.|t|

R + |t|

(∫ ∞

R+|t|h(t, r)8r3dr

)1/8

.


Now |t|R+|t| ≤ 1 and t 7→

(∫∞R+|t| h(t, r)

8r3dr)1/8

is in L2 in time, concluding this

step.

Step 2. In this step, we consider a positive function f in L2(R, L8(R4)), and define

M(f)(t, x) = supI∋t

1

|I|

∫

I

f(τ, x)dτ,

where the supremum is taken over all finite intervals I that contain t, and we prove

(C.1) ‖M(f)‖L2L8 . ‖f‖L2L8 .

Indeed, if f ∈ L2L2 then

(C.2) ‖M(f)‖L2L2 . ‖f‖L2L2

by Fubini and Hardy-Littlewood maximal theorem in the time variable. Further-more, if f ∈ L2L∞, one can prove:

(C.3) ‖M(f)‖L2L∞ . ‖f‖L2L∞ .

Indeed,

M(f)(t, x) . supI∋t

1

|I|

∫

I

‖f(τ)‖L∞xdτ

and thus

‖Mf(t)‖L∞x

≤ supI∋t

1

|I|

∫

I

‖f(τ)‖L∞xdτ.

Using Hardy-Littlewood maximum theorem in the t variable, we obtain (C.3). In-terpolation gives (C.1).

Step 3. Let w ∈ (L2L8)(R). Let

f(t, r) = |w(t, r)|11r>t+R.

Then, for r > |t|+R, we have

1

|t|

∣∣∣∣∫ t

0

w(τ, r)dτ

∣∣∣∣ ≤1

|t|

∫ |t|

−|t|11r>τ+R|w(τ, r)|dτ ≤ 2M(f)(t, r),

since r > |t|+R ≥ |τ | +R for 0 ≤ |τ | ≤ |t|. As a consequence, for r > |t|+R,

|A(w)(t, r)| . |t|r2M(f)(t, r).

By Step 1 with h(t, r) =M(f)(t, r), then Step 2, we have

‖A(w)‖(L2L8/3)(R) . ‖M(f)‖(L2L8)(R) . ‖w‖(L2L8)(R).

Lemma C.2. Let R ≥ 1, λ > 1/R. For w ∈ (L3L6)(R), we define

(Bw)(t, r) =t

r4 log(rλ)1/2

∫ t

0

w(τ, r)dτ.

Then B is bounded from (L3L6)(R) to (L1L2)(R) and

‖Bw‖(L1L2)(R) .1

log(λR)1/3‖w‖(L3L6)(R)


Proof. We just work with t > 0. By Minkowski in time,

(∫ ∞

R+t

(∫ t

0

w(τ, r)t

r4 log(λr)dτ

)2

r3dr

)1/2

≤∫ t

0

∥∥∥∥w(τ, r)t

r4 log(λr)

∥∥∥∥L2(R+t)

dτ.

Furthermore, by Holder

∥∥∥∥w(τ, r)t

r4 log(λr)

∥∥∥∥L2(R+t)

= t

(∫ ∞

R+t

|w(τ, r)|2 r3

r8 log2(λr)dr

)1/2

≤ t

(∫ ∞

R+t

|w(τ, r)|6r3dr)1/6(∫ ∞

R+t

r3

r12 log3(λr)dr

)1/3

.t

(R+ t)8/3 log(λ(R + t))

∫ t

0

(∫ ∞

R+t

|w(τ, r)|6r3dr)1/6

dτ

Integrating, we obtain

‖11t>0B(w)‖(L1L2)(R)

.

∫ ∞

0

t

(R + t)8/3 log(λ(R + t))

∫ t

0

(∫ ∞

R+t

|w(τ, r)|6r3dr)1/6

dτdt

By Fubini,

‖11t>0B(w)‖(L1L2)(R)

.

∫ ∞

0

(∫ ∞

R+τ

|w(τ, r)|6r3dr)1/6 ∫ ∞

τ

t

(R+ t)8/3 log(λ(R + t))dt dτ

.

∫ ∞

0

(∫ ∞

R+τ

|w(τ, r)|6r3dr)1/6 ∫ ∞

τ

dt

(R + t)5/3 log(λ(R + t))dτ

.

∫ ∞

0

(∫ ∞

R+τ

|w(τ, r)|6r3dr)1/6

dτ

(R+ τ)2/3 log(λ(R + τ))

.

(∫ ∞

0

(∫ ∞

R+τ

|w(τ, r)|6r3dr)1/2

dτ

)1/3 (∫ ∞

0

dτ

(R+ τ) log(λ(R + τ))3/2dτ

)2/3

,

where at the last line we have used Holder’s inequality. The desired conclusionfollows easily.

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