Construction of multi-solitons for the energy-critical wave equation in dimension 5
aa r X i v : . [ m a t h . A P ] A p r CONSTRUCTION OF MULTI-SOLITONS FOR THE ENERGY-CRITICALWAVE EQUATION IN DIMENSION 5
YVAN MARTEL AND FRANK MERLE
Abstract.
We construct -solitons of the focusing energy-critical nonlinear wave equationin space dimension , i.e. solutions u of the equation such that u ( t ) − [ W ( t ) + W ( t )] → as t → + ∞ in the energy space, where W and W are Lorentz transforms of the explicit standing soliton W ( x ) = (1 + | x | / − / , with any speeds ℓ = ℓ ( | ℓ k | < ). The existence result alsoholds for the case of K -solitons, for any K ≥ , assuming that the speeds ℓ k are collinear.The main difficulty of the construction is the strong interaction between the solitonsdue to the slow algebraic decay of W ( x ) as | x | → + ∞ . This is in contrast with previousconstructions of multi-solitons for other nonlinear dispersive equations (like generalized KdVand nonlinear Schrödinger equations in energy subcritical cases), where the interactions areexponentially small in time due to the exponential decay of the solitons. Introduction
Statement of the main result.
We consider the focusing energy-critical nonlinear waveequation in dimension ( ∂ t u − ∆ u − | u | u = 0 , ( t, x ) ∈ [0 , ∞ ) × R ,u | t =0 = u ∈ ˙ H , ∂ t u | t =0 = u ∈ L . (1.1)Recall that the Cauchy problem for equation (1.1) is locally well-posed in the energy space ˙ H × L , using suitable Strichartz estimates. See e.g. [26, 11, 16, 29, 30, 28, 12, 14]. Notethat equation (1.1) is invariant by the ˙ H scaling: if u ( t, x ) is solution of (1.1), then u λ ( t, x ) = 1 λ / u (cid:18) tλ , xλ (cid:19) is also solution of (1.1) and k u λ k ˙ H = k u k ˙ H . For ˙ H × L solution, the energy E ( u ( t ) , ∂ t u ( t )) and momentum M ( u ( t ) , ∂ t u ( t )) are conserved, where E ( u, v ) = 12 Z v + 12 Z |∇ u | − Z | u | , M ( u, v ) = Z v ∇ u. Recall that the function W defined by W ( x ) = (cid:18) | x | (cid:19) − , ∆ W + W = 0 , x ∈ R , (1.2)is a stationary solution, called soliton , of (1.1). Using the Lorentz transformation on W , weobtain traveling solitons: for ℓ ∈ R , with | ℓ | < , let W ℓ ( x ) = W p − | ℓ | − ! ℓ ( ℓ · x ) | ℓ | + x ! ; (1.3) then u ( t, x ) = ± W ℓ ( x − ℓ t ) is solution of (1.1).Recall that an important conjecture in the field says that any global solution of (1.1)decomposes as t → + ∞ as a finite sum of (rescaled and translated) solitons plus a radiation(solution of the linear wave equation). Such a classification was achieved in the radial case in[8] (in space dimension 3) but is still widely open in the nonradial case (see [9] and referencestherein).In this paper, we address the question of the construction of non trivial asymptotic behaviorsin the nonradial case. In this context, multi-solitons are canonical objects behaving as t → ∞ exactly as the sum of several solitons in the energy space. The main result of this paper is theexistence of -solitons for (1.1) and of K -solitons for K ≥ for collinear speeds. Theorem 1 (Existence of multi-solitons) . Let K ≥ . For k ∈ { , . . . , K } , let λ ∞ k > , y ∞ k ∈ R , ι k = ± and ℓ k ∈ R with | ℓ k | < , ℓ k = ℓ k ′ for k ′ = k .Assume that one of the following assumptions holds (A) Two-solitons ( K = 2 ).(B) Collinear speeds. For all k ∈ { , . . . , K } , ℓ k = ℓ k e where ℓ k ∈ ( − , .Then, there exist T > and a solution u of (1.1) on [ T , + ∞ ) in the energy space such that lim t → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u ( t ) − K X k =1 ι k ( λ ∞ k ) / W ℓ k (cid:18) . − ℓ k t − y ∞ k λ ∞ k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H = 0 , (1.4) lim t → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ t u ( t ) + K X k =1 ι k ( λ ∞ k ) / ( ℓ k · ∇ W ℓ k ) (cid:18) . − ℓ k t − y ∞ k λ ∞ k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L = 0 . (1.5)The question of existence and properties of multi-solitons for nonlinear models has a longhistory starting with the celebrated works of Fermi, Pasta and Ulam [10] and Kruskal andZabusky [32], and closely related to the study of integrable equations by the inverse scatteringtransform. We refer in particular to the review work of Miura [24] on multi-solitons for theKorteweg-de Vries equation and to Zakharov and Shabat [33] for multi-solitons of the 1D cubicSchrödinger equation. Recall that in integrable cases, these solutions are very special: theyare explicit and behave exactly as the sum of several solitons both at t → + ∞ and t → −∞ .In particular, they describe the collision and interaction of several solitons globally in time,i.e. for all t ∈ ( −∞ , + ∞ ) .Apart from works on integrable models, there have been several proofs of existence ofmulti-solitons for nonlinear dispersive equations, starting with [22] for the L critical nonlin-ear Schrödinger equation and [17] for the subcritical and critical generalized Korteweg-de Vriesequations. Note that [17] also contains a uniqueness result in the energy space, whose proof isspecific to KdV type equations. Concerning existence, the general strategy of these works isto build backwards in time a sequence of approximate solutions satisfying uniform estimatesand then to use a compactness argument. In [17] and also in [18], concerning the subcriti-cal nonlinear Schrödinger equation, uniform estimates are deduced from long time stabilityarguments, adapted from the previous works [31] (for single solitons) and [20] (for several de-coupled solitons). Later, the strategy of these works was extended to the case of exponentiallyunstable solitons, see [4] for the construction of multi-solitons and [3] for the classification of allmulti-solitons of the supercritical generalized KdV equation. In these papers, the exponentialinstability is controled through a simple topological argument. ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 3
For the Klein-Gordon equation, the strategy was adapted by Cote and Munoz [5] (for realand unstable solitons) and Bellazzini, Ghimenti and Le Coz [1] (for complex, stable solitons).For the water-waves system, see the recent work of Ming, Rousset and Tzvetkov [23].Note that all the works mentioned before are for exponentially decaying solitons, and thusexponentially small interactions as t → + ∞ . The main difficulty of constructing multi-solitonsfor (1.1) is due to the algebraic decay of W , which implies that the solitons have strong in-teractions, of order t − . For the Benjamin-Ono equation, multi-solitons exist with solitonsbehaving algebraically at ∞ , but they are obtained explicitly using the integrability of theequation (see e.g. [21] and [25]). Stability and asymptotic stability of such multi-solitons isproved in [13], but relying on specific monotonicity formulas for KdV type equations. In [15],devoted to the construction of multi-solitons for the Hartree equation, solitons are also decay-ing algebraically. However, in that case, the potential related to the soliton is exponentiallydecaying, which allows a decoupling facilitating the construction of an approximate solutionat order t − M for arbitrarily large M . For M > M large enough, an actual solution can thenbe constructed close to this approximate solution. Such decoupling is not present in the caseof the energy critical wave equation (1.1) and it seems delicate to construct sharp approximatemulti-solitons (i.e. at order t − M for large M ).1.2. Comments on Theorem 1. (1) Each soliton being exponentially unstable, it can bederived as a consequence of the proof that the multi-solitons constructed in Theorem 1 areunstable. Uniqueness of multi-soliton in the energy space, up to the unstable directions, is anopen problem as for the nonlinear Schrödinger equation. The uniqueness statements in [17]and [3] are specific to KdV-type equations.The global behavior of u ( t ) i.e. for t < T is an open problem. We conjecture that it doesnot have the multi-soliton behavior as t → −∞ . We refer to [19] for the proof of nonexistenceof pure multi-solitons in the case of the (non integrable) quartic generalized Korteweg de Vriesequation for a certain range of speeds.(2) Dimension N ≥ . We expect that Theorem 1 still holds true for the energy-criticalwave equation for space dimensions N ≥ . Indeed, at the formal level, all the importantcomputations of this paper can be reproduced for N ≥ . However, the lack of regularity ofthe nonlinearity create several additional technical difficulties, which we choose not to treat inthis paper. Recall that such difficulties were overcome for the Cauchy problem in the energyspace in [2].(3) Dimension 3 and 4. We conjecture that in this case, there exists no multi-soliton in thesense (1.4)–(1.5), for any value of K ≥ . Heuristically, from the asymptotics as | x | → ∞ , W ( x ) ∼ | x | − N in dimension N , the interaction between two solitons of different speeds is t − N , i.e. t − in dimension 3, and t − in dimension 4. Following our method, these interactionsare too strong and create diverging terms in the construction. However, to prove nonexistenceof multi-soliton rigorously, one would need a priori information on any multi-soliton, whichis an open problem for any dimension N ≥ .1.3. Strategy of the proof.
First, we note that Theorem 1 in case (A) follows from case(B) with K = 2 and the Lorentz transformation. See Section 5 for a detailed proof, inspiredby arguments in [14, 9].The proof of Theorem 1 in case (B) follows the strategy by uniform estimates and com-pactness introduced in [17] and [18], but due to the algebraic decay of the solitons, proving Y. MARTEL AND F. MERLE uniform estimates is more delicate. For k ∈ { , . . . , K } , let λ ∞ k > , y ∞ k ∈ R and ℓ k ∈ R with | ℓ k | < , ℓ k = ℓ k ′ for k ′ = k .Let S n → + ∞ as n → ∞ and, for each n , let u n be the (backwards) solution of (1.1) withdata at time S n u n ( S n , x ) ∼ K X k =1 ι k ( λ ∞ k ) / W ℓ k (cid:18) x − ℓ k S n − y ∞ k λ ∞ k (cid:19) , (1.6) ∂ t u ( S n , x ) ∼ − K X k =1 ι k ( λ ∞ k ) / ( ℓ k · ∇ W ℓ k ) (cid:18) x − ℓ k S n − y ∞ k λ ∞ k (cid:19) . (1.7)(See (4.1) for a precise definition of ( u n ( S n ) , ∂ t u n ( S n )) . The goal is to prove the followinguniform estimates on the time interval [ T , S n ] , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u n ( t ) − K X k =1 ι k ( λ ∞ k ) / W ℓ k (cid:18) . − ℓ k t − y ∞ k λ ∞ k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H . t , (1.8) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂ t u n ( t ) + K X k =1 ι k ( λ ∞ k ) / ℓ k · ∇ W ℓ k (cid:18) . − ℓ k t − y ∞ k λ ∞ k (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L . t . (1.9)for T large independent of n . Indeed, the existence of a multi-soliton then follows easily fromstandard compactness arguments (note that we also obtain bounds on weighted higher orderSobolev norms for ( u n , ∂ t u n ) which facilitate the convergence). Thus, we now focus on theproof of (1.8)–(1.9). Note first that such long time stability estimates cannot be true for anyinitial data of the form (1.6)–(1.7); indeed, to take into account the exponential instability ofeach soliton W ℓ k , we need to adjust the initial condition ( u n ( S n ) , ∂ t u n ( S n )) . This adjustmentrelies on a simple topological argument on K scalar parameters, first introduced in a similarcontext in [4].We introduce ε = u n − X k W k , η = ∂ t u n + X k ( ℓ k · ∇ W k ) , where W k ( t, x ) = ι k λ / k ( t ) W ℓ k (cid:18) x − ℓ k t − y k ( t ) λ k ( t ) (cid:19) . By a standard procedure, in the definition of W k , the modulation parameters λ k ( t ) and y k ( t ) are chosen close to λ ∞ k and y ∞ k in order to obtain suitable orthogonality conditions on ( ε, η ) .The equation of ( ε, η ) is thus coupled by equations on λ k and y k . See Lemma 3.1.The general strategy of the proof of the uniform estimates (1.8)–(1.9) is to use globalfunctionals that are locally of the form Z x ∼ ℓ k t + y k ( t ) |∇ ε | + | η | + 2 ( ℓ k · ∇ ε ) η − | W k | ε , around each soliton W k , i.e. in regions x ∼ ℓ k t + y k ( t ) . Note that the coercivity of suchfunctional under usual orthogonality conditions on ( ε, η ) is standard. The difficulty is to “glue”these K functionals to obtain a unique global functional on ( ε, η ) which is locally adapted toeach soliton W k . ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 5
In case (B) of Theorem 1, we assume ℓ k = ℓ k e and − < ℓ < . . . < ℓ K < . To prove(1.8)-(1.9), we introduce the following energy functional H K = Z E K + 2 Z ( χ K ( t, x ) ∂ x ε ) η, where E K is the following “linearized energy density” E K = |∇ ε | + | η | − (cid:16) | P k W k + ε | − | P k W k | − | P k W k | ( P k W k ) ε (cid:17) , (1.10)and the bounded function χ K ( t, x ) is equal to ℓ k in a neighborhood of the soliton W k andclose to x t in “transition regions” between two solitons (see (4.15) for a precise definition).Note that the functional H K is inspired by the ones used in [17] and [18] for the constructionof multi-solitons for (gKdV) and (NLS) equations in energy subcritical cases.The functional H K has the following two important properties (see Proposition 4.2 for moreprecise statements):(1) H K is coercive, in the sense that (up to unstable directions, to be controled separately),it controls the size of ( ε, η ) in the energy space H K ∼ k ε k H + k η k L . (2) The variation of H K is controled on [ T , S n ] in the following (weak) sense − ddt (cid:0) t H K (cid:1) . t − . (1.11)Note that the term t − in the right-hand side is related to interactions between solitons.Therefore, integrating (1.11) on [ t, S n ] , from (1.6)–(1.7), we find the uniform bound, for any t ∈ [ T , S n ] , k ε k ˙ H + k η k L . t − . By time integration of the equations of the parameters, the above estimate implies | y k ( t ) − y ∞ k | . t − , | λ k ( t ) − λ ∞ k | . t − , and (1.8)–(1.9) follow. Acknowledgements.
This work was partly supported by the project ERC 291214 BLOWDISOL.2.
Preliminaries
Notation.
We denote ( g, ˜ g ) L = Z g ˜ g, k g k L = Z | g | , ( g, ˜ g ) ˙ H = Z ∇ g · ∇ ˜ g, k g k H = Z |∇ g | . For ~g = (cid:18) gh (cid:19) , ~ ˜ g = (cid:18) ˜ g ˜ h (cid:19) , set (cid:16) ~g, ~ ˜ g (cid:17) L = ( g, ˜ g ) L + (cid:16) h, ˜ h (cid:17) L , (cid:16) ~g, ~ ˜ g (cid:17) E = ( g, ˜ g ) ˙ H + (cid:16) h, ˜ h (cid:17) L , k ~g k E = k g k H + k h k L . Y. MARTEL AND F. MERLE
When x is seen as a specific coordinate, denote x = ( x , . . . , x ) , ∇ g = ( ∂ x g, . . . , ∂ x g ) , ∆ g = X j =2 ∂ x j g. For − < ℓ < , ( g, ˜ g ) ˙ H ℓ = (1 − ℓ ) Z ∂ x g∂ x ˜ g + Z ∇ g · ∇ ˜ g, k g k H ℓ = ( g, g ) ˙ H ℓ More generally, for ℓ ∈ R such that | ℓ | < , ( g, ˜ g ) ˙ H ℓ = Z [ ∇ g · ∇ ˜ g − ( ℓ · ∇ g )( ℓ · ∇ ˜ g )] , k g k H ℓ = k g k H − k ℓ · ∇ g k L . Observe that if we define g ℓ ( x ) = g p − | ℓ | − ! ℓ ( ℓ · x ) | ℓ | + x ! , and similarly ˜ g , ˜ g ℓ , then ( g ℓ , ˜ g ℓ ) ˙ H ℓ = (1 − | ℓ | ) ( g, ˜ g ) ˙ H . (2.1)Let Λ and e Λ be the ˙ H and L scaling operators defined as follows Λ g = 32 g + x · ∇ g, e Λ g = 52 g + x · ∇ g, e Λ ∇ = ∇ Λ , ~ Λ = (cid:18) e ΛΛ (cid:19) . (2.2)Let J = (cid:18) − (cid:19) . Recall the Hardy and Sobolev inequaliies, for any v ∈ ˙ H , Z | v | | x | . Z |∇ v | , (2.3) k v k L / . k∇ v k L . (2.4)Set h x i = (1 + | x | ) and k v k Y = Z (cid:0) | v ( x ) | + |∇ v ( x ) | (cid:1) h x i dx, k v k Y = Z (cid:0) |∇ v ( x ) | + |∇ v ( x ) | (cid:1) h x i dx. If g ∈ C ([ t , t ] , Y ) then the unique solution v ∈ C ([ t , t ] , ˙ H ) of ∂ t v − ∆ v = g with v ( t ) = 0 and ∂ t v ( t ) = 0 , satisfies ( v, v t ) ∈ C ([ t , t ] , Y × Y ) and k ( v, v t )( t ) k Y × Y ≤ Z tt k g ( s ) k Y ds. (2.5)Moreover, the following estimate holds, for all v ∈ Y , k| v | v k Y . k v k ˙ H k v k Y . (2.6)Thus, it follows from a standard argument (fixed point) that (1.1) is locally well-posed in thespace Y × Y with a time of existence depending only on the size of the Y × Y norm ofthe initial data. ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 7
For initial data in the energy space ˙ H × L , the Cauchy problem is also locally well-posed ina certain sense, using suitable Strichartz estimates ; we refer to section 2 of [14] and referencestherein.Denote f ( u ) = | u | u, F ( u ) = 310 | u | . Energy linearization around W . Let L = − ∆ − f ′ ( W ) , ( Lg, g ) L = Z |∇ g | − f ′ ( W ) g ,H = (cid:18) L
00 Id (cid:19) , ( H~g, ~g ) L = ( Lg, g ) L + k h k L . Let ~g be small in the energy space. Then, expanding, integrating by parts, using the equationof W and (2.4), one has E ( W + g, h ) = E ( W, − Z (∆ W + f ( W )) g + 12 (cid:18)Z |∇ g | − f ′ ( W ) g (cid:19) + 12 Z h − Z (cid:18) F ( W + g ) − F ( W ) − f ( W ) g − f ′ ( W ) g (cid:19) = E ( W,
0) + 12 (
Lg, g ) L + 12 k h k L + O ( k g k H ) . (2.7)In this paper addressing the case of several solitons, it is crucial to be able to spacially splitthe solitons. For some < α ≪ to be fixed, set ϕ ( x ) = (1 + | x | ) − α (2.8)We gather here some properties of the operator L . Lemma 2.1 (Spectral properties of L ) . (i) Spectrum. The operator L on L with domain H is a self-adjoint operator with essential spectrum [0 , + ∞ ) , no positive eigenvalue and only onenegative eigenvalue − λ , with a smooth radial positive eigenfunction Y ∈ S ( R ) . Moreover, L (Λ W ) = L ( ∂ x j W ) = 0 , for any j = 1 , . . . , . (2.9) There exists µ > such that, for all g ∈ ˙ H , the following holds. (ii) Coercivity with W orthogonality (Appendix D of [27]). ( Lg, g ) L ≥ µ k g k H − µ ( g, Λ W ) H + X j =1 (cid:0) g, ∂ x j W (cid:1) H + ( g, W ) H (2.10)(iii) Coercivity with Y orthogonality. ( Lg, g ) L ≥ µ k g k H − µ ( g, Λ W ) H + X j =1 (cid:0) g, ∂ x j W (cid:1) H + ( g, Y ) L (2.11)(iv) Localized coercivity. For α > small enough, Z |∇ g | ϕ − f ′ ( W ) g ≥ µ Z |∇ g | ϕ − µ ( g, Λ W ) H + X j =1 (cid:0) g, ∂ x j W (cid:1) H + ( g, Y ) L (2.12) Y. MARTEL AND F. MERLE
Proof. (i) contains well-known facts on L that are easily checked directly. We refer to AppendixD of [27] for the proof of (2.10). The proof of (iii) is standard since ( LY, Y ) < .Proof of (2.12). By direct computations Z |∇ ( gϕ ) | = Z |∇ g | ϕ − Z | g | ϕ ∆ ϕ. Note that (here the space dimension is ) ∆ ϕ = − α (cid:0) (3 − α ) | x | + 5 (cid:1) ϕ (1 + | x | ) , and thus | ∆ ϕ | ≤ α ϕ h x i , and thus by (2.3), Z | g | ϕ | ∆ ϕ | ≤ α Z | g | ϕ h x i ≤ δ ( α ) Z |∇ ( gϕ ) | , where δ ( α ) → as α → . This implies the following estimate (cid:12)(cid:12)(cid:12)(cid:12)Z |∇ g | ϕ − Z |∇ ( gϕ ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ ( α ) Z |∇ ( gϕ ) | . (2.13)We check that | ( g (1 − ϕ ) , Λ W ) ˙ H | + | (cid:0) g (1 − ϕ, ∂ x j W (cid:1) ˙ H | + | ( g (1 − ϕ ) , Y ) L | ≤ δ ( α ) k gϕ k ˙ H . (2.14)Indeed, by the Cauchy-Schwarz inequality, the decay properties of W and Hardy inequality, ( g (1 − ϕ ) , Λ W ) H = ( g (1 − ϕ ) , ∆(Λ W )) L ≤ Z ( gϕ ) h x i Z | ∆(Λ W ) | | − ϕ | h x i ϕ ≤ δ ( α ) k gϕ k H ; the rest of the proof of (2.14) is similar. We also have Z W g (1 − ϕ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) − ϕ ϕ h x i W (cid:13)(cid:13)(cid:13)(cid:13) L ∞ Z ( gϕ ) h x i . δ ( α ) k gϕ k H . (2.15)By (2.11) applied to gϕ and then (2.14), for α small, ( L ( gϕ ) , gϕ ) L ≥ µ k gϕ k H − µ ( gϕ, Λ W ) H + X j =1 (cid:0) gϕ, ∂ x j W (cid:1) H + ( gϕ, W ) H ≥ ( µ − δ ( α )) k gϕ k H − µ ( g, Λ W ) H + X j =1 (cid:0) g, ∂ x j W (cid:1) H + ( g, W ) H Finally, using (2.13) and (2.15) we get (2.12), for α small enough. (cid:3) Energy linearization around W ℓ . For − < ℓ < , let W ℓ ( x ) = W (cid:18) x √ − ℓ , x (cid:19) , (1 − ℓ ) ∂ x W ℓ + ∆ W ℓ + W ℓ = 0 , (2.16)so that u ( t, x ) = W ℓ ( x − ℓt, ¯ x ) is a solution of (1.1). Note that E ( W ℓ , − ℓ∂ x W ℓ ) − ℓ Z | ∂ x W ℓ | = (1 − ℓ ) E ( W, . (2.17) ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 9
Let L ℓ = − (1 − ℓ ) ∂ x − ∆ − f ′ ( W ℓ ) , (2.18) ( L ℓ g, g ) L = (1 − ℓ ) Z | ∂ x g | + Z (cid:0) |∇ g | − f ′ ( W ℓ ) g (cid:1) , (2.19) H ℓ = (cid:18) − ∆ − f ′ ( W ℓ ) − ℓ∂ x ℓ∂ x Id (cid:19) , ( H ℓ ~g, ~g ) L = ( L ℓ g, g ) L + k ℓ∂ x g + h k L . (2.20)As before, L ℓ and H ℓ are related to the linearization of the energy around W ℓ . Indeed,proceeding as in (2.7), E ( W ℓ + g, − ℓ∂ x W ℓ + h ) + ℓ Z ∂ x ( W ℓ + g )( − ℓ∂ x W ℓ + h )= E ( W ℓ , − ℓ∂ x W ℓ ) − ℓ Z ( ∂ x W ℓ ) − Z (∆ W ℓ ) g − Z f ( W ℓ ) g − ℓ Z ( ∂ x W ℓ ) h + ℓ Z ( ∂ x W ℓ ) g + ℓ Z ( ∂ x W ℓ ) h + 12 Z | h | + 12 Z (cid:0) |∇ g | − f ′ ( W ℓ ) g (cid:1) + ℓ Z h∂ x g + O ( k g k H ) . and thus, using (2.16) and (2.17), E ( W ℓ + g, − ℓ∂ x W ℓ + h ) + ℓ Z ∂ x ( W ℓ + g )( − ℓ∂ x W ℓ + h )= (1 − ℓ ) E ( W,
0) + 12 ( H ℓ ~g, ~g ) L + O ( k g k H ) . The following functions appear when studying the properties of the operators H ℓ and H ℓ J~Z Λ ℓ = (cid:18) Λ W ℓ − ℓ∂ x Λ W ℓ (cid:19) , ~Z ∇ j ℓ = (cid:18) ∂ x j W ℓ − ℓ∂ x ∂ x j W ℓ (cid:19) , ~Z Wℓ = (cid:18) W ℓ − ℓ∂ x W ℓ (cid:19) ,Y ℓ ( x ) = Y (cid:18) x √ − ℓ , x (cid:19) , ~Z ± ℓ = (cid:16) ℓ∂ x Y ℓ ± √ λ √ − ℓ Y ℓ (cid:17) e ± ℓ √ λ √ − ℓ x Y ℓ e ± ℓ √ λ √ − ℓ x . We gather below several technical facts.
Claim 1.
The following hold for any − < ℓ < , (i) Properties of L ℓ . L ℓ (Λ W ℓ ) = L ℓ ( ∂ x j W ℓ ) = 0 , L ℓ Y ℓ = − λ Y ℓ , L ℓ W ℓ = − W ℓ , (2.21)(ii) Properties of H ℓ and H ℓ J . H ℓ ~Z Λ ℓ = H ℓ ~Z ∇ j ℓ = 0 , H ℓ ~Z Wℓ = − W ℓ ! , (2.22) (cid:16) H ℓ ~Z Wℓ , ~Z Wℓ (cid:17) L = − Z W ℓ , − H ℓ J ( ~Z ± ℓ ) = ± p λ (1 − ℓ ) ~Z ± ℓ , (2.23) (cid:16) ~Z Λ ℓ , ~Z Wℓ (cid:17) E = (cid:16) ~Z ∇ j ℓ , ~Z Wℓ (cid:17) E = 0 , (cid:16) ~Z Λ ℓ , ~Z ± ℓ (cid:17) L = (cid:16) ~Z ∇ j ℓ , ~Z ± ℓ (cid:17) L = 0 . (2.24) (iii) Antecedents. There exist ~z ± ℓ such that H ℓ ~z ± ℓ = ~Z ± ℓ , (cid:0) H ℓ ~z ± ℓ , ~z ± ℓ (cid:1) L = 0 , (cid:16) ~z ± ℓ , ~Z Λ ℓ (cid:17) E = (cid:16) ~z ± ℓ , ~Z ∇ j ℓ (cid:17) E = 0 (2.25) Proof.
The proof of (2.21) follows from the same properties at ℓ = 0 .Next, note that for any function g , H ℓ (cid:18) g − ℓ∂ x g (cid:19) = (cid:18) L ℓ g (cid:19) , (cid:18) H ℓ (cid:18) g − ℓ∂ x g (cid:19) , (cid:18) g − ℓ∂ x g (cid:19)(cid:19) L = ( L ℓ g, g ) L . (2.26) Proof of (2.22) . First, by (2.26) and (2.21), H ℓ ( ~Z Λ ℓ ) = H ℓ ( ~Z ∇ j ℓ ) = 0 . The identity concern-ing ~Z Wℓ also follows directly from (2.26) and (2.21). Proof of (2.23) . Note that − H ℓ J = − ℓ∂ x ∆ + W ℓ Id − ℓ∂ x ! . On the one hand, − ℓ∂ x (cid:18) ℓ∂ x Y ℓ ± √ λ √ − ℓ Y ℓ (cid:19) e ± ℓ √ λ √ − ℓ x ! + ∆ Y ℓ e ± ℓ √ λ √ − ℓ x ! + 73 W ℓ Y ℓ e ± ℓ √ λ √ − ℓ x = − ( L ℓ Y ℓ ) e ± ℓ √ λ √ − ℓ x ± (1 − ℓ ) ℓ √ λ √ − ℓ ( ∂ x Y ℓ ) e ± ℓ √ λ √ − ℓ x = ± p λ (1 − ℓ ) (cid:16) ± p λ (1 − ℓ ) − Y ℓ + ℓ ( ∂ x Y ℓ ) (cid:17) e ± ℓ √ λ √ − ℓ x . On the other hand, (cid:18) ℓ∂ x Y ℓ ± √ λ √ − ℓ Y ℓ (cid:19) e ± ℓ √ λ √ − ℓ x − ℓ∂ x Y ℓ e ± ℓ √ λ √ − ℓ x ! = ± p λ (1 − ℓ ) Y ℓ e ± ℓ √ λ √ − ℓ x Thus, − H ℓ J ( ~Z ± ℓ ) = ±√ λ (1 − ℓ ) ~Z ± ℓ . Proof of (2.24) . Since (cid:0) ∂ x j Λ W, ∂ x j W (cid:1) L = 0 ( ˙ H scaling) and (cid:16) ∂ x j ∂ x j ′ W, ∂ x j W (cid:17) L = 0 (by parity), we have (cid:16) ~Z Λ ℓ , ~Z Wℓ (cid:17) E = (cid:16) ~Z ∇ k ℓ , ~Z Wℓ (cid:17) E = 0 . Next, from (2.24), the fact that H ℓ isself-adjoint in L and (2.22), we have ∓ p λ (1 − ℓ ) (cid:16) ~Z Λ ℓ , ~Z ± ℓ (cid:17) L = (cid:16) ~Z Λ ℓ , H ℓ J ( ~Z ± ℓ ) (cid:17) L = (cid:16) H ℓ ~Z Λ ℓ , J ( ~Z ± ℓ ) (cid:17) L = 0 . The identity (cid:16) ~Z ∇ j ℓ , ~Z ± ℓ (cid:17) L = 0 is proved in a similar way. Proof of (2.25) . We set ~z ± ℓ = ∓ J ~Z ± ℓ √ λ (1 − ℓ ) / + α Λ , ± ~Z Λ ℓ + X j =1 α ∇ j , ± ~Z ∇ j ℓ , where α Λ , ± and α ∇ j , ± are chosen so that (cid:16) ~z ± ℓ , ~Z Λ ℓ (cid:17) E = (cid:16) ~z ± ℓ , ~Z ∇ j ℓ (cid:17) E = 0 . ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 11
By (2.22) and (2.23), we have H ℓ ~z ± ℓ = ~Z ± ℓ . Finally, H ℓ being self-adjoint, we have (cid:0) H ℓ ~z ± ℓ , ~z ± ℓ (cid:1) L = ∓ H ℓ ~z ± ℓ , − J ~Z ± ℓ √ λ (1 − ℓ ) ! L = ∓ √ λ (1 − ℓ ) / (cid:16) ~Z ± ℓ , J ~Z ± ℓ (cid:17) L = 0 . (cid:3) We claim the following coercivity results with ~Z ± k orthogonalities. Lemma 2.2.
Let − < ℓ < . There exists µ > such that, for all ~g ∈ ˙ H × L , the followingholds. (i) Coercivity of H ℓ with Z ± ℓ orthogonalities. ( H ℓ ~g, ~g ) L ≥ µ k ~g k E − µ ( g, Λ W ℓ ) H ℓ + X j =1 (cid:0) g, ∂ x j W ℓ (cid:1) H ℓ + (cid:16) ~g, ~Z + ℓ (cid:17) L + (cid:16) ~g, ~Z − ℓ (cid:17) L . (2.27)(ii) Localized coercivity. For α > small enough, Z (cid:0) |∇ g | ϕ − f ′ ( W ℓ ) g + h ϕ + 2 ℓ ( ∂ x g ) hϕ (cid:1) ≥ µ Z (cid:0) |∇ g | + h (cid:1) ϕ − µ ( g, Λ W ℓ ) H + X j =1 (cid:0) g, ∂ x j W ℓ (cid:1) H + (cid:16) ~g, ~Z + ℓ (cid:17) L + (cid:16) ~g, ~Z − ℓ (cid:17) L . (2.28) Proof. Proof of (2.27) . By a standard argument, it is equivalent to prove ( g, Λ W ℓ ) ˙ H ℓ = (cid:0) g, ∂ x j W ℓ (cid:1) ˙ H ℓ = (cid:16) ~g, ~Z ± ℓ (cid:17) L = 0 ⇒ ( H ℓ ~g, ~g ) L ≥ µ k ~g k E . (2.29)Note that the proof of (2.29) is largely inspired by Proposition 2 in [5], Lemma 5.1 in [7], andProposition 5.5 in [6]. Case ℓ = 0 . Note that in this case ~Z ± = (cid:18) ±√ λ YY (cid:19) , and g as in (2.29) thus satisfiesthe orthogonality conditions ( g, Λ W ) ˙ H = (cid:0) g, ∂ x j W (cid:1) ˙ H = ( g, Y ) L = 0 . Then, (2.29) followsfrom (2.11). Case ℓ = 0 . Note that (2.27) is thus equivalent to ( g, Λ W ℓ ) ˙ H ℓ = (cid:0) g, ∂ x j W ℓ (cid:1) ˙ H ℓ = (cid:0) H ℓ ~g, ~z ± ℓ (cid:1) L = 0 ⇒ ( H ℓ ~g, ~g ) L ≥ µ ℓ k ~g k E . (2.30)We decompose g and ~z ± ℓ as follows ~g = a ~Z Wℓ + ~g ⊥ , ~z ± ℓ = a ± ~Z Wℓ + ~z ± , ⊥ ℓ , ~g ⊥ = (cid:18) g ⊥ h ⊥ (cid:19) , ~z ±⊥ ℓ = z ±⊥ ℓ, z ±⊥ ℓ, ! , (2.31)where a and a ± are chosen so that (cid:16) g ⊥ , W ℓ (cid:17) ˙ H ℓ = (cid:16) z ± , ⊥ ℓ, , W ℓ (cid:17) ˙ H ℓ = 0 . (2.32)We still have (cid:16) g ⊥ , Λ W ℓ (cid:17) ˙ H ℓ = (cid:16) g ⊥ , ∂ x j W ℓ (cid:17) ˙ H ℓ = 0 , (cid:16) ~z ± , ⊥ ℓ, , Λ W ℓ (cid:17) ˙ H ℓ = (cid:16) ~z ± , ⊥ ℓ, , ∂ x j W ℓ (cid:17) ˙ H ℓ = 0 . Note that since (see (2.22) and (2.16)) H ℓ ~Z Wℓ = − W ℓ ! = 43 (cid:18) (1 − ℓ ) ∂ x W ℓ + ∆ W ℓ (cid:19) , (2.32) is equivalent to (cid:16) H ℓ ~g ⊥ , ~Z Wℓ (cid:17) L = (cid:16) H ℓ ~z ± , ⊥ ℓ , ~Z Wℓ (cid:17) L = 0 . (2.33)The decompositions (2.31) being orthogonal with respect to ( H ℓ ., . ) L , we have ( H ℓ ~g, ~g ) L = a (cid:16) H ℓ ~Z Wℓ , ~Z Wℓ (cid:17) L + (cid:16) H ℓ ~g ⊥ , ~g ⊥ (cid:17) L , (cid:0) H ℓ ~z ± ℓ , ~z ± ℓ (cid:1) L = ( a ± ) (cid:16) H ℓ ~Z Wℓ , ~Z Wℓ (cid:17) L + (cid:16) H ℓ ~z ± , ⊥ ℓ , ~z ± , ⊥ ℓ (cid:17) L , (cid:0) H ℓ ~g, ~z ± ℓ (cid:1) L = aa ± (cid:16) H ℓ ~Z Wℓ , ~Z Wℓ (cid:17) L + (cid:16) H ℓ ~g ⊥ , ~z ± , ⊥ ℓ (cid:17) L , (2.34)which imply (recall that (cid:16) H ℓ ~Z Wℓ , ~Z Wℓ (cid:17) L < ), from (2.23), ( H ℓ ~g, ~g ) L = − (cid:16) H ℓ ~g ⊥ , ~z − , ⊥ ℓ (cid:17) L (cid:16) H ℓ ~g ⊥ , ~z + , ⊥ ℓ (cid:17) L r(cid:16) H ℓ ~z − , ⊥ ℓ , ~z − , ⊥ ℓ (cid:17) L (cid:16) H ℓ ~z + , ⊥ ℓ , ~z + , ⊥ ℓ (cid:17) L + (cid:16) H ℓ ~g ⊥ , ~g ⊥ (cid:17) L . (2.35)Let A = sup ~ω ∈ Span( ~z + , ⊥ ℓ ,~z − , ⊥ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) H ℓ ~ω, ~z − , ⊥ ℓ (cid:17) L r(cid:16) H ℓ ~z − , ⊥ ℓ , ~z − , ⊥ ℓ (cid:17) L ( H ℓ ~ω, ~ω ) L (cid:16) H ℓ ~ω, ~z + , ⊥ ℓ (cid:17) L r(cid:16) H ℓ ~z + , ⊥ ℓ , ~z + , ⊥ ℓ (cid:17) L ( H ℓ ~ω, ~ω ) L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Since ( H ℓ ., . ) is positive definite on Span(∆ W ℓ , ∆Λ W ℓ , ∆ ∂ x j W ℓ ) ⊥ , applying Cauchy-Schwarzinequality to each of the term of the product above, we find A ≤ . Moreover, A = 1 wouldimply that ~z − , ⊥ ℓ and ~z + , ⊥ ℓ are proportional, which is clearly not true for ℓ = 0 (for example,due to different behavior at ∞ of ~Z ± k ). Thus, A < . As a consequence, we also obtain thatfor all ~ω ∈ Span(∆ W ℓ , ∆Λ W ℓ , ∆ ∂ x j W ℓ ) ⊥ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) H ℓ ~ω, ~z − , ⊥ ℓ (cid:17) L r(cid:16) H ℓ ~z − , ⊥ ℓ , ~z − , ⊥ ℓ (cid:17) L ( H ℓ ~ω, ~ω ) L (cid:16) H ℓ ~ω, ~z + , ⊥ ℓ (cid:17) L r(cid:16) H ℓ ~z + , ⊥ ℓ , ~z + , ⊥ ℓ (cid:17) L ( H ℓ ~ω, ~ω ) L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ A ( H ℓ ~ω, ~ω ) L . Thus, by (2.35) and then (2.10) (after change of variables), ( H ℓ ~g, ~g ) L ≥ (1 − A ) (cid:16) H ℓ ~g ⊥ , ~g ⊥ (cid:17) L ≥ c (cid:13)(cid:13)(cid:13) ~g ⊥ (cid:13)(cid:13)(cid:13) E . The result then follows from | a | . k ~g ⊥ k E from (2.34). ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 13
Proof of (2.28). First, we apply (2.27) on ~gϕ : ( H ℓ ( ~gϕ ) , ~gϕ ) L ≥ µ k ~gϕ k E − µ ( ~gϕ, Λ W ℓ ) H ℓ + X j =1 ( ~gϕ, ∂ x j W ℓ ) H ℓ + (cid:16) ~gϕ, ~Z + ℓ (cid:17) L + (cid:16) ~gϕ, ~Z − ℓ (cid:17) L . Recall that ( H ℓ ( ~gϕ ) , ~gϕ ) L = Z |∇ ( gϕ ) | − Z W ℓ g ϕ + 2 ℓ Z ∂ x ( gϕ )( hϕ ) + Z h ϕ . Note that ∂ x ϕ = − αx | x | ϕ and so (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ x ( gϕ )( gϕ ) − Z ( ∂ x g ) hϕ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z gh ( ∂ x ϕ ) ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cα Z | g || h | ϕ h x i≤ Cα (cid:18)Z ( gϕ ) h x i (cid:19) (cid:18)Z | h | ϕ (cid:19) ≤ Cα Z |∇ ( gϕ ) | + Cα Z | h | ϕ . Thus, using (2.13), (cid:12)(cid:12)(cid:12)(cid:12) ( H ℓ ( ~gϕ ) , ~gϕ ) L − Z (cid:0) |∇ g | − f ′ ( W ℓ ) g + h + 2 ℓ ( ∂ x g ) h (cid:1) ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ ( α ) k ~gϕ k E . To complete the proof, we just notice that as in (2.14) (cid:12)(cid:12)(cid:12)(cid:16) ~g (1 − ϕ ) , ~Z ± ℓ (cid:17) L (cid:12)(cid:12)(cid:12) ≤ δ ( α ) k ~gϕ k E , (2.36)and similarly for the other scalar products appearing in (2.28), and as in (2.15), Z W ℓ g (1 − ϕ ) . δ ( α ) k gϕ k H . (2.37)Combining these estimates, we obtain (2.28), for α small enough. (cid:3) Energy linearization around W ℓ . We only define some notation generalizing the pre-vious section. For ℓ ∈ R such that | ℓ | < , W ℓ defined in (1.3) solves ∆ W ℓ − ℓ · ∇ ( ℓ · ∇ W ℓ ) + W ℓ = 0 . (2.38)The following operators are related to the linearization of the energy around W ℓ L ℓ = − ∆ − ℓ · ∇ ( ℓ · ∇ ) − f ′ ( W ℓ ) , H ℓ = (cid:18) − ∆ − f ′ ( W ℓ ) − ℓ · ∇ ℓ · ∇ Id (cid:19) . Set ~Z Λ ℓ = (cid:18) Λ W ℓ − ℓ · ∇ (Λ W ℓ ) (cid:19) , ~Z ∇ j ℓ = (cid:18) ∂ x j W ℓ − ℓ · ∇ ( ∂ x W ℓ ) (cid:19) , ~Z W ℓ = (cid:18) W ℓ − ℓ · ∇ W ℓ (cid:19) ,Y ℓ = Y p − | ℓ | − ! ℓ ( ℓ · x ) | ℓ | + x ! , ~Z ± ℓ = (cid:18) ℓ · ∇ Y ℓ ± √ λ √ −| ℓ | Y ℓ (cid:19) e ± √ λ √ −| ℓ | ℓ · x Y ℓ e ± √ λ √ −| ℓ | ℓ · x . Note from (2.24) and (2.23), (cid:16) ~Z Λ ℓ , ~Z ± ℓ (cid:17) L = (cid:16) ~Z ∇ j ℓ , ~Z ± ℓ (cid:17) L = 0 , (2.39) − H ℓ J ~Z ± ℓ = ± p λ (1 − | ℓ | ) ~Z ± ℓ . (2.40)3. Decomposition around the sum of K solitons We prove in this section a general decomposition around K solitons. Let K ≥ and forany k ∈ { , . . . , K } , let λ ∞ k > , y ∞ k ∈ R , ℓ k ∈ R , | ℓ k | < with ℓ k ′ = ℓ k for k ′ = k .First, for ~G = ( G, H ) , set ( θ ∞ k G )( t, x ) = ι k ( λ ∞ k ) / G (cid:18) x − ℓ k t − y ∞ k λ ∞ k (cid:19) , ~θ ∞ k ~G = θ ∞ k Gθ ∞ k λ ∞ k H . In particular, set W ∞ k = θ ∞ k W ℓ k , ~W ∞ k = θ ∞ k W ℓ k − ℓ k λ ∞ k · θ ∞ k ( ∇ W ℓ k ) . Second, for C functions λ k ( t ) > , y k ( t ) ∈ R to be chosen, let ( θ k G )( t, x ) = ι k λ / k ( t ) G (cid:18) x − ℓ k t − y k ( t ) λ k ( t ) (cid:19) , ~θ k ~G = θ k Gθ k λ k H , ~ ˜ θ k ~G = θ k λ k Gθ k H . (3.1)In particular, set W k = θ k W ℓ k , ~W k = θ k W ℓ k − ℓ k λ k · θ k ( ∇ W ℓ k ) . (3.2)In what follows P Kk =1 is often simply denoted by P k . Lemma 3.1 (Properties of the decomposition) . There exist T ≫ and < δ ≪ such thatif u ( t ) is a solution of (1.1) on [ T , T ] , where T ≤ T < T , such that ∀ t ∈ [ T , T ] , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − X k ~W ∞ k ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H × L ≤ δ , (3.3) then there exist C functions λ k > , y k on [ T , T ] such that, ~ε ( t ) being defined by ~ε = (cid:18) εη (cid:19) , ~u = (cid:18) uu t (cid:19) = X k ~W k + ~ε, (3.4) the following hold on [ T , T ] . (i) First properties of the decomposition. ( ε, θ k (Λ W ℓ k )) ˙ H ℓ k = (cid:0) ε, θ k ( ∂ x j W ℓ k ) (cid:1) ˙ H ℓ k = 0 , (3.5) | λ k ( t ) − λ ∞ k | + | y k ( t ) − y ∞ k | + k ~ε k E . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − X k ~W ∞ k ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H × L (3.6) ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 15 (ii) Equation of ~ε . ε t = η + Mod ε η t = ∆ ε + f X k W k + ε ! − f X k W k ! + R W + Mod η (3.7) where R W = f X k W k ! − X k f ( W k ) , (3.8) Mod ε = X k ˙ λ k λ k θ k (Λ W ℓ k ) + X k ˙ y k λ k · θ k ( ∇ W ℓ k ) (3.9) Mod η = − X k ˙ λ k λ k ℓ k · θ k ( ∇ Λ W ℓ k ) − X k ˙ y k λ k · θ k ( ∇ ( ℓ k · ∇ W ℓ k )) . (3.10)(iii) Parameters equations. X k | ˙ λ k ( t ) | + | ˙ y k ( t ) | . k ~ε ( t ) k E . (3.11)(iv) Unstable directions. Let z ± k ( t ) = (cid:16) ~ε ( t ) , ~ ˜ θ k ~Z ± ℓ k (cid:17) L . (3.12) Then, (cid:12)(cid:12)(cid:12)(cid:12) ddt z ± k ( t ) ∓ √ λ λ k (1 − | ℓ k | ) z ± k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) . k ~ε ( t ) k E + k ~ε ( t ) k E t + 1 t . (3.13) Proof.
Step 1.
Decomposition. Let T ≫ , fix t ≥ T and assume that (3.3) holds for t . Let Γ ∞ = ( λ ∞ k , y ∞ k ) k ∈{ ,...,K } , Γ = ( λ k , y k ) k ∈{ ,...,K } ∈ ((0 , + ∞ ) × R ) K , where λ k and y k are to be found (depending on t ). Consider the map Φ : ˙ H × ((0 , + ∞ ) × R ) K → R K ( ω, Γ) ( ω + X k ′ W ∞ k ′ − X k ′ θ k ′ W ℓ k ′ , θ k (Λ W ℓ k )) ˙ H ℓ k , ( ω + X k ′ W ∞ k ′ − X k ′ θ k ′ W ℓ k ′ , θ k ( ∂ x W ℓ k )) ˙ H ℓ k ,. . . , ( ω + X k ′ W ∞ k ′ − X k ′ θ k ′ W ℓ k ′ , θ k ( ∂ x W ℓ k )) ˙ H ℓ k ! k ∈{ ,...,K } , where θ k is defined in (3.1). By explicit computations, we have (cid:16) d Γ Φ(0 , Γ ∞ ) · ˜Γ (cid:17) k = X k ′ ˜ λ k ′ λ ∞ k ′ (cid:0) θ ∞ k ′ (Λ W ℓ k ′ ) , θ ∞ k (Λ W ℓ k ) (cid:1) ˙ H ℓ k + X k ′ ˜ y k ′ λ ∞ k ′ · (cid:0) θ ∞ k ′ ( ∇ W ℓ k ′ ) , θ ∞ k (Λ W ℓ k ) (cid:1) ˙ H ℓ k , X k ′ ˜ λ k ′ λ ∞ k ′ (cid:0) θ ∞ k ′ (Λ W ℓ k ′ ) , θ ∞ k ( ∂ x W ℓ k ) (cid:1) ˙ H ℓ k + X k ′ ˜ y k ′ λ ∞ k ′ · (cid:0) θ ∞ k ′ ( ∇ W ℓ k ′ ) , θ ∞ k ( ∂ x W ℓ k ) (cid:1) ˙ H ℓ k , . . . , X k ′ ˜ λ k ′ λ ∞ k ′ (cid:0) θ ∞ k ′ (Λ W ℓ k ′ ) , θ ∞ k ( ∂ x W ℓ k ) (cid:1) ˙ H ℓ k + X k ′ ˜ y k ′ λ ∞ k ′ · (cid:0) θ ∞ k ′ ( ∇ W ℓ k ′ ) , θ ∞ k ( ∂ x W ℓ k ) (cid:1) ˙ H ℓ k , ! Thus, by parity property, (cid:16) ∂ x j W, ∂ x ′ j W (cid:17) ˙ H = 0 and the decay properties of W , (cid:16) d Γ Φ(0 , Γ ∞ ) · ˜Γ (cid:17) k = ˜ λ k λ ∞ k ( θ ∞ k (Λ W ℓ k ) , θ ∞ k (Λ W ℓ k )) ˙ H ℓ k , ˜ y k, λ ∞ k ( θ ∞ k ( ∂ x W ℓ k ) , θ ∞ k ( ∂ x W ℓ k )) ˙ H ℓ k , . . . , X k ˜ y k, λ ∞ k ( θ ∞ k ( ∂ x W ℓ k ) , θ ∞ k ( ∂ x W ℓ k )) ˙ H ℓ k ! + E · ˜Γ , where kEk . T . Hence, d Γ Φ(0 , Γ ∞ ) is invertible for T large enough, with a lower bounduniform in Γ ∞ . Moreover, Φ(0 , Γ ∞ ) = 0 . Therefore, by the implicit function theorem (in fact,a uniform variant of the IFT), there exist < δ ≪ , < δ ≪ , and a continuous map Ψ : B ˙ H (0 , δ ) → B ((0 , + ∞ ) × R ) K (Γ ∞ , δ ) , such that for all ω ∈ B ˙ H (0 , δ ) and all Γ ∈ B ((0 , + ∞ ) × R ) K (Γ ∞ , δ ) , Φ( ω, Γ) = 0 if and only if
Γ = Ψ( ω ) . Moreover, | Ψ( ω ) − Γ ∞ | . k ω k ˙ H . This defines a continuous map t ∈ [ T , T ] ( λ k ( t ) , y k ( t )) k ∈{ ,...,K } such that | λ k ( t ) − λ ∞ k | + | y k ( t ) − y ∞ k | . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u ( t ) − X k W ∞ k ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H and such that ~ε ( t ) defined by (3.4) satisfies the orthogonality conditions (3.5). Since k W k ( t ) − W ∞ k ( t ) k ˙ H . | λ k ( t ) − λ ∞ k | + | y k ( t ) − y ∞ k | , (3.14)we have k ~ε ( t ) k E . X k k W k ( t ) − W ∞ k ( t ) k ˙ H + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − X k ~W ∞ k ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H × L . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − X k ~W ∞ k ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H × L , and (3.6) is proved. ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 17
For future reference, note that (cid:13)(cid:13)(cid:13) h x i / ∇ ( W k ( t ) − W ∞ k ( t )) (cid:13)(cid:13)(cid:13) L . t ( | λ k ( t ) − λ ∞ k | + | y k ( t ) − y ∞ k | ) . (3.15)Thus, if ( u ( t ) , ∂ t u ( t )) ∈ Y × Y , then we have kh x i / ∇ ε ( t ) k L + kh x i / η ( t ) k L . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h x i / ∇ u ( t ) − X k W ∞ k ( t ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h x i / ∂ t u ( t ) + X k ( ℓ k · ∇ W ∞ k )( t ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L + t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − X k ~W ∞ k ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H × L , and also k ~ε ( t ) k Y × Y . t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − X k ~W ∞ k ( t ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y × Y . (3.16) Step 2.
Equation of ~ε and parameter estimates. We formally derive the equations of ~ε ( t ) , λ k ( t ) and y k ( t ) from the equation of u . First, ε t = u t − X k ∂ t W k = η − X k ℓ k λ k · θ k ( ∇ W ℓ k ) − X k ∂ t ( θ k W ℓ k )= η + X k ˙ λ k λ k θ k (Λ W ℓ k ) + X k ˙ y k λ k · θ k ( ∇ W ℓ k ) , (3.17)since, by direct computations, ∂ t ( θ k W ℓ k ) = − ℓ k λ k · θ k ( ∇ W ℓ k ) − ˙ λ k λ k θ k (Λ W ℓ k ) − ˙ y k λ k · θ k ( ∇ W ℓ k ) . (3.18)Second (using (2.2)) η t = u tt + ∂ t X k ℓ k λ k · θ k ( ∇ W ℓ k ) ! = ∆ u + | u | u − X k ℓ k λ k · θ k ( ∇ ( ℓ k · ∇ W ℓ k )) − X k ˙ λ k λ k ℓ k · θ k ( ∇ Λ W ℓ k ) − X k ˙ y k λ k · θ k ( ∇ ( ℓ k · ∇ W ℓ k )) . Using u = P k θ k W ℓ k + ε , we have ∆ u = X k θ k λ k (∆ W ℓ k ) + ∆ ε, (3.19)and | u | u = f ( u ) = X k f ( θ k W ℓ k ) + X k f ′ ( θ k W ℓ k ) ! ε + R NL + R W , (3.20)where R W is defined in (3.8) and R NL = f X k W k + ε ! − f X k W k ! − f ′ X k W k ! ε Since f ( θ k W ℓ k ) = θ k λ k f ( W ℓ k ) , f ′ ( θ k W ℓ k ) = θ k λ k f ′ ( W ℓ k ) , we obtain ∆ u + | u | u = X k θ k λ k (cid:18) ∆ W ℓ k + W ℓ k (cid:19) + ∆ ε + 73 X k θ k λ k W ℓ k ε + R NL + R W . Using (2.38), we obtain η t = ∆ ε + 73 X k θ k λ k W ℓ k ε + R NL + R W − X k ˙ λ k ℓ k λ k · θ k ( ∇ Λ W ℓ k ) − X k ˙ y k λ k · θ k ( ∇ ( ℓ k · ∇ W ℓ k )) . In conclusion for ~ε , we obtain ~ε t = ~ L ~ε + ~ Mod + ~R NL + ~R W , (3.21)where ~ L = (cid:18)P k θ k λ / k W / ℓ k (cid:19) , ~R NL = (cid:18) R NL (cid:19) ~R W = (cid:18) R W (cid:19) , (3.22)and ~ Mod = X k ˙ λ k λ k ~θ k ~Z Λ ℓ k + X k ˙ y k λ k · ~θ k ~Z ∇ ℓ k . (3.23) Step 3.
Now, we derive the equations of λ k and y k from the orthogonality (3.5). First, ddt ( ε, θ (Λ W ℓ )) ˙ H ℓ = ( ε t , θ (Λ W ℓ )) ˙ H ℓ + ( ε, ∂ t ( θ (Λ W ℓ ))) ˙ H ℓ = 0 Thus, using (3.17), η, θ (Λ W ℓ )) ˙ H ℓ − (cid:18) ε, ℓ λ · θ ( ∇ (Λ W ℓ )) (cid:19) ˙ H ℓ + ˙ λ λ (cid:18) ( θ (Λ W ℓ ) , θ (Λ W ℓ )) ˙ H ℓ − (cid:0) ε, (cid:0) θ (Λ W ℓ ) (cid:1)(cid:1) ˙ H ℓ (cid:19) + (cid:18) ˙ y λ · θ ( ∇ W ℓ ) , θ (Λ W ℓ ) (cid:19) ˙ H ℓ − (cid:18) ε, ˙ y λ · θ ( ∇ Λ W ℓ ) (cid:19) ˙ H ℓ + K X k =2 ˙ λ k λ k ( θ k (Λ W ℓ k ) , θ (Λ W ℓ )) ˙ H ℓ + (cid:18) ˙ y k λ k · θ ( ∇ W ℓ ) , θ (Λ W ℓ ) (cid:19) ˙ H ℓ . (3.24)By the decay properties of W ℓ and integration by parts, we note that (cid:12)(cid:12)(cid:12)(cid:12) ( η, θ (Λ W ℓ )) ˙ H ℓ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ε, ℓ λ · θ ( ∇ (Λ W ℓ )) (cid:19) ˙ H ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k ~ε k E . (3.25) ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 19
Next, by (2.1), ( θ (Λ W ℓ ) , θ (Λ W ℓ )) ˙ H ℓ − (cid:0) ε, (cid:0) θ (Λ W ℓ ) (cid:1)(cid:1) ˙ H ℓ = (1 − | ℓ | ) k Λ W k H + O ( k ~ε k E ) , and by parity, (cid:18) ˙ y λ · θ ( ∇ W ℓ ) , θ (Λ W ℓ ) (cid:19) ˙ H ℓ = 0 , (cid:18) ε, ˙ y λ · θ ( ∇ Λ W ℓ ) (cid:19) ˙ H ℓ = O ( | ˙ y |k ~ε k E ) . Concerning the last terms, we claim, for k ∈ { , . . . , K } , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ λ k λ k ( θ k (Λ W ℓ k ) , θ (Λ W ℓ )) ˙ H ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) ˙ y k λ k · θ ( ∇ W ℓ ) , θ (Λ W ℓ ) (cid:19) ˙ H ℓ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ λ k λ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˙ y k λ k (cid:12)(cid:12)(cid:12)(cid:12)! . (3.26)Indeed, estimate (3.26) is a direct consequence of the following technical result. Claim 2.
Let < r ≤ r be such that r + r > . For t large, the following hold.– If r > then Z | W | r | W | r . t − r , (3.27) – If r ≤ then Z | W | r | W | r . t − r + r ) . (3.28) Proof of Claim 2.
Estimates written in this proof are for t large enough, and all constantsmay depend on ℓ k . For convenience, we denote ρ k = x − ℓ k t − y k ( t ) , Ω k ( t ) = { x such that | ρ k | < | ℓ − ℓ | t/ } . Note that, for t large, for x ∈ Ω , | W ( x ) | . h ρ i . h ρ i + t ) , for x ∈ Ω C , | W ( x ) | . h ρ i + t ) . t , for x ∈ Ω C , | W ( x ) | . h ρ i + t ) . t , Case r > , r > . Then, Z Ω | W | r | W | r . t − r Z | W | r . t − r , Z Ω C | W | r | W | r . t − r Z | W | r . t − r . Case r > , < r ≤ . In this case, Z Ω | W | r | W | r . Z h ρ i + t ) r h ρ i r dx . Z h x i + t ) r dx h x i r . t − r + r )+5 . t − r , and Z Ω C | W | r | W | r . t − r Z | W | r . t − r . Case < r ≤ , < r ≤ , r + r > . First, as before, Z Ω | W | r | W | r . t − r + r )+5 , Z Ω | W | r | W | r . t − r + r )+5 . Next, by Holder inequality, Z (Ω ∪ Ω ) C | W | r | W | r . Z Ω C h ρ i + t ) r + r ) ! r r r Z Ω C h ρ i + t ) r + r ) ! r r r . t − r + r )+5 . The claim is proved (cid:3)
In conclusion of the previous estimates, the orthogonality condition ( ε, θ (Λ W ℓ )) ˙ H ℓ = 0 ,gives the following | ˙ λ | . k ~ε k E + | ˙ y | k ~ε k E + 1 t K X k =1 (cid:16) | ˙ λ k | + | ˙ y k | (cid:17) . (3.29)Using the other orthogonality conditions, we obtain similarly, for k = 1 , . . . , , | ˙ λ k | . k ~ε k E + | ˙ y k | k ~ε k E + 1 t K X k ′ =1 (cid:16) | ˙ λ k ′ | + | ˙ y k ′ | (cid:17) , (3.30) | ˙ y k | . k ~ε k E + | ˙ λ k | k ~ε k E + 1 t K X k ′ =1 (cid:16) | ˙ λ k ′ | + | ˙ y k ′ | (cid:17) . (3.31)Combining these estimates, we find (3.11). Note that equation (3.24) and the correspondingformula for ˙ λ k and ˙ y k for k ≥ , where ~ε is replaced by ~u − P k ~W k form a nondegenerate firstorder differential system, whose unique solution is ( λ k , y k ) k , which justifies the C regularityof the parameters. Step 4.
Unstable directions. Recall that the quantities z ± k are defined through the L scalar product z ± k ( t ) = (cid:16) ~ε ( t ) , ~ ˜ θ k ~Z ± ℓ k (cid:17) L . Recall also that ~Z ± ℓ k ∈ S . By (3.21), we have ddt z ± = ddt (cid:16) ~ε, ~ ˜ θ ~Z ± ℓ (cid:17) L = (cid:16) ~ε t , ~ ˜ θ ~Z ± ℓ (cid:17) L + (cid:16) ~ε, ∂ t (cid:16) ~ ˜ θ ~Z ± ℓ (cid:17)(cid:17) L = (cid:16) ~ L ~ε, ~ ˜ θ ~Z ± ℓ (cid:17) L + ℓ λ · (cid:16) ~ε, ~ ˜ θ ∇ ~Z ± ℓ (cid:17) L + ˙ λ λ (cid:16)(cid:16) ~θ ~Z Λ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L − (cid:16) ~ε, ~ ˜ θ ~ Λ ~Z ± ℓ (cid:17) L (cid:17) + ˙ y λ · (cid:16)(cid:16) ~θ ~Z ∇ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L − (cid:16) ~ε, ~ ˜ θ ∇ ~Z ± ℓ (cid:17) L (cid:17) + K X k =2 ˙ λ k λ k (cid:16) ~θ k ~Z Λ ℓ k , ~ ˜ θ ~Z ± ℓ (cid:17) L + ˙ y k λ k · (cid:16) ~θ k ~Z ∇ ℓ k , ~ ˜ θ ~Z ± ℓ (cid:17) L ! + (cid:16) ~R NL + ~R W , ~ ˜ θ ~Z ± ℓ (cid:17) L . ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 21
First, by direct computations, using (2.40), (cid:16) ~ L ~ε, ~ ˜ θ ~Z ± ℓ (cid:17) L − ℓ λ · (cid:16) ~ε, ~ ˜ θ ∇ ~Z ± ℓ (cid:17) L = 1 λ (cid:16) ~ε, ~ ˜ θ (cid:16) − H ℓ J ~Z ± ℓ (cid:17)(cid:17) L + X k ≥ (cid:16) ε, f ′ ( θ k W ℓ k )( θ Z ± ℓ , ) (cid:17) L = ± √ λ λ (1 − | ℓ | ) z ± + X k ≥ (cid:16) ε, f ′ ( θ k W ℓ k )( θ Z ± ℓ , ) (cid:17) L . Note that by the decay properties of ~Z ± ℓ and Claim 2, for k ≥ , (cid:12)(cid:12)(cid:12)(cid:16) ε, f ′ ( θ k W ℓ k )( θ Z ± ℓ , ) (cid:17) L (cid:12)(cid:12)(cid:12) . k ε k ˙ H t . (3.32)By (2.39), we have (cid:16) ~θ ~Z Λ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L = (cid:16) ~Z Λ ℓ , ~Z ± ℓ (cid:17) L = 0 , and thus, by (3.11), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ λ λ (cid:16)(cid:16) ~θ ~Z Λ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L − (cid:16) ~ε, ~ ˜ θ ~ Λ ~Z ± ℓ (cid:17) L (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | ˙ λ | k ~ε k E . k ~ε k E . (3.33)Similarly, (cid:12)(cid:12)(cid:12)(cid:12) ˙ y λ · (cid:16)(cid:16) ~θ ~Z ∇ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L + (cid:16) ~ε, ~ ˜ θ ∇ ~Z ± ℓ (cid:17) L (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . k ~ε k E . (3.34)Next, by Claim 2, we have (cid:12)(cid:12)(cid:12)(cid:16) ~θ ~Z Λ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) ~θ ~Z ∇ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L (cid:12)(cid:12)(cid:12) . t . Thus, by (3.11), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ λ λ (cid:16) ~θ ~Z Λ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ˙ y λ · (cid:16) ~θ ~Z ∇ ℓ , ~ ˜ θ ~Z ± ℓ (cid:17) L (cid:12)(cid:12)(cid:12)(cid:12) . k ~ε k E t . (3.35)Finally, we claim (cid:12)(cid:12)(cid:12)(cid:16) ~R W , ~ ˜ θ ~Z ± ℓ (cid:17) E (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) ~R NL , ~ ˜ θ ~Z ± ℓ (cid:17) E (cid:12)(cid:12)(cid:12) . t + k ε k ˙ H t + k ε k H . (3.36)Proof of (3.36). Note the following estimate, for any p > , | R W | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f X k W k ! − X k f ( W k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . X k = k ′ | W k | | W k ′ | . (3.37)Thus, using Claim 2, (cid:12)(cid:12) ( R W , θ ( − ℓ ∂ x Λ W ℓ )) L (cid:12)(cid:12) . Z X k = k ′ | W k | | W k ′ | | W | . t . (3.38) Next, we decompose R NL = R ε, + R ε, , where R ε, = f ′ X k W k ! − X k f ′ ( W k ) ! ε,R ε, = f X k W k + ε ! − f X k W k ! − f ′ X k W k ! ε. First, | R ε, | ≤ X k ′ = k | W k ′ | | W k | | ε | . Thus, using Claim 2 and (2.3) (cid:12)(cid:12) ( R ε, , θ ( − ℓ ∂ x Λ W ℓ )) L (cid:12)(cid:12) . Z X k = k ′ | W k ′ | | W k | | W | | ε | (3.39) . (cid:18)Z | ε | | W | (cid:19) Z W X k ′ = k | W k ′ | | W k | . t k ε k ˙ H . (3.40)Finally, we have | R ε, | . (cid:16)P k | W k | (cid:17) | ε | + | ε | , and thus, by (2.3) and (2.4), (cid:12)(cid:12) ( R ε, , θ ( − ℓ ∂ x Λ W ℓ )) L (cid:12)(cid:12) . Z X k | W k | ! | ε | + | ε | ! | W | . k ε k H + k ε k ˙ H . The proof of (3.36) is complete.Extending this computation to z ± k for any k , we obtain in conclusion (cid:12)(cid:12)(cid:12)(cid:12) ddt z ± k ( t ) ∓ √ λ λ k ( t ) (1 − | ℓ k | ) z ± k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) . k ~ε ( t ) k E + k ~ε ( t ) k E t + 1 t . (3.41)The proof of Lemma 3.1 is complete. (cid:3) Proof of Theorem 1 case (B)
In this section, we prove the existence of a solution u ( t ) of (1.1) satisfying (1.4)–(1.5) incase (B) of Theorem 1. We argue by compactness and obtain u ( t ) as the limit of suitableapproximate multi-solitons u n ( t ) .Let K ≥ and for all k ∈ { , . . . , K } , let λ ∞ k > , y ∞ k ∈ R and ℓ k ∈ R . Let S n → + ∞ .For ζ ± k,n ∈ R small to be determined later (see statements of Proposition 4.1, Claim 3 andLemma 4.2), we consider the solution u n of ∂ t u n − ∆ u n − | u n | u n = 0( u n ( S n ) , ∂ t u n ( S n )) = X k h ( ~θ ∞ k ~W ℓ k )( S n ) + ζ + k,n ( ~θ ∞ k ~Z + ℓ k )( S n ) + ζ − k,n ( ~θ ∞ k ~Z − ℓ k )( S n ) i (4.1)Note that since ( u n ( S n ) , ∂ t u n ( S n )) ∈ Y × Y , the solution u n is well-defined in Y × Y atleast on a small interval of time around S n (see section 2.1). ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 23
Now, we state the main uniform estimates on u n . Proposition 4.1.
Under the assumptions of Theorem 1, case (B), there exist n > and T > such that, for any n ≥ n , there exist ( ζ ± k,n ) k ∈{ ,...,K } ∈ R K , with K X k =1 | ζ ± k,n | . S n , (4.2) and such that the solution ~u n = ( u n , ∂ t u n ) of (4.1) is well-defined in Y × Y on the timeinterval [ T , S n ] and satisfies ∀ t ∈ [ T , S n ] , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u n ( t ) − K X k =1 ~W ∞ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H × L . t , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u n ( t ) − K X k =1 ~W ∞ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y × Y . t . (4.3)4.1. Proof of Theorem 1 case (B), assuming Proposition 4.1.
In view of the uniformbounds obtained in (4.3) at t = T , up to the extraction of a subsequence, ( u n ( T ) , ∂ t u n ( T )) converges strongly in ˙ H × L to some ( u , u ) as n → + ∞ . Consider the solution u ( t ) of(1.1) associated to the initial data ( u , u ) at t = T . Then, by the uniform bounds (4.3)and the continuous dependence of the solution of (1.1) with respect to its initial data in theenergy space ˙ H × L (see e.g. [14] and references therein), the solution u is well-defined inthe energy space on [ T , ∞ ) and satisfies (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ~u ( t ) − K X k =1 W ∞ k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H × L . t . (4.4)This finishes the proof of Theorem 1 in case (B), assuming Proposition 4.1.The rest of this section is devoted to the proof of Proposition 4.1.4.2. Bootstrap setting.
We denote by B R K ( ρ ) (respectively, S R K ( ρ ) ) the ball (respectively,the sphere) of R K of center and of radius ρ > , for the usual norm | ( ξ k ) k | = (cid:16)P Kk =1 ξ k (cid:17) / .For t = S n and for t < S n as long as u ( t ) is well-defined in ˙ H × L and satisfies (3.3), wedecompose u n ( t ) as in Lemma 3.1. In particular, we denote by ( ε, η ) , ( λ k ) k , ( y k ) k , ( z ± k ) k theparameters of the decomposition of u n . We also set W K = K X k =1 W k , f W K = K X k =1 | W k | . (4.5)We start with a technical result similar to Lemma 3 in [4]. This claim will allow us to adjustthe initial values of ( z ± k ( S n )) k from the choice of ζ ± k,n in (4.1). Claim 3 (Choosing the initial unstable modes) . There exist n > and C > such that,for all n ≥ n , for any ( ξ k ) k ∈{ ,...,K } ∈ B R K ( S − / n ) , there exists a unique ( ζ ± k,n ) k ∈{ ,...,K } ∈ B R K ( CS − / n ) such that the decomposition of u n ( S n ) satisfies z − k ( S n ) = ξ k , z + k ( S n ) = 0 , (4.6) | λ k ( S n ) − λ ∞ k | + | y k ( S n ) − y ∞ k | + k ~ε ( S n ) k E . S − / n , (4.7) k ~ε ( S n ) k Y × Y . S − n . (4.8) Sketch of the proof of Claim 3.
The proof of existence of ( ζ ± k,n ) k in Claim 3 is similar toLemma 3 in [4] and we omit it. Estimates in (4.7) are consequences of (3.6), (4.8) followsfrom (3.16). (cid:3) From now on, for any ( ξ k ) k ∈ B R K ( S − / n ) , we fix ( ζ ± k,n ) k as given by Claim 3 and thecorresponding solution u n of (4.1).The proof of Proposition 4.1 is based on the following bootstrap estimates: for C ∗ > tobe chosen, K X k =1 | λ k ( t ) − λ ∞ k | + | y k ( t ) − y ∞ k | ≤ ( C ∗ ) t , K X k =1 | z ± k ( t ) | ≤ t k ~ε ( t ) k E ≤ C ∗ t , k ~ε ( t ) k Y × Y ≤ ( C ∗ ) t (4.9)Set T ∗ = T ∗ n (( ξ k ) k ) = inf { t ∈ [ T , S n ] ; u n satisfies (3.3) and (4.9) holds on [ t, S n ] } . (4.10)Note that by Claim 3, estimate (4.9) is satisfied at t = S n . Moreover, if (4.9) is satisfied on [ τ, S n ] for some τ ≤ S n then by the well-posedness theory in Y × Y and continuity, u n ( t ) is well-defined and satisfies the decomposition of Lemma 3.1 on [ τ ′ , S n ] , for some τ ′ < τ . Inparticular, the definition of T ∗ makes sense and it will suffice to strictly improve (4.9) on [ T ∗ , S n ] to prove T ∗ = T for some ( ξ k ) k . Note also that we will prove that T ∗ = S n for ( ξ k ) k ∈ S R K ( S − / n ) (see proof of Lemma 4.2).In what follows, we will prove that there exists T large enough and at least one choice of ( ξ k ) k ∈ B R K ( S − / n ) so that T ∗ = T , which is enough to finish the proof of Proposition 4.1.For this, we derive general estimates for any ( ξ k ) k ∈ B R K ( S − / n ) (see Lemma 4.1) and use atopological argument (see Lemma 4.2) to control the instable directions, in order to strictlyimprove estimates in (4.9) and thus prove that they cannot be saturated on [ T , S n ] .4.3. Energy functional.
One of the main points of the proof of Proposition 4.1 is to derivesuitable estimates in the energy norm that will strictly improve the bound on k ~ε ( t ) k E from(4.9); the other estimates then follow easily.We claim the following proposition in case (B) of Theorem 1. This is the only place in thepaper where we need the restriction of collinear speeds. Proposition 4.2.
Under the assumptions of Theorem 1, case (B), there exist µ > and afunction H K ( t ) on [ T ∗ , S n ] , which satisfies the following properties. (i) Bound. |H K ( t ) | ≤ k ~ε k E µ . (4.11)(ii) Coercivity. H K ( t ) ≥ µ k ~ε k E − t − µ . (4.12)(iii) Time variation. − ddt (cid:0) t H K (cid:1) ( t ) . C ∗ t − . (4.13) ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 25
Proof of Proposition 4.2.
We consider the case where the K solitons are moving in the samedirection. In particular, by rotation invariance, we assume ∀ k ∈ { , . . . , K } , ℓ k = ℓ k e where ℓ k ∈ ( − , . (4.14)Moreover, without loss of generality, − < ℓ < . . . < ℓ K < . Fix max k ( | β k | ) < ℓ < . For < σ <
110 min( ℓ k +1 − ℓ k ) small enough to be fixed, we setfor k = 1 , . . . , K − , ℓ + k = ℓ k + σ ( ℓ k +1 − ℓ k ) , for k = 2 , . . . , K, ℓ − k = ℓ k − σ ( ℓ k − ℓ k − ) , and for t > , Ω( t ) = (( ℓ +1 t, ℓ − t ) ∪ . . . ∪ ( ℓ + K − t, ℓ − K t )) × R , Ω C ( t ) = R \ Ω( t ) . We consider the continuous function χ K ( t, x ) = χ K ( t, x ) defined as follows, for all t > , χ K ( t, x ) = ℓ for x ∈ ( −∞ , ℓ +1 t ] ,χ K ( t, x ) = ℓ k for x ∈ [ ℓ − k t, ℓ + k t ] , for k ∈ { , . . . , K − } , χ K ( t, x ) = ℓ K for x ∈ [ ℓ − K t, + ∞ ) ,χ K ( t, x ) = x (1 − σ ) t − σ − σ ( ℓ k +1 + ℓ k ) for x ∈ [ ℓ + k t, ℓ − k +1 t ] , k ∈ { , . . . , K − } . (4.15)In particular, ∂ t χ K ( t, x ) = 0 , ∇ χ K ( t, x ) = 0 , on Ω C ( t ) ,∂ x χ K ( t, x ) = 1(1 − σ ) t for x ∈ Ω( t ) ,∂ t χ K ( t, x ) = − t x (1 − σ ) t for x ∈ Ω( t ) . (4.16)We define H K ( t ) = Z E K ( t, x ) dx + 2 Z ( χ K ( t, x ) ∂ x ε ( t, x )) η ( t, x ) dx, where E K = |∇ ε | + | η | − F ( W K + ε ) − F ( W K ) − f ( W K ) ε ) . (4.17)Note that from (4.9) and (3.11), we have X k (cid:16) | ˙ λ k | + | ˙ y k | (cid:17) . k ~ε ( t ) k E . C ∗ t . In particular, from (3.9) and (3.10), for all p ∈ N (here | p | = P j p j ), | ∂ px Mod ε ( t ) | . C ∗ t f W | p | K , | ∂ px Mod η ( t ) | . C ∗ t f W + | p | K . (4.18) Proof of (4.11) . Since | F ( W K + ε ) − F ( W K ) − f ( W K ) ε | . | ε | + f W K | ε | , the estimate (4.11) on H K follows from Hölder inequality, (2.4) and (4.9). Proof of (4.12) . Set N Ω ( t ) = Z Ω (cid:0) |∇ ε ( t ) | + η ( t ) + 2( χ K ( t ) ∂ x ε ( t )) η ( t ) (cid:1) and N Ω C ( t ) = Z Ω C (cid:0) |∇ ε ( t ) | + η ( t ) (cid:1) . Note that, since | χ K | < ℓ , N Ω = ℓ Z Ω (cid:12)(cid:12)(cid:12)(cid:12) χ K ℓ ∂ x ε + η (cid:12)(cid:12)(cid:12)(cid:12) + Z Ω |∇ ε | + Z Ω (cid:18) − χ K ℓ (cid:19) ( ∂ x ε ) + (1 − ℓ ) Z η ≥ ℓ Z Ω (cid:12)(cid:12)(cid:12)(cid:12) χ K ℓ ∂ x ε + η (cid:12)(cid:12)(cid:12)(cid:12) + (1 − ℓ ) Z Ω (cid:0) |∇ ε | + η (cid:1) . (4.19)To obtain (4.12), we will actually prove the following stronger property H K ( t ) ≥ N Ω ( t ) + µ N Ω C ( t ) − t − µ − t − α µ k ~ε k E − µ k ~ε k E . (4.20)We decompose H K = f + f + f , where f = Z |∇ ε | − Z X k f ′ ( W k ) ! ε + Z η + 2 Z ( χ K ∂ x ε ) η, f = − Z (cid:18) F ( W K + ε ) − F ( W K ) − f ( W K ) ε − f ′ ( W K ) ε (cid:19) , f = Z X k f ′ ( W k ) − f ′ ( W K ) ! ε , We claim the following estimates f ≥ N Ω + µ N Ω C − t − µ − t − α µ k ~ε k E , (4.21) | f | + | f | . k ~ε k E + k ~ε k E t . (4.22)Note that combining these estimates with (4.9) and taking T large enough (depending on C ∗ ), we obtain (4.20) and then (4.12) for some other µ > .Proof of (4.21). The main ingredient in the proof of (4.21) is Lemma 2.2. For ϕ defined in(2.8), set ϕ k ( t, x ) = ϕ (cid:18) x − ℓ k e t − y k ( t ) λ k ( t ) (cid:19) . ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 27
We decompose f as follows f = N Ω + X k (cid:18)Z |∇ ε | ϕ k − Z f ′ ( W k ) ε + Z η ϕ k + 2 Z ( χ K ∂ x ε ) ηϕ k (cid:19) + Z Ω C (cid:0) |∇ ε | + η + 2 χ K ( ∂ x ε ) η (cid:1) − X k ϕ k ! − Z Ω (cid:0) |∇ ε | + η + 2 χ K ( ∂ x ε ) η (cid:1) X k ϕ k ! + 2 X k Z ( χ K − ℓ k )( ∂ x ε ) ηϕ k = N Ω + f , + f , + f , + f , . By Lemma 2.2, the orthogonality conditions on ~ε and a change of variable, we have f , ≥ µ Z (cid:0) |∇ ε | + η (cid:1) X k ϕ k ! − µ X k (cid:0) ( z − k ) + ( z + k ) (cid:1) . Thus, using (4.9), f , ≥ µ Z (cid:0) |∇ ε | + η (cid:1) X k ϕ k ! − µ t ≥ µ Z Ω C (cid:0) |∇ ε | + η (cid:1) X k ϕ k ! − µ t . Next, note that if x is such that ϕ k ( t, x ) > , then ϕ k ′ ( x ) . t − α for k ′ = k . Thus, − X k ϕ k & − t − α . By direct computations (with the notation v + = max(0 , v ) ), f , = ℓ Z Ω C (cid:12)(cid:12)(cid:12)(cid:12) χ K ℓ ∂ x ε + η (cid:12)(cid:12)(cid:12)(cid:12) − X k ϕ k ! + Z Ω C |∇ ε | − X k ϕ k ! + Z Ω C (cid:18) − χ K ℓ (cid:19) | ∂ x ε | − X k ϕ k ! + (1 − ℓ ) Z Ω C η − X k ϕ k ! ≥ (1 − ℓ ) Z Ω C (cid:0) |∇ ε | + η (cid:1) − X k ϕ k ! + − k ~ε k E t α . Also, we see easily that | f , | . t − α k ~ε k E . Finally, by the definition of χ K in (4.15), the decay property of ϕ and (4.9) (for a boundon y k ), we have k ( χ K − ℓ k ) ϕ k k L ∞ ≤ t − α . Thus, | f , | . t − α k ~ε k E . Therefore, for some µ > , and T large enough, we have f , + f , + f , + f , ≥ µ N Ω C − µ t − t − α k ~ε k E . Proof of (4.22). Using Hölder inequality, (2.4) and (4.9), we have | f | . Z | ε | + | ε | f W K . k ~ε k E . Next, since by the decay property of W , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ′ ( W K ) − X k f ′ ( W k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . f W K t , using (2.3), we obtain | f | . t Z | ε | f W K . k ~ε k E t . Proof of (4.13) . Step 1.
First estimates. We decompose ddt H K = Z ∂ t E K + 2 Z χ K ∂ t (( ∂ x ε ) η ) + 2 Z ( ∂ t χ K )( ∂ x ε ) η = g + g + g . We claim the following estimates g = 2 Z ε (cid:0) − ∆Mod ε − f ′ ( W K )Mod ε (cid:1) + 2 Z η Mod η + 2 Z X k ℓ k ∂ x W k ! (cid:0) f ( W K + ε ) − f ( W K ) − f ′ ( W K ) ε (cid:1) + O (cid:18) C ∗ t (cid:19) , (4.23) g = − − σ ) t Z Ω (cid:0) η + ( ∂ x ε ) − |∇ ε | (cid:1) − Z χ K ( ∂ x W K ) (cid:0) f ( W K + ε ) − f ( W K ) − f ′ ( W K ) ε (cid:1) + 2 Z ( χ K ∂ x Mod ε ) η − Z εχ K ∂ x Mod η + O (cid:18) C ∗ t (cid:19) , (4.24) g = − − σ ) t Z Ω x t ∂ x εη. (4.25) Estimate on g . From direct computations and the definition of
Mod ε in (3.9), we have g = 2 Z ( ∇ ε t · ∇ ε + η t η − ε t ( f ( W K + ε ) − f ( W K )))+ 2 Z X k ℓ k ∂ x W k ! (cid:0) f ( W K + ε ) − f ( W K ) − f ′ ( W K ) ε (cid:1) + 2 Z Mod ε (cid:0) f ( W K + ε ) − f ( W K ) − f ′ ( W K ) ε (cid:1) = g , + g , + g , . Using (3.7) and integration by parts, g , = 2 Z ηR W + 2 Z ( ∇ ε · ∇ Mod ε − ( f ( W K + ε ) − f ( W K )) Mod ε + η Mod η ) ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 29
By Cauchy-Schwarz inequality, (3.37) and then (4.9), (cid:12)(cid:12)(cid:12)(cid:12)Z ηR W (cid:12)(cid:12)(cid:12)(cid:12) . k η k L k R W k L . k η k L t . C ∗ t . Thus, g , + g , = 2 Z ε (cid:0) − ∆Mod ε − f ′ ( W K )Mod ε (cid:1) + 2 Z η Mod η + O (cid:18) C ∗ t (cid:19) , and (4.23) follows. Estimate on g . g = 2 Z ( χ K ∂ x ε t ) η + 2 Z ( χ K ∂ x ε ) η t = 2 Z ( χ K ∂ x η ) η + 2 Z ( χ K ∂ x ε ) (∆ ε + ( f ( W K + ε ) − f ( W K )) + R W )+ 2 Z ( χ K ∂ x Mod ε ) η + 2 Z ( χ K ∂ x ε )Mod η . Note that by integration by parts and (4.16) Z ( χ K ∂ x η ) η + 2 Z ( χ K ∂ x ε )∆ ε = − Z ∂ x χ K (cid:0) η + ( ∂ x ε ) − |∇ ε | (cid:1) = − − σ ) t Z Ω (cid:0) η + ( ∂ x ε ) − |∇ ε | (cid:1) . Next, we observe Z ( χ K ∂ x ε ) ( f ( W K + ε ) − f ( W K ) ε ) = Z χ K ∂ x ( F ( W K + ε ) − F ( W K ) − f ( W K ) ε ) − Z χ K ( ∂ x W K ) (cid:0) f ( W K + ε ) − f ( W K ) − f ′ ( W K ) ε (cid:1) . Moreover, integrating by parts and using (4.16), − Z χ K ∂ x ( F ( W K + ε ) − F ( W K ) − f ( W K ) ε )= 1(1 − σ ) t Z Ω ( F ( W K + ε ) − F ( W K ) − f ( W K ) ε ) . Thus, by (4.9) and the decay of W , (cid:12)(cid:12)(cid:12)(cid:12)Z χ K ∂ x ( F ( W K + ε ) − F ( W K ) − f ( W K ) ε ) (cid:12)(cid:12)(cid:12)(cid:12) . t Z Ω (cid:18) | ε | + W K | ε | (cid:19) . t . Last, integrating by parts, Z ( χ K ∂ x ε )Mod η = − Z ( χ K ε ) ∂ x Mod η − Z ( ∂ x χ K ) ε Mod η = − Z ( χ K ε ) ∂ x Mod η + O (cid:18) t (cid:19) . Indeed, by (4.9), (4.16), (4.18) and (2.3), (cid:12)(cid:12)(cid:12)(cid:12)Z ( ∂ x χ K ) ε Mod η (cid:12)(cid:12)(cid:12)(cid:12) . C ∗ t Z Ω | ε | f W K . C ∗ t (cid:18)Z | ε | f W K (cid:19) (cid:18)Z Ω f W K (cid:19) . ( C ∗ ) t . t . Estimate on g . (4.25) is a consequence of (4.16). Step 2.
Using cancellations and conclusion. In conclusion of estimates (4.23)–(4.25), ddt H K = h + h + h + h + O (cid:18) C ∗ t (cid:19) , where h = − − σ ) t Z Ω (cid:16) η + ( ∂ x ε ) + 2 x t ( ∂ x ε ) η − |∇ ε | (cid:17) , h = 2 Z X k ( ℓ k − χ K ) ∂ x W k ! (cid:0) f ( W K + ε ) − f ( W K ) − f ′ ( W K ) ε (cid:1) , h = 2 Z η (Mod η + χ K ∂ x Mod ε ) , h = 2 Z ε (cid:0) − ∆Mod ε − χ K ∂ x Mod η − f ′ ( W K )Mod ε (cid:1) . First, by (4.19) and the definition of χ K in (4.15), − ((1 − σ ) t ) h ≤ ℓ Z Ω (cid:12)(cid:12)(cid:12)(cid:12) χ K ℓ ∂ x ε + η (cid:12)(cid:12)(cid:12)(cid:12) + Z Ω (cid:18) − χ K ℓ (cid:19) ( ∂ x ε ) + (1 − ℓ ) Z η + 2 Z Ω (cid:16) x t − χ K (cid:17) ∂ x εη ≤ N Ω + Cσ Z (cid:0) | ∂ x ε | + η (cid:1) ≤ (1 + Cσ ) N Ω . Second, we observe that by the definition of χ K in (4.16) and the decay of ∂ x W and W , | ( ℓ k − χ K ) ∂ x W k | . | ℓ k − χ K | | W k | / . t | W k | / . Thus, by (2.3) and (2.4), | h | . t Z (cid:18) | ε | + | ε | f W K (cid:19) . ( C ∗ ) t . t . Denote M k = ˙ λ k λ k Λ W k + ˙ y k · ∇ W k so that Mod ε = X k M k , Mod η = − X k ℓ k ∂ x M k (see the definition of Mod ε and Mod η in (3.9)–(3.10)). Using (4.18), the definition of χ K (see(4.16)) and the decay of W , | ( ℓ k − χ K ) ∂ x M k | . C ∗ t t | W k | . (4.26)In particular, | Mod η + χ K ∂ x Mod ε | . C ∗ t f W K , ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 31 and thus, since f W K is bounded in L , | h | = (cid:12)(cid:12)(cid:12)(cid:12)Z η (Mod η + χ K ∂ x Mod ε ) (cid:12)(cid:12)(cid:12)(cid:12) . C ∗ t k η k L . ( C ∗ ) t . t . Finally, we see that by (2.21), − ∆ M k + ℓ k ∂ x M k − f ′ ( W k ) M k = 0 . Thus, as before, (cid:12)(cid:12) − ∆ M k + ℓ k χ K ∂ x M k − f ′ ( W K ) M k (cid:12)(cid:12) . (cid:12)(cid:12) ( χ K − ℓ k ) ∂ x M k (cid:12)(cid:12) + (cid:12)(cid:12) f ′ ( W K ) − f ′ ( W k ) (cid:12)(cid:12) | M k | . C ∗ t | W k | . Therefore, (cid:12)(cid:12) − ∆Mod ε − χ K ∂ x Mod η − f ′ ( W K )Mod ε (cid:12)(cid:12) . C ∗ t f W K . It follows that (by (2.3)), | h | . C ∗ t (cid:13)(cid:13)(cid:13) ε f W / K (cid:13)(cid:13)(cid:13) L . C ∗ t k ε k ˙ H . ( C ∗ ) t . t . In conclusion, using (4.20), for σ small, and T large, − ddt H K ≤ (1 + Cσ ) t N Ω + O (cid:18) C ∗ t (cid:19) ≤ t H K + O (cid:18) C ∗ t (cid:19) . The proof of Proposition 4.2 is complete. (cid:3)
End of the proof of Proposition 4.1.
The following result, mainly based on Propo-sition 4.2, improves all the estimates in (4.9), except the ones on ( z − k ) k . Lemma 4.1 (Closing estimates except ( z − k ) k ) . For C ∗ > large enough, for all t ∈ [ T ∗ , S n ] , | λ k ( t ) − λ ∞ k | + | y k ( t ) − y ∞ k | ≤ ( C ∗ ) t , K X k =1 | z + k ( t ) | ≤ t k ~ε ( t ) k E ≤ C ∗ t , k ~ε ( t ) k Y × Y ≤ ( C ∗ ) t (4.27)The control of the directions ( z − k ) k , related to the dynamical instability of W , requires aspecific argument used in [4] in a similar context. Lemma 4.2 (Control of unstable directions) . There exist ( ξ k,n ) k ∈ B R K ( S − / n ) such that,for C ∗ > large enough, T ∗ (( ξ k,n ) k ) = T . In particular, let ( ζ ± n ) be given by Claim 3 fromsuch ( ξ k,n ) k , then the solution u n of (4.1) satisfies (4.3) . Note that Lemma 4.2 completes the proof of Proposition 4.1.
Proof of Lemma 4.1.
Step 1.
We prove that for C ∗ large enough, for all t ∈ [ T ∗ , S n ] , k ~ε k Y × Y ≤ ( C ∗ ) t . (4.28)The system (3.7) of equations of ε and η can be written under the form (cid:26) ε t = η + Mod ε η t = ∆ ε + R ε + R W + Mod η , where | R ε | . | ε | / + | ε | f W / K , |∇ R ε | . |∇ ε | (cid:16) | ε | / + f W / K (cid:17) + | ε | f W / K . In particular, by (2.6) k R ε k Y . k ε k ˙ H k ε k Y + t k ε k ˙ H . C ∗ t − / . Moreover, k R W k Y . t − / , and by (4.18), k Mod ε k Y + k Mod η k Y . C ∗ t − / . Using (4.8) and (2.5), we obtain k ~ε ( t ) k Y × Y . k ~ε ( S n ) k Y × Y + Z S n t (cid:0) k R ε ( t ′ ) k Y + k R W ( t ′ ) k Y + k Mod ε ( t ′ ) k Y + k Mod η ( t ′ ) k Y (cid:1) dt ′ . C ∗ t . In particular, taking C ∗ large enough, we obtain (4.28). Step 2.
Estimates on parameters. The estimates on | λ k ( t ) − λ ∞ k | and | y k ( t ) − y ∞ k | followfrom integration of (3.11) using (4.9) and (4.7), and possibly taking a larger C ∗ .Now, we prove the bound on z + k ( t ) . Let c k = √ λ λ ∞ k (1 − | ℓ k | ) / > . Then, from (3.13) and(4.9), ddt (cid:2) e − c k t z + k (cid:3) . e − c k t C ∗ t . Integrating on [ t, S n ] and using (4.6), we obtain − z + k ( t ) . C ∗ t − . Doing the same for − e − c k t z + k ,we obtain the conclusion for T large enough. Step 3.
Bound on the energy norm. Finally, to prove the estimate on k ~ε ( t ) k E , we useProposition 4.2. Recall from (4.7) and then (4.11) that H K ( S n ) . S − n . (4.29)Integrating (4.13) on [ t, S n ] , and using (4.29), we obtain, for all t ∈ [ T ∗ , S n ] , H K . C ∗ t − . Using (4.12), we conclude that k ~ε k E . C ∗ t − . (cid:3) Proof of Lemma 4.2.
Step 1.
Choice of ( ζ k ) . We follow the strategy of Lemma 6 in [4].The proof is by contradiction, we assume that for any ( ξ k ) k ∈{ ,...,K } ∈ B R K ( S − / n ) , T ∗ (( ξ k ) k ) defined by (4.10) satifies T ∗ ∈ ( T , S n ) . In this case, by Lemma 4.1 and continuity, it holdsnecessarily K X k =1 | z − k ( T ∗ ) | = 1( T ∗ ) . (4.30)We claim the following transversality property at T ∗ ddt t K X k =1 | z − k ( t ) | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = T ∗ < − c < . (4.31) ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 33
Let c k = √ λ λ ∞ k (1 − | ℓ k | ) / > and c = min k c k . From (3.13) and (4.9), for all t ∈ [ T ∗ , S n ] , ddt (cid:16) t (cid:0) z − k (cid:1) (cid:17) = 2 t z − k ddt z − k + 5 t (cid:0) z − k (cid:1) ≤ − t c k (cid:0) z − k (cid:1) + CC ∗ t ≤ − ct (cid:0) z − k (cid:1) + CC ∗ t . Thus, from (4.30) ddt t K X k =1 (cid:0) z − k (cid:1) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = T ∗ ≤ − c + CC ∗ ( T ∗ ) < − c, for T large enough (depending on C ∗ , but independent of n ).As a consequence of (4.31), we observe that the map T ∗ ( ξ k ) k ∈{ ,...,K } ∈ B R K ( S − / n ) T ∗ (( ξ k ) k ) is continuous. Indeed, if T ∗ < S n , by (4.31), it is clear that for all σ > small enough, thereexists δ > so that for all t ∈ [ T ∗ + σ, S n ] , t P k (cid:0) z − k ( t ) (cid:1) < (1 − δ ) . In particular, for ( ˜ ξ k ) k ∈ B R K ( S − / n ) close enough to ( ξ k ) k , it follows that for all t ∈ [ T ∗ + σ, S n ] , t P k (cid:0) ˜ z − k ( t ) (cid:1) < (1 − δ ) , and thus ˜ T ∗ < T ∗ + σ . By similar arguments, for ( ˜ ξ k ) k ∈ B R K ( S − / n ) close enoughto ( ξ k ) k , we also have ˜ T ∗ > T ∗ − σ .We define M : B R K ( S − / n ) → S R K ( S − / n )( ξ k ) k (cid:18) T ∗ S n (cid:19) / ( z − k ( T ∗ )) k From what precedes, M is continuous. Moreover, from (4.30) and (4.31), M restricted to S R K ( S − / n ) is the identity (since in this case T ∗ = S n and z − k ( S n ) = ξ k from (4.6)). Theexistence of such a map is contradictory with Brouwer’s fixed point theorem. Step 2.
Conclusion. Proof of (4.3). These estimates follow directly from the estimates(4.9) on ε ( t ) , λ k ( t ) , y k ( t ) and (3.14), (3.15). (cid:3) Proof of Theorem 1 case (A) by Lorentz transformation
Let λ ∞ , λ ∞ > , y ∞ , y ∞ ∈ R , ι = ± , ι = ± . Let ℓ , ℓ ∈ R with ℓ = ℓ and | ℓ k | < for k = 1 , . We claim that there exists a solution u of (1.1) in the energy space, ona time interval [ S , + ∞ ) such that (1.4) and (1.5) hold. Step 1.
Reduction of the problem by rotation. We change coordinates in R so that byinvariance of (1.1) by rotation, we reduce with loss of generality to the following case: ℓ · e = ℓ , ℓ · e = ℓ , ℓ · e = ℓ · e := β, ℓ · e j = ℓ · e j = 0 , for j = 3 , , . (5.1)Indeed, it suffices to take as first vector of the new orthonormal basis B ′ of R , the vector e ′ = ℓ − ℓ | ℓ − ℓ | , and as second vector e ′ = a ℓ + b ℓ , where a and b are chosen so that e ′ · e ′ = 0 and | e ′ | = 1 . Then, ℓ · e ′ = ℓ · e ′ . The basis B ′ is then completed in any way.Let x = ( x , x , x ) . Note that if β = 0 , then ℓ k = ℓ k e for k = 1 , and then we are reduced to case (B) ofTheorem 1 for K = 2 . Now, we consider the general case < β < . Set ˜ ℓ k = ℓ k p − β , | ˜ ℓ k | < , k = 1 , . (5.2)Also set ( k = 1 , ) ˜ y ∞ k ∈ R such that ˜ y ∞ k, = y ∞ k, + βℓ − β y ∞ k, , ˜ y ∞ k, = y ∞ k, p − β , ˜ y ∞ k,j = y ∞ k,j , for j = 3 , , . (5.3)For k = 1 , , let ˜ W ∞ k ( t, x ) = ι k ( λ ∞ k ) / W ˜ ℓ k x − ˜ ℓ k e t − ˜ y ∞ k λ ∞ k ! , ~ ˜ W ∞ k = ( ˜ W ∞ k , ∂ t ˜ W ∞ k ) . Let ˜ u ( t ) be the solution of (1.1) satisfying (cid:13)(cid:13)(cid:13) ~ ˜ u ( t ) − h ~ ˜ W ∞ ( t ) + ~ ˜ W ∞ ( t ) i(cid:13)(cid:13)(cid:13) ˙ H × L = 0 (5.4)given by Theorem 1, case (B). Define the Lorentz transform with parameter β e of the solution ˜ u , i.e. u ( s, y ) = ˜ u s − βy p − β , y , y − βs p − β , y ! . (5.5)We claim that u ( s, y ) is a -soliton of (1.1) in the sense of Theorem 1 with parameters λ ∞ k , y ∞ k and speeds ℓ k e + β e .First, from the arguments of the proof of Lemma 6.1 in [9], since ˜ u ( t, x ) is well-defined on [ T , + ∞ ) it is well-defined everywhere on the space-time domain R × R except possibly in ahalf cone of the form t − t − < −| x − x − | , for some t − ∈ R and x − ∈ R . Thus, there exists S ∈ R such that u ( s ) defined by (5.5) makes sense on R for all s > S (see also Lemma 5.1below). Moreover, from the arguments of section 6 in [9] (see also section 2 of [14]), u is afinite energy solution of (1.1) on [ S , + ∞ ) .To prove the claim, we consider separately the regions “far from the solitons” and “close tothe solitons”. Step 2.
Estimate far from the solitons. We claim that for all δ > , there exists A δ > such that for all s ≥ S δ , k ( u ( s ) , ∂ t u ( s )) k ( ˙ H × L )( | y − ( ℓ k e + β e ) | >A δ ) . δ. (5.6)Let δ > and T δ > be such that sup t>T δ (cid:13)(cid:13)(cid:13) ~ ˜ u ( t ) − h ~ ˜ W ∞ ( t ) + ~ ˜ W ∞ ( t ) i(cid:13)(cid:13)(cid:13) ˙ H × L < δ. (5.7)Moreover, let A δ > large enough so that for k = 1 , , sup t ∈ R (cid:13)(cid:13)(cid:13) ~ ˜ W ∞ k ( t ) (cid:13)(cid:13)(cid:13) ( ˙ H × L )( | x − ˜ ℓ k e t | >A δ / < δ. (5.8) ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 35
We recall the following result from section 2 of [14], Claim 6.7 and proof of Lemma 6.1 of[9] (and references therein for the small data Cauchy theory).
Lemma 5.1 (Small scattering solutions and Lorentz transform [9]) . There exists δ > suchthat the following holds. (i) For all ( w , w ) ∈ ˙ H × L such that k ( w , w ) k ˙ H × L < δ , there exists a globalscattering solution ( w ( t ) , ∂ t w ( t )) of (1.1) with initial data ( w , w ) .Moreover, sup t ∈ R k ( w ( t ) , ∂ t w ( t )) k ˙ H × L . δ . (ii) For ( w, ∂ t w ) as in (i) and β ∈ ( − , , the function w β ( s, y ) defined by w β ( s, y ) = w s − βy p − β , y , y − βs p − β , y ! (5.9) is a global scattering solution of (1.1) . Moreover, for some constant C β > , sup t ∈ R k ( w β , ∂ t w β )( t ) k ˙ H × L ≤ C β k ( w , w ) k ˙ H × L . (5.10)We defined a cutoff function ζ ∈ C ∞ ( R ) such that ζ ( x ) = 1 for | x | > , ζ ( x ) = 0 for | x | < . For t > T δ to be chosen later, we also define ζ ext ( x ) = ζ x − ˜ ℓ e t A δ ! ζ x − ˜ ℓ e t A δ ! . Define u ext ( t ) the solution of (1.1) corresponding to the following initial data at t = t , u ext ( t , x ) = ˜ u ( t , x ) ζ ext ( x ) , ∂ t u ext ( t , x ) = ( ∂ t ˜ u ( t , x )) ζ ext ( x ) . By (5.7) and (5.8), choosing δ > small enough (compared to δ , given by Lemma 5.1), wehave k ( u ext ( t ) , ∂ t u ext ( t )) k ˙ H × L ≤ δ < δ . By Lemma 5.1, u ext ( t ) is thus a global scattering solution of (1.1) on R × R , and satisfies sup t ∈ R k ( u ext ( t ) , ∂ t u ext ( t )) k ˙ H × L . δ. Moreover, if we define u ext β ( s, y ) as the Lorentz transform with parameter β e of u ext (as in(5.9)), then u ext β is also a global scattering solution of (1.1) satisfying sup s ∈ R k ( u ext β ( s ) , ∂ t u ext β ( s )) k ˙ H × L . δ. (5.11)Now, we deduce consequences of these observations on ˜ u and u . Indeed, since u ext ( t , x ) = u ( t , x ) , and ∂ t u ext ( t , x ) = ∂ t u ( t , x ) , for a.e. ( t, x ) such that | x − ˜ ℓ k e t | > A δ for k = 1 , ,it follows from finite speed of propagation that u ext ( t, x ) = ˜ u ( t, x ) , ∂ t u ext ( t, x ) = ∂ t ˜ u ( t, x ) a.e. on C A δ ( t ) , where C A δ ( t ) = { ( t, x ) such that | x − ˜ ℓ e t | > A δ + | t − t | and | x − ˜ ℓ e t | > A δ + | t − t |} . by global scattering solution, we mean a solution defined for all time t ∈ R and behaving in the energyspace as a free solution both as t → + ∞ and t → −∞ Then, by the definitions of u and u ext β , for almost every ( s, y ) such that s − βy p − β , y , y − βs p − β , y ! ∈ C A δ ( t ) , we have u ext ( s, y ) = u ( s, y ) , ∂ s u ext ( s, y ) = ∂ t u ( s, y ) . (5.12)Now, let s ≥ S δ := T δ √ − β and choose t = p − β s . By (5.11) and (5.12), k ( u β ( s ) , ∂ t u β ( s )) k ( ˙ H × L )(Ω Aδ ( s )) = k ( u ext β ( s ) , ∂ t u ext β ( s )) k ( ˙ H × L )(Ω Aδ ( s )) ≤ k ( u ext β ( s ) , ∂ t u ext β ( s )) k ˙ H × L . δ, (5.13)where Ω A δ ( s ) = ( y such that s − βy p − β , y , y − βs p − β , y ! ∈ C A δ ( t ) ) . For C β = −| β | , let Γ A δ ( s ) = (cid:8) y such that (cid:0) | y − ℓ k s | + | y − βs | + | y | (cid:1) > C β A δ for k = 1 and . (cid:9) We claim that Ω A δ ( s ) ⊃ Γ A δ ( s ) . (5.14)Indeed, for y ∈ Γ A δ ( s ) , by the choice of t , for k = 1 , , (cid:18) | y − ˜ ℓ k t | + 11 − β | y − βs | + | y | (cid:19) / = (cid:18) | y − ℓ k s | + 11 − β | y − βs | + | y | (cid:19) / ≥ (1 − | β | ) (cid:0) | y − ℓ k s | + | y − βs | + | y | (cid:1) / + | β | p − β | y − βs | > A δ + | β | p − β | y − βs | = A δ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s − βy p − β − t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Thus, y ∈ Ω A δ ( s ) .Now, we observe that (5.14) and (5.13) prove (5.6). Step 3.
Estimate close to the solitons. First, we compute W ∞ k ( s, y ) , the Lorentz transformwith parameter β e of ˜ W k ( t, x ) . From the definition of ˜ W ∞ k , (5.2) and (5.3), W ∞ k ( s, y ) = ˜ W ∞ k s − βy p − β , y , y − βs p − β , y ! = ι k ( λ ∞ k ) / W y − ˜ ℓ k (cid:18) s − βy √ − β (cid:19) − ˜ y ∞ k, λ ∞ k q − ˜ ℓ k , y − βs √ − β − ˜ y ∞ k, λ ∞ k , y − ˜ y ∞ k λ ∞ k = ι k ( λ ∞ k ) / W (cid:16) y − ℓ k s − y ∞ k, (cid:17) + βℓ k − β ( y − βs − y ∞ k, ) λ ∞ k q − ℓ − β , y − βs − y ∞ k, p − β λ ∞ k , y − y ∞ k λ ∞ k . ULTI-SOLITONS FOR CRITICAL WAVE EQUATION 37
By the radial symmetry of W , i.e. W ( x ) = W ( | x | ) , we have W ∞ k ( s, y ) = ι k ( λ ∞ k ) / W ℓ k e + β e (cid:18) y − ( ℓ k e + β e ) s − y ∞ k λ ∞ k (cid:19) . Therefore, the Lorentz transform with parameter β e of ˜ v = ˜ u − [ ˜ W ∞ + ˜ W ∞ ] is v = u − [ W ∞ + W ∞ ] and to finish the proof of Theorem 1 in case (A), we only have to prove that,for S δ large enough, sup s>S δ k ( v, ∂ s v )( s ) k ˙ H × L . δ. (5.15)By (5.6) and the decay properties of W , we know that for S δ large, sup s>S δ k ( v, ∂ s v )( s ) k ( ˙ H × L )(Γ Aδ ( s )) . δ. (5.16)We now concentrate on an estimate for v ( s ) close to the soliton centers.First, we claim that for any δ > , for any B > , for S δ ( δ, B ) large enough, and any s > S δ , Z Z | s − s | + | y − ( ℓ k e + β e ) s | s √ − β − CB Z (cid:0) | ˜ v t | + | ˜ v x | (cid:1) ( t ) dx . δ, for S δ ( δ, B ) large enough by (5.4). Proceeding similarly for |∇ v | , we obtain (5.17).It follows from (5.17) and (5.16) that for any s > S δ , there exists s ∈ [ s , s + 1] , suchthat k ( v, ∂ s v )( s ) k H × L . δ. (5.18)Now, we use the equation of v to obtain an energy estimate for all large time. Note that v satisfies v tt − ∆ v + f ( v + W ∞ + W ∞ ) − f ( W ∞ ) − f ( W ∞ ) = 0 . (5.19) Using the equation of v , the properties of W ∞ k and standard small data Cauchy theory (byStrichartz estimates, see e.g. section 2 of [14]), taking δ > small enough, and for S δ largeenough, we obtain from (5.18), sup [ s − ,s +1] Z (cid:0) |∇ v | + | ∂ s v | (cid:1) ( s, y ) dyds . δ. Thus, (5.15) is proved.This completes the proof of Theorem 1 in case (A).
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