A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies
aa r X i v : . [ m a t h - ph ] D ec A NEW PROOF OF EXISTENCE OF SOLUTIONS FORFOCUSING AND DEFOCUSING GROSS-PITAEVSKIIHIERARCHIES
THOMAS CHEN AND NATAˇSA PAVLOVI´C
Abstract.
We consider the cubic and quintic Gross-Pitaevskii (GP) hierar-chies in d ≥ Introduction
In this note, we investigate the existence of solutions to the Gross-Pitaevskii(GP) hierarchy, with focusing and defocusing interactions. We present a new proofof existence of solutions that does not require the a priori bound on the spacetimenorm used in our earlier work, [6], which we adopted from the work of Klainermanand Machedon, [19].The GP hierarchy is a system of infinitely many coupled linear PDE’s describinga Bose gas of infinitely many particles, interacting via delta interactions. SomeGP hierarchies (defocusing energy subcritical and focusing L -subcritical) can beobtained as limits of BBGKY hierarchies of N -particle Schr¨odinger systems of iden-tical bosons, in the limit N → ∞ . In the recent literature on this topic, there isa particular interest in the special class of factorized solutions to GP hierarchies,which are parametrized by solutions of a nonlinear Schr¨odinger (NLS) equation. Inthis context, the NLS is interpreted as the mean field limit of an infinite system ofinteracting bosons in the so-called Gross-Pitaevskii limit. We refer to [10, 11, 12, 20,19, 24] and the references therein, and also to [1, 3, 5, 9, 13, 14, 15, 17, 16, 18, 26].For recent mathematical developments focusing on the related problem of Bose-Einstein condensation, we refer to [2, 21, 22, 23] and the references therein.In a landmark series of works, Erd¨os, Schlein, and Yau [10, 11, 12] provided thederivation of the cubic NLS as a dynamical mean field limit of an interacting Bosegas for a very general class of systems. The construction requires two main steps:(i) Derivation of the GP hierarchy as the N → ∞ limit of the BBGKY hier-archy of density matrices associated to an N -body Schr¨odinger equation.The latter is defined for a scaling where the particle interaction potentialtends to a delta distribution, and where total kinetic and total interactionenergy have the same order of magnitude in powers of N . (ii) Proof of the uniqueness of solutions for the GP hierarchy. It is subsequentlyverified that for factorized initial data, the solutions of the GP hierarchyare determined by a cubic NLS, for systems with 2-body interactions.The proof of the uniqueness of solutions of the GP hierarchy is the most difficultpart of this program, and it is obtained in [10, 11, 12] by use of highly sophisticatedFeynman graph expansion methods inspired by quantum field theory.In [19], Klainerman and Machedon presented a different method to prove theuniqueness of solutions for the cubic GP hierarchy in d = 3, in a different space ofsolutions than in [10, 11]. Their approach uses Strichartz-type spacetime bounds onmarginal density matrices, and a sophisticated combinatorial result, obtained via acertain “boardgame argument” (which is a reformulation of a method developed in[10, 11]). The analysis of Klainerman and Machedon requires the assumption of ana priori spacetime bound which is not proven in [19]. In [20], Kirkpatrick, Schlein,and Staffilani proved that this a priori spacetime bound is satisfied, locally in time,for the cubic GP hierarchy in d = 2, by exploiting the conservation of energy in theBBGKY hierarchy, in the limit as N → ∞ . In [5], we proved that the analogous apriori spacetime bound holds for the quintic GP hierarchy in d = 1 , k -particle marginal density matrices G = { Γ = ( γ ( k ) ( x , . . . , x k ; x ′ , . . . , x ′ k ) ) k ∈ N | Tr γ ( k ) < ∞ } and invoke a contraction mapping argument. Accordingly, we prove in [6] localwell-posedness for the cubic and quintic GP hierarchies, in various dimensions.In [6], we use the Klainerman-Machedon a priori assumption on the boundednessof a certain spacetime norm for both the uniqueness and the existence parts of theproof (which are obtained in the same step, via the contraction mapping argument).In this paper, we give a new proof of the existence of solutions for focusing anddefocusing p -GP hierarchies, without assuming any priori spacetime bounds. How-ever, we prove as an a posteriori result that the Klainerman-Machedon spacetimebound is indeed satisfied by this solution. Organization of the paper.
In Section 2 we introduce the GP hierarchy and thespaces that we use to analyze the hierarchy. In Section 3 we state the main resultof this paper, Theorem 3.1, and we present a proof of this theorem. The proof usesa free Strichartz estimate, as well as an iterated version of the Strichartz estimate,both of which are presented in Section 4.2.
The model
In this section, we introduce the mathematical model analyzed in this paper.We will mostly adopt the notations and definitions from [6], and we refer to [6] formotivations and more details.
XISTENCE OF SOLUTIONS FOR THE GP HIERARCHY 3
The spaces.
We introduce the space G := ∞ M k =1 L ( R dk × R dk )of sequences of density matricesΓ := ( γ ( k ) ) k ∈ N where γ ( k ) ≥
0, Tr γ ( k ) = 1, and where every γ ( k ) ( x k , x ′ k ) is symmetric in allcomponents of x k , and in all components of x ′ k , respectively, i.e. γ ( k ) ( x π (1) , ..., x π ( k ) ; x ′ π ′ (1) , ..., x ′ π ′ ( k ) ) = γ ( k ) ( x , ..., x k ; x ′ , ..., x ′ k ) (2.1)holds for all π, π ′ ∈ S k .For brevity, we will denote the vector ( x , · · · , x k ) by x k and similarly the vector( x ′ , · · · , x ′ k ) by x ′ k .The k -particle marginals are assumed to be hermitean, γ ( k ) ( x k ; x ′ k ) = γ ( k ) ( x ′ k ; x k ) . (2.2)We call Γ = ( γ ( k ) ) k ∈ N admissible if γ ( k ) = Tr k +1 γ ( k +1) , that is, γ ( k ) ( x k ; x ′ k ) = Z dx k +1 γ ( k +1) ( x k , x k +1 ; x ′ k , x k +1 )for all k ∈ N .Let 0 < ξ <
1. We define H αξ := n Γ ∈ G (cid:12)(cid:12)(cid:12) k Γ k H αξ < ∞ o (2.3)where k Γ k H αξ = ∞ X k =1 ξ k k γ ( k ) k H αk ( R dk × R dk ) , with k γ ( k ) k H αk := (cid:0) Tr( | S ( k,α ) γ ( k ) | ) (cid:1) (2.4)where S ( k,α ) := Q kj =1 (cid:10) ∇ x j (cid:11) α (cid:10) ∇ x ′ j (cid:11) α .2.2. The GP hierarchy.
We introduce cubic, quintic, focusing, and defocusingGP hierarchies, using notations and definitions from [6].Let p ∈ { , } . The p -GP (Gross-Pitaevskii) hierarchy is given by i∂ t γ ( k ) = k X j =1 [ − ∆ x j , γ ( k ) ] + µB k + p γ ( k + p ) (2.5)in d dimensions, for k ∈ N . Here, B k + p γ ( k + p ) = B + k + p γ ( k + p ) − B − k + p γ ( k + p ) , (2.6) T. CHEN AND N. PAVLOVI´C where B + k + p γ ( k + p ) = k X j =1 B + j ; k +1 ,...,k + p γ ( k + p ) , and B − k + p γ ( k + p ) = k X j =1 B − j ; k +1 ,...,k + p γ ( k + p ) , with (cid:16) B + j ; k +1 ,...,k + p γ ( k + p ) (cid:17) ( t, x , . . . , x k ; x ′ , . . . , x ′ k )= Z dx k +1 · · · dx k + p dx ′ k +1 · · · dx ′ k + p k + p Y ℓ = k +1 δ ( x j − x ℓ ) δ ( x j − x ′ ℓ ) γ ( k + p ) ( t, x , . . . , x k + p ; x ′ , . . . , x ′ k + p ) , and (cid:16) B − j ; k +1 ,...,k + p γ ( k + p ) (cid:17) ( t, x , . . . , x k ; x ′ , . . . , x ′ k )= Z dx k +1 · · · dx k + p dx ′ k +1 · · · dx ′ k + p k + p Y ℓ = k +1 δ ( x ′ j − x ℓ ) δ ( x ′ j − x ′ ℓ ) γ ( k + p ) ( t, x , . . . , x k + p ; x ′ , . . . , x ′ k + p ) . The operator B k + p γ ( k + p ) accounts for p + 1-body interactions between the Boseparticles. We remark that for factorized solutions γ ( k ) ( t, x , . . . , x k ; x ′ , . . . , x ′ k ) = k Y j =1 φ ( t, x j ) ¯ φ ( t, x ′ j ) , (2.7)the corresponding 1-particle wave function satisfies the p -NLS i∂ t φ = − ∆ φ + µ | φ | p φ which is focusing if µ = −
1, and defocusing if µ = +1.As in [6, 7], we refer to (2.5) as the cubic GP hierarchy if p = 2, and as the quintic GP hierarchy if p = 4. For µ = 1 or µ = − p -GP hierarchy can be rewritten in the following compact manner: i∂ t Γ + b ∆ ± Γ = µ b B ΓΓ(0) = Γ , (2.8)where b ∆ ± Γ := ( ∆ ( k ) ± γ ( k ) ) k ∈ N , with ∆ ( k ) ± = k X j =1 (cid:16) ∆ x j − ∆ x ′ j (cid:17) , and b B Γ := ( B k + p γ ( k + p ) ) k ∈ N . (2.9) XISTENCE OF SOLUTIONS FOR THE GP HIERARCHY 5
Also in this paper we will use the notation b B + Γ := ( B + k + p γ ( k + p ) ) k ∈ N , b B − Γ := ( B − k + p γ ( k + p ) ) k ∈ N . We refer to [6] for more detailed explanations.3.
Statement and proof or the main Theorem
In our earlier work [6], we proved local existence and uniqueness of solutions tothe p -GP hierarchy in the space of solutions W αξ ( I ) := n Γ (cid:12)(cid:12)(cid:12) Γ ∈ L ∞ t ∈ I H αξ , b B + Γ , b B − Γ ∈ L t ∈ I H αξ o (3.1)where I := [0 , T ]. The requirement on the spacetime norm of b B ± Γ corresponds tothe Klainerman-Machedon a priori condition used in [19]. For our proof, we useda contraction mapping argument, based on which both existence and uniquenesswere obtained in the same process. However, the question remained whether thecondition that b B Γ ∈ L t ∈ I H αξ for some ξ is necessary for both the existence anduniqueness of solutions. In the present paper, we prove that for the existence part,this a priori assumption is not required. However, as an a posteriori result, we showthat the solution obtained in this paper has the property that b B Γ ∈ L t ∈ I H αξ .We remark that for regularity α > n , the result of this paper follows in an easierway by employing the estimate k b B Γ k H αξ ≤ C k Γ k H αξ (3.2)instead of Strichartz estimates, which do not need to be invoked. The bound (3.2)for quintic GP was proved in our earlier work [5] (see Theorem 4.3), and wasemployed to give a short proof of uniqueness for the quintic GP hierarchy (seeTheorem 6.1 in [5]). A bound of the type (3.2) for the cubic GP was proved in arecent paper of Chen and Liu [8], and was used to prove local well-posedness forthe GP hierarchy in the case when α > n . Theorem 3.1.
Let α ∈ A ( d, p ) where A ( d, p ) := ( , ∞ ) if d = 1( d − p − , ∞ ) if d ≥ d, p ) = (3 , (cid:2) , ∞ ) if ( d, p ) = (3 , , (3.3) Assume that Γ ∈ H αξ ′ . Then, there exists a solution of the p -GP hierarchy Γ ∈ L ∞ t ∈ I H αξ satisfying Γ( t ) = U ( t )Γ + i µ Z t U ( t − s ) b B Γ( s ) ds , (3.4) for < ξ < ξ ′ sufficiently small (it is sufficient that ξ < η ξ ′ where the constant η is specified in Lemma 4.4 below).In particular, this solution has the property that b B Γ ∈ L t ∈ I H αξ . T. CHEN AND N. PAVLOVI´C
Remark 3.2.
We note that the presence of two different energy scales ξ, ξ ′ has thefollowing interpretation on the level of the NLS. Let R := ( ξ ′ ) − / and R := ξ − / .Then, the local well-posedness result in Theorem 3.1, applied to factorized initialdata Γ = Γ φ and the associated solution Γ( t ) = Γ φ ( t ) (of the form (2.7) ), isequivalent to the following statement: For k φ k H ( R n ) < R , there exists a uniquesolution k φ k L ∞ t ∈ I H ( R n ) < R , with R > R , in the space { φ ∈ L ∞ t ∈ I H ( R n ) | k| φ | p φ k L t H < ∞} . This version of local well-posedness, specified for balls B R (0) , B R (0) ⊂ H ( R n ) ,contains the less specific formulation of local well-posedness where only finitenessis required, k φ k H ( R n ) < ∞ and k φ k L ∞ t ∈ I H ( R n ) < ∞ .Proof. The p -GP hierarchy is given by i∂ t γ ( n ) = n X j =1 [ − ∆ x j , γ ( n ) ] + µB n + p γ ( n + p ) (3.5)for all n ∈ N .Let P ≤ N denote the projection operator P ≤ N : G → G Γ = ( γ (1) , γ (2) , . . . ) ( γ (1) , . . . , γ ( N ) , , , . . . ) , (3.6)and P >N = 1 − P ≤ N . We consider the solution Γ N ( t ) of the p -GP hierarchy, i∂ t Γ N = b ∆ ± Γ N + µ b B Γ N , (3.7)for the truncated initial dataΓ N (0) = P ≤ N Γ = ( γ (1)0 , . . . , γ ( N )0 , , , . . . ) (3.8)for an arbitrary, large, fixed N ∈ N .We observe that (3.5) determines a closed, infinite sub-hierarchy, for initial data γ ( n ) N (0) = 0, for n > N , which has the trivial solution γ ( n ) N ( t ) = 0 , t ∈ I = [0 , T ] , n > N . (3.9)Without invoking uniqueness, it is not possible to conclude that this is the uniquesolution of the sub-hierarchy for n > N with zero initial data.However, for the construction of a solution of (3.5) (without any statement onuniqueness), we are free to choose γ ( n ) N ( t ) = 0 for n > N . In particular, it thenfollows that for n ≥ N − p + 1, i∂ t γ ( n ) N ( t ; x N ; x ′ N ) = k X j =1 [ − ∆ x j , γ ( n ) N ]( t ; x N ; x ′ N ) (3.10)solves the free evolution equation, since B n + p γ ( n + p ) = 0, and thus, γ ( n ) N ( t ) = U ( n ) ( t ) γ ( n ) N (0) . (3.11)On the other hand, for n ≤ N − p , γ ( n ) N ( t ) satisfies the p -GP hierarchy in the fullform (3.5). XISTENCE OF SOLUTIONS FOR THE GP HIERARCHY 7
In conclusion, the solution of (3.5) constructed above is given byΓ N ( t ) = U ( t )Γ N (0) + i µ Z t U ( t − s ) b B Γ N ( s ) ds (3.12)for initial data Γ N (0) = P ≤ N Γ .We introduce three parameters ξ, ξ ′′ , ξ ′ satisfying ξ < η ξ ′′ < η ξ ′ , (3.13)where the constant 0 < η < b B Γ N ) N ∈ N is Cauchy in L t ∈ I H αξ ′′ . That is, forany ǫ >
0, there exists N ( ǫ ) ∈ N such that k b B (Γ N − Γ N ) k L t ∈ I H αξ ′′ ≤ C ( ξ ′ , ξ ′′ ) k P >N Γ ( n ) (0) k H αξ ′ < ǫ (3.14)holds for all N , N > N ( ǫ ). This is because by assumption, k Γ(0) k H αξ ′ < ∞ , whichis a power series in ξ ′ > k P >N Γ ( n ) (0) k H αξ ′ = X k>N ( ξ ′ ) k k γ ( k ) (0) k H α → N → ∞ , so that (3.14) follows.Accordingly, there exists a strong limitΘ = lim N →∞ b B Γ N ∈ L t ∈ I H αξ ′′ . (3.16)We claim that Θ( t ) = b BU ( t )Γ(0) + i µ Z t b BU ( t − s )Θ( s ) ds . (3.17)In order to prove this claim, we observe that (cid:13)(cid:13)(cid:13) Θ( t ) − b BU ( t )Γ(0) − i µ Z t b BU ( t − s )Θ( s ) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ≤ (cid:13)(cid:13)(cid:13) Θ( t ) − b B Γ N ( t ) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ (3.18)+ (cid:13)(cid:13)(cid:13) b BU ( t )(Γ(0) − Γ N (0)) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ (3.19)+ (cid:13)(cid:13)(cid:13) Z t ds b BU ( t − s )(Θ( s ) − b B Γ N ( s )) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ (3.20)+ (cid:13)(cid:13)(cid:13) b B Γ N ( t ) − b BU ( t )Γ N (0) − i Z t b BU ( t − s ) b B Γ N ( s ) ds (cid:13)(cid:13)(cid:13) L t ∈ I H αξ . (3.21)First, we notice that (3.21) is identically zero because Γ N is a solution of the p -GPhierarchy, (3.5). Since ξ < ξ ′′ we can estimate the term (3.18) as follows (cid:13)(cid:13)(cid:13) Θ( t ) − b B Γ N ( t ) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ≤ k Θ( t ) − b B Γ N ( t ) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ′′ = o N (1) , (3.22) T. CHEN AND N. PAVLOVI´C where the last line follows from (3.16). For (3.19), since ξ < η ξ ′′ we can use thefree Strichartz estimate (4.1) as follows: (cid:13)(cid:13)(cid:13) b BU ( t ) (Γ(0) − Γ N (0)) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ≤ k Γ(0) − Γ N (0) k H αξ ′′ ≤ k Γ(0) − Γ N (0) k H αξ ′ = k P >N Γ(0) k H αξ ′ . (3.23)For the term (3.20), we have (cid:13)(cid:13)(cid:13) Z t ds b BU ( t − s )(Θ( s ) − b B Γ N ( s )) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ≤ (cid:13)(cid:13)(cid:13) Z t ds (cid:13)(cid:13)(cid:13) b BU ( t − s )(Θ( s ) − b B Γ N ( s )) (cid:13)(cid:13)(cid:13) H αξ (cid:13)(cid:13)(cid:13) L t ∈ I ≤ Z T ds (cid:13)(cid:13)(cid:13) b BU ( t − s )(Θ( s ) − b B Γ N ( s )) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ≤ C ( T, ξ, ξ ′′ ) Z T ds (cid:13)(cid:13)(cid:13) Θ( s ) − b B Γ N ( s ) (cid:13)(cid:13)(cid:13) H αξ ′′ (3.24) ≤ C ( T, ξ, ξ ′′ ) T / (cid:13)(cid:13)(cid:13) Θ( s ) − b B Γ N ( s ) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ′′ (3.25)= o N (1) . (3.26)Here, to obtain (3.24) we use the free Strichartz estimate (4.1) in a manner similarto the T − T ∗ argument for the Schr¨odinger equation. To obtain (3.25) we usedH¨older estimate and to get the last line (3.26), we used (3.16). In conclusion, (cid:13)(cid:13)(cid:13) Θ( t ) − b BU ( t )Γ(0) − i µ Z t b BU ( t − s )Θ( s ) (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ≤ o N (1) + C ( T, ξ ) k P >N Γ(0) k H αξ ′ → N → ∞ ) . (3.27)Therefore, taking the limit N → ∞ , we find that Θ satisfiesΘ( t ) = b BU ( t )Γ(0) + i µ Z t b BU ( t − s )Θ( s ) ds (3.28)as claimed.Moreover, we observe that for t ∈ I = [0 , T ] we have: k Γ N ( t ) − Γ N ( t ) k H αξ ≤ k U ( t )(Γ N (0) − Γ N (0)) k H αξ + k Z t ds U ( t − s ) b B (Γ N ( s ) − Γ N ( s )) k H αξ ≤ k Γ N (0) − Γ N (0) k H αξ + Z T ds k b B (Γ N ( s ) − Γ N ( s )) k H αξ ≤ k Γ N (0) − Γ N (0) k H αξ ′ + Z T ds k b B (Γ N ( s ) − Γ N ( s )) k H αξ ′′ (3.29) ≤ k Γ N (0) − Γ N (0) k H αξ ′ + T k b B (Γ N ( s ) − Γ N ( s )) k L s ∈ I H αξ ′′ ≤ C ( T, ξ ′ , ξ ′′ ) k P >N Γ k H αξ ′ , XISTENCE OF SOLUTIONS FOR THE GP HIERARCHY 9 using the relation ξ < ηξ ′′ < η ξ ′ to obtain (3.29), and (3.14) to pass to the lastline. Thus, similarly as in (3.14), there exists for every ǫ > N ( ǫ ) ∈ N such that k Γ N ( t ) − Γ N ( t ) k H αξ < ǫ , (3.30)for all N , N > N ( ǫ ). This implies that (Γ N ) N ∈ N is a Cauchy sequence in L ∞ t ∈ I H αξ ,thus we obtain the strong limitΓ = lim N →∞ Γ N ∈ L ∞ t ∈ I H αξ , (3.31)given the initial data Γ ∈ H αξ ′ .Next, we claim that Γ satisfiesΓ( t ) = U ( t )Γ + i µ Z t U ( t − s )Θ( s ) ds . (3.32)Indeed, we have for t ∈ I that (cid:13)(cid:13)(cid:13) Γ( t ) − U ( t )Γ − i µ Z t U ( t − s )Θ( s ) ds (cid:13)(cid:13)(cid:13) H αξ ≤ k Γ( t ) − Γ N ( t ) k H αξ (3.33)+ k U ( t )(Γ − Γ N (0)) k H αξ (3.34)+ (cid:13)(cid:13)(cid:13) Z t U ( t − s )(Θ( s ) − b B Γ N ( s )) ds (cid:13)(cid:13)(cid:13) H αξ (3.35)+ (cid:13)(cid:13)(cid:13) Γ N ( t ) − U ( t )Γ N (0) − i µ Z t U ( t − s ) b B Γ N ( s ) ds (cid:13)(cid:13)(cid:13) H αξ . (3.36)Here, we note that (3.36) is identically zero because Γ N ( t ) is a solution of the p -GPhierarchy, (3.5). Moreover, we have(3.35) ≤ Z T ds (cid:13)(cid:13)(cid:13) Θ( s ) − b B Γ N ( s ) (cid:13)(cid:13)(cid:13) H αξ ≤ T (cid:13)(cid:13)(cid:13) Θ − b B Γ N (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ≤ T (cid:13)(cid:13)(cid:13) Θ − b B Γ N (cid:13)(cid:13)(cid:13) L t ∈ I H αξ ′′ . (3.37)Therefore, (cid:13)(cid:13)(cid:13) Γ( t ) − U ( t )Γ − i µ Z t U ( t − s )Θ( s ) ds (cid:13)(cid:13)(cid:13) L ∞ t ∈ I H αξ ≤ k Γ − Γ N k L ∞ t ∈ I H αξ + k Γ − Γ N (0) k H αξ + T (cid:13)(cid:13) Θ − b B Γ N (cid:13)(cid:13) L t ∈ I H αξ ′′ , (3.38)where the right hand side tends to zero as N → ∞ , due to the convergence (3.31)and (3.16). This implies (3.32).Finally, we observe that b B Γ( t ) = b BU ( t )Γ + i µ Z t b BU ( t − s )Θ( s ) ds (3.39) while Θ( t ) = b BU ( t )Γ + i µ Z t b BU ( t − s )Θ( s ) ds . (3.40)Comparing the right hand sides, we infer that b B Γ = Θ ∈ L t ∈ I H αξ . This concludesthe proof. (cid:3) Iterated Duhamel formula and boardgame argument
In this section, it is our main goal to prove Lemma 4.4 below. We first summarizesome results established in [5, 6, 19], which are related to Strichartz estimates forthe GP hierarchy.We first reformulate the Strichartz estimate for the free evolution U ( t ) = e it b ∆ ± =( U ( n ) ( t )) n ∈ N proven in [6, 19]. Lemma 4.1.
Let α ∈ A ( d, p ) . Assume that Γ ∈ H αξ ′ for some < ξ ′ < . Then,for any < ξ < ξ ′ , there exists a constant C ( ξ, ξ ′ ) such that the Strichartz estimatefor the free evolution k b BU ( t )Γ k L t ∈ R H αξ ≤ C ( ξ, ξ ′ ) k Γ k H αξ ′ (4.1) holds.Proof. From Theorem 1.3 in [19] and Proposition A.1 in [6], we have, for α ∈ A ( d, p ),that k B ( k + p ) U ( k + p ) ( t ) γ ( k + p ) k L t ∈ R H αk ≤ k X j =1 k B + j ; k +1 ,...,k + p U ( k + p ) ( t ) γ ( k + p ) k L t ∈ R H αk ≤ C k k γ ( k + p ) k H αk + p . (4.2)Then for any 0 < ξ < ξ ′ , we have: k b BU ( t )Γ k L t ∈ R H αξ ≤ X k ≥ ξ k k B ( k + p ) U ( k + p ) ( t ) γ ( k + p ) k L t ∈ R H αk ≤ C X k ≥ k ξ k k γ ( k + p )0 k H αk + p (4.3)= C ( ξ ′ ) − p X k ≥ k (cid:18) ξξ ′ (cid:19) k ( ξ ′ ) ( k + p ) k γ ( k + p )0 k H αk + p ≤ C ( ξ ′ ) − p sup k ≥ k (cid:18) ξξ ′ (cid:19) k X k ≥ ( ξ ′ ) ( k + p ) k γ ( k + p )0 k H αk + p ≤ C ( ξ, ξ ′ ) k Γ k H αξ ′ , where to obtain (4.3) we used (4.2). (cid:3) XISTENCE OF SOLUTIONS FOR THE GP HIERARCHY 11
Definition 4.2.
Let e Γ = ( e γ ( n ) ) n ∈ N denote a sequence of arbitrary Schwartz classfunctions e γ ( n ) ∈ S ( R × R nd × R nd ) . Then, we define the associated sequence Duh j ( e Γ) of j -th level iterated Duhamel terms , with components given by Duh j ( e Γ) ( n + p ) ( t ) (4.4):= ( − iµ ) j Z t dt · · · Z t j − dt j − e i ( t − t )∆ ( n + p ± B n + p e i ( t − t )∆ ( n + 2 p ± B n + p · · · · · · B n + jp e it j − ∆ ( n + jp ± e γ ( n + jp ) ( t j − ) . As usual, the definition is given for Schwartz class functions, and can be extendedto the spaces in discussion by density arguments. The fact that Duh j ( e Γ) ( n ) ∈S ( R × R nd × R nd ) holds under the above conditions, for all n , can be easily verified.Using the boardgame strategy of [19] (which is a reformulation of a combinatorialargument developed in [10, 11]), one obtains: Lemma 4.3.
Let α ∈ A ( d, p ) and p ∈ { , } . Then, for e Γ = ( e γ ( n ) ) n ∈ N as above, k B n + p Duh j ( e Γ) ( n + p ) ( t ) k L t ∈ I H α ( R nd × R nd ) (4.5) ≤ nC n ( c T ) j k B n + jp U ( n + p ) ( · ) e γ ( n + jp ) k L t ∈ I H α ( R ( n + jp d × R ( n + jp d ) , where the constants c , C depend only on d, p . For the proof in the cubic case, p = 2, we refer to [6, 19], and for the proof inthe quintic case, p = 4, to [5].We then observe that any solution Γ N of (3.5) with initial data Γ N (0) = P ≤ N Γ satisfies the equation (obtained from acting with b B on (3.12)) b B Γ N ( t ) = b BU ( t )Γ N (0) + i Z t b BU ( t − s ) b B Γ N ( s ) ds (4.6)and by iteration,( b B Γ N ) ( n ) ( t ) = k − X j =1 B n + p Duh j (Γ N (0)) ( n + p ) ( t )+ B n + p Duh k ( b B Γ N ) ( n + p ) ( t ) , (4.7)obtained from iterating the Duhamel formula k times for the n -th component of b B Γ. Since Γ ( m ) N ( t ) = 0 for all m > N , the remainder term on the last line is zerowhenever n + kp > N . Thus,( b B Γ N ) ( n ) ( t ) = ⌈ N − n ) /p ⌉ X j =1 B n + p Duh j (Γ N (0)) ( n + p ) ( t ) , (4.8)where each term on the right explicitly depends only on the initial data Γ N (0)(there is no implicitly dependence on the solution Γ N ( t )). Lemma 4.4.
Assume that α ∈ A ( d, p ) and p ∈ { , } , and that Γ ∈ H αξ ′ forsome < ξ ′ < . Let N , N ∈ N , where N < N . Then, there exists a constant < η = η ( d, p ) < such that the estimate k b B (Γ N − Γ N ) k L t ∈ I H αξ ≤ C ( T, ξ, ξ ′ ) k P >N Γ k H αξ ′ (4.9) holds whenever ξ < ηξ ′ (we note that it suffices to let η < C − where the constant C = C ( d, p ) is specified in Lemma 4.3).Proof. For simplicity of notation, we shall present the explicit arguments for the(cubic) case p = 2. The (quintic) case p = 4 is completely analogous.We have( b B (Γ N − Γ N )) ( n ) ( t ) = N − n X j =1 B n +1 Duh j ((Γ N (0) − Γ N (0)) ( n +1) ( t ) − N − n X j = N − n +1 B n +1 Duh j (Γ N (0)) ( n +1) ( t ) , (4.10)using the fact that γ ( n + j ) N = 0 for j > N − n , see (4.4).Since γ ( n ) N (0) = γ ( n ) N (0) for 1 ≤ n ≤ N , the first sum on the rhs of (4.10) isidentically zero.For the second term on the rhs of (4.10), we have for the summation index that j ≥ N − n + 1. Thus, the components of Γ N (0) occurring in (4.10) are given by γ ( n + j ) N with j ≥ N − n + 1, that is, γ ( m ) N with m > N .Using Lemma 4.3 and the free Strichartz estimate (4.2), we therefore find that k ( b B (Γ N − Γ N )) ( n ) ( t ) k L t ∈ I H α ≤ N − n X j = N − n +1 k B n +1 Duh j (Γ N (0)) ( n +1) ( t ) k L t ∈ I H α ≤ N − n X j = N − n +1 nC n ( c T ) j k B n + j U ( n + j ) ( t ) γ n + jN (0) k L t ∈ I H α ≤ ( ξ ′ ) − n n C n N − n X j = N − n +1 ( c T ( ξ ′ ) − ) j ( ξ ′ ) n + j k γ n + jN (0) k H α ≤ ( ξ ′ ) − n n C n C ( T, ξ ′ ) k P >N Γ N (0) k H αξ ′ , (4.11)for T > c T ( ξ ′ ) − ≤
1. Hence, X n ∈ N ξ n k ( b B (Γ N − Γ N )) ( n ) ( t ) k L t ∈ I H α ≤ C ( T, ξ ′ ) (cid:16) X n ∈ N n C n ( ξ/ξ ′ ) n (cid:17) k P >N Γ N (0) k H αξ ′ ≤ C ( T, ξ, ξ ′ ) k P >N Γ N (0) k H αξ ′ , (4.12)for ξ < ηξ ′ where η < C − , noting that C = C ( d, p ). XISTENCE OF SOLUTIONS FOR THE GP HIERARCHY 13
This proves the claim for the case p = 2. The case p = 4 is completely analogous,and we shall omit a repetition of arguments. (cid:3) Acknowledgements.
We are grateful to Igor Rodnianski for pointing out an errorin an earlier version of this work, and for useful comments. The work of T.C. issupported by NSF grant DMS 0704031 / DMS-0940145 and DMS-1009448. Thework of N.P. is supported by NSF grant number DMS 0758247 and an Alfred P.Sloan Research Fellowship.
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T. Chen, Department of Mathematics, University of Texas at Austin.
E-mail address : [email protected] N. Pavlovi´c, Department of Mathematics, University of Texas at Austin.
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