A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies

We consider the cubic and quintic Gross-Pitaevskii (GP) hierarchies in $d\geq1$ dimensions, for focusing and defocusing interactions. We present a new proof of existence of solutions that does not require the a priori bound on the spacetime norm, which was introduced in the work of Klainerman and Machedon, \cite{klma}, and used in our earlier work, \cite{chpa2}.


Introduction
In the present paper, we continue our investigation of the Cauchy problem for the Gross-Pitaevskii (GP) hierarchy, with focusing and defocusing interactions. In particular, we present a new proof of local well-posedness for the cubic and quintic GP hierarchy in d dimensions.
The GP hierarchy is a quantum field theory describing a Bose gas with infinitely many particles, interacting via delta interactions. Some defocusing GP hierarchies can be obtained as limits of BBGKY hierarchies of N -particle Schrödinger systems of identical bosons, in the limit N → ∞. Of particular interest have been special solutions to the defocusing GP hierarchies, of factorized type, which are parametrized by solutions of a nonlinear Schrödinger (NLS) equation. We refer to [11,12,13,21,20,25] and the references therein, and also to [1,3,6,10,14,15,16,18,17,19,27]. For recent mathematical developments on the related problem of Bose-Einstein condensation, we refer to [2,22,23,24] and the references therein.
To be more precise, Erdös, Schlein, and Yau [11,12,13] developed a method to derive the cubic NLS as a dynamical mean field limit of an interacting Bose gas. Roughly speaking the method consists of two steps: (i) Deriving the GP hierarchy as the limit as N → ∞ of the BBGKY hierarchy of density matrices for particle number N , for a scaling where the particle interaction potential tends to a delta distribution, and where total kinetic and total interaction energy have the same order of magnitude. (ii) Proving uniqueness of solutions for the GP hierarchy. It is subsequently verified that for factorized initial data, the solutions of the GP hierarchy are 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 difficult part of this program, and is obtained in [11,12,13] by use of highly sophisticated Feynman graph expansion methods inspired by quantum field theory. Recently, in [20], Klainerman and Machedon introduced an alternative method to prove the uniqueness of solutions for the cubic GP hierarchy in d = 3, based on spacetime bounds on marginal density matrices and a sophisticated combinatorial result, obtained via a certain "boardgame argument". The analysis of Klainerman and Machedon requires the assumption of an a priori spacetime bound which is not proven in [20]. In [21], 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 the BBGKY hierarchy, in the limit as N → ∞. In [7], we prove the a priori bound conjectured in [20]. For the proof, we introduce a natural topology on the space of sequences of k-particle marginal density matrices < ∞ } and invoke a Picard fixed point argument. We handle the relevant combinatorics by an application of the "boardgame argument" of [20]. Accordingly, we prove in [7] local well-posedness for the cubic and quintic GP hierarchies, in various dimensions. Our results in [7] do not assume any factorization of the initial data.
In this paper, we give a new proof of local well-posedness for focusing and defocusing p-GP hierarchies, and hence uniqueness of solutions, without assuming any factorization of the initial data. As a key ingredient of the proof, we use a T − T * type argument in order to establish a Strichartz-like inequality. On the other hand, in order to formulate this T − T * argument, we prove a new Strichartz estimate for the homogeneous GP hierarchy, stated in the language of Sobolev type spaces H α ξ defined on elements of G, which is stronger than the one we proved in [7]. The proof of this Strichartz estimate builds upon the Strichartz estimate for the free evolution that was originally proved in the context of the cubic GP-hierarchy in [20], and in the context of the quintic GP hierarchy in [7]. The approach developed in the work at hand is much simpler than the one in [7]; in particular, the new proof presented in this article does not use the "boardgame argument" at all. Moreover, the well-posedness results in [7] required the introduction of two different energy scales, one for the initial data, and a different one for the solution; this cumbersome condition is eliminated in the work at hand.
Organization of the paper. In Section 2, we introduce the model that is studied in this paper and the relevant notation. In Section 3, we state the main result of this paper: the local well-posedness theorem. Then, in Section 4, we state and prove a new Strichartz estimate for the homogeneous GP hierarchy. The proof of the local well-posedness theorem is presented in Section 5.

The model
In this section, we introduce the mathematical model analyzed in this paper. We will mostly adopt the notations and definitions from [7], and we refer to [7] for motivations and more details.
2.1. The spaces. We introduce the space of sequences of density matrices holds for all π, π ′ ∈ S k .
Below and throughout the paper we will denote the vector (x 1 , · · · , x k ) by x k and similarly the vector ( The k-particle marginals are assumed to be hermitean, Let 0 < ξ < 1. We define We remark that similar spaces are used in the isospectral renormalization group analysis of spectral problems in quantum field theory, [4].
The operator B k+ p 2 γ (k+ p 2 ) accounts for p 2 + 1-body interactions between the Bose particles. We remark that for factorized solutions , the corresponding 1-particle wave function satisfies the p-NLS which is focusing if µ = −1, and defocusing if µ = +1.
As in [7,9], we refer to (2.6) as the cubic GP hierarchy if p = 2, and as the quintic GP hierarchy if p = 4. For µ = 1 or µ = −1 we refer to the corresponding GP hierarchies as being defocusing or focusing, respectively.
The p-GP hierarchy can be rewritten in the following compact manner: where Also in this paper we will use the notation We refer to [7] for more detailed explanations.

Statement of the main Theorem
The main result of this paper is the proof of the local well-posedness of the Cauchy problem for focusing and defocusing p-GP hierarchies as formulated in Theorem 3.1 below.
In the work at hand, we are improving our result in [7] on several levels. To begin with, the local well-posedness obtained in [7] states that for initial data Γ 0 ∈ H α ξ1 , with ξ 1 > 0, there exists a unique solution Γ(t) for t ∈ [0, T ], in the space W([0, T ], ξ 2 ) defined in (3.3) below, for some 0 < ξ 2 < ξ 1 . In the current work, we eliminate this undesirable requirement of two different energy scales ξ 1 , ξ 2 , in all parts of our proof.
Moreover, as noted in the introduction, our method in this paper crucially involves a step analogous to the "T − T * argument" as used in the context of the NLS, [28]. As a key novelty, we do not need to make use of any high order iterations of the Duhamel formula any longer, in contrast to all previous works on this problem known to us. Expansion methods of this type necessitate very sophisticated combinatorial arguments, such as the Feynman graph expansions of [11,12], or the "boardgame argument" of [20], to control the large number of terms.
In order to be able to apply a T −T * type argument, we establish a new Strichartz estimate for the free evolution in Proposition 4.1 below, which significantly improves an analogous result in [7]. It is formulated for the spaces H α ξ , and the key improvement over [7] again is the reduction from two energy scales relevant for the problem down to a single one.
A main advantage of this new approach is its simplicity and brevity. Our improved local well-posedness result is used in [8] to obtain global well-posedness of the Cauchy problem for defocusing GP hierarchies.

Global Strichartz estimates for the free evolution
In this section, we prove a new Strichartz estimate for the free evolution The key improvement of this result over its predecessor in [7] is the fact that on both sides of (4.1), we are using the norm on the same space H α ξ . In [7], we prove a similar result, however, we used the H α ξ2 -norm on the left hand side, and the H α ξ1 -norm on the right hand side, for 0 < ξ 2 < ξ 1 .
Proposition 4.1. Let α ∈ A(d, p). Assume that Γ 0 ∈ H α ξ for some 0 < ξ < 1 (see Remark 3.2). Then, the following Strichartz estimate for the free evolution holds: where the constant C 0 is independent of ξ and (4.2) Proof. From Theorem 1.3 in [20] and Proposition A.1 in [7], we have, for α ∈ A(d, p), that We therefore conclude that (4.4) We will prove below the inequality for any monotonically increasing sequence (a k ) k , a k+1 ≥ a k > 0, where the constant C ξ is determined in (4.16) below.
Before giving a proof of (4.5), we apply it to prove (4.1). First of all, by the admissibility of Γ(t), one has which implies that the sequence γ is monotone increasing. This fact, combined with (4.4) and (4.5) implies that where the constant C 0 is independent of ξ. Hence, (4.1) follows from (4.5).
To prove (4.5), we assume that k≥1 ξ k a k < ∞, otherwise there is nothing to prove. Accordingly, we find that Therefore, there exists We thus have, using the monotonicity of (a k ) and the definition of L * , (4.12) On the other hand, it is clear that Thus, which implies By the definition of L * as a minimum, we hence obtain the upper bound We thus have that as claimed in (4.5). This concludes the proof.

Short proof of local well-posedness via T − T * argument
In this section, we present a new proof of the local well-posedness of the Cauchy problem for focusing and defocusing p-GP hierarchies. As noted previously, we entirely avoid any high order Duhamel expansions, but instead employ a T − T * type argument, similarly as for the NLS; see also [28]. Moreover, in contrast to our results in [7], only a single energy scale appears in Theorem 5.1, parametrized by 0 < ξ < 1. The improved Strichartz estimate of Proposition 4.1 plays a crucial role in our argument. (5.9) below. Then, there exists a unique solution Γ ∈ L ∞ t∈I H α ξ of the p-GP hierarchy, with in the space for the initial condition Γ(0) = Γ 0 .
Proof. Let I := [0, T ]. For Γ ∈ W(I, ξ), we have We recall that B = B + − B − , and both terms B + and B − can be treated in the same way. We shall discuss B + Γ.
From the proof of (4.1), where the constant C 0 is independent of ξ and C ξ is given by (4.2).
On the other hand, by the admissibility of Γ(t), one has which, together with the unitarity of the free evolution, implies that the sequence is monotone increasing. Hence, we obtain by use of the arguments presented in the proof of Proposition 4.1.
We next apply a T − T * type argument to find that where the constant C 1 = C 1 (d, p) is independent of ξ. We note that here, we have used (5.4) and (5.6) to obtain (5.7), and Hölder's estimate to obtain (5.8).