---
abstract: |
    We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\cH_\xi^\alpha$ of sequences of marginal density matrices. The regularity is accounted for by $$\alpha \,  \left\{
        \begin{array}{rcl}
        > &\frac12& {\rm if} \; d=1 \\ 
        >&\frac d2-\frac{1}{2(p-1)} & {\rm if} \; d\geq2 \; {\rm and} \; (d,p)\neq(3,2)\\
        \geq & 1 & {\rm if} \; (d,p)=(3,2) \,,
        \end{array}
        \right.$$ where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $\xi>0$ is arbitrary and determines the energy scale of the problem. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in $d=3$. In the defocusing case, we prove the existence and uniqueness of solutions globally in time for the cubic GP hierarchy for $1\leq d\leq3$, and of the quintic GP hierarchy for $1\leq d\leq 2$, in an appropriate space of Sobolev type, and under the assumption of an a priori energy bound. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. Also pseudoconformal invariance is established in the cases corresponding to $L^2$ criticality, both in the focusing and defocusing context. All of these results hold without the assumption of factorized initial conditions.
address:
- 'T. Chen, Department of Mathematics, University of Texas at Austin.'
- 'N. Pavlović, Department of Mathematics, University of Texas at Austin.'
author:
- Thomas Chen
- Nataša Pavlović
title: 'On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies'
---

Introduction
============

The derivation of the nonlinear Schrödinger equation as the dynamical mean field limit of the manybody quantum dynamics of interacting Bose gases is a research area that is recently experiencing remarkable progress, see [@esy1; @esy2; @ey; @kiscst; @klma; @rosc] and the references therein, and also [@adgote; @eesy; @frgrsc; @frknpi; @frknsc; @he; @sp]. A main motivation to investigate this problem is to understand the dynamical behavior of Bose-Einstein condensates. For recent developments in the mathematical analysis of Bose gases and their condensation, we refer to the fundamental work of Lieb, Seiringer, Yngvason, et al.; see [@ailisesoyn; @lise; @lisesoyn; @liseyn] and the references therein.

The procedure developed in the landmark works of Erdös, Schlein, and Yau, [@esy1; @esy2; @ey], to obtain the dynamical mean field limit of an interacting Bose gas, comprises the following main ingredients. One determines the BBGKY hierarchy of marginal density matrices for particle number $N$, and derives the Gross-Pitaevskii (GP) hierarchy in the limit $N\rightarrow\infty$, for a scaling where the particle interaction potential tends to a delta distribution; see also [@kiscst; @sc]. For factorized initial data, the solutions of the GP hierarchy are governed by a cubic NLS for systems with 2-body interactions, [@esy1; @esy2; @ey; @kiscst], and quintic NLS for systems with 3-body interactions, [@chpa]. The proof of the uniqueness of solutions of the GP hierarchy is the most difficult part of this analysis, and is obtained in [@esy1; @esy2; @ey] by use of highly sophisticated Feynman graph expansion methods inspired by quantum field theory.

Recently, an alternative method to prove the uniqueness of solutions in the $d=3$ case has been developed by Klainerman and Machedon in [@klma], using spacetime bounds on the density matrices in the GP hierarchy; this result makes the assumption of a particular a priori spacetime bound on the density matrices which has so far remained conjectural. In the work [@kiscst] of Kirkpatrick, Schlein, and Staffilani, the corresponding problem in $d=2$ is solved, and the assumption made in [@klma] is replaced by a spatial a priori bound which is proven in [@kiscst]. Alternative methods to obtain dynamical mean field limits of interacting Bose gases using operator-theoretic methods are developed by Fröhlich et al in [@frgrsc; @frknpi; @frknsc].

All of the above mentioned works discuss Bose gases with [*repulsive*]{} interactions; it is currently not known how to obtain a GP hierarchy from the $N\rightarrow\infty$ limit of a BBGKY hierarchy with attractive interactions. In the work at hand, we have nothing to add to this issue. Instead, we start here directly from the level of the GP hierarchy, and are thus free to also consider [*attractive*]{} interactions within this context. Accordingly, we will refer to the corresponding GP hierarchies as [*cubic*]{}, [*quintic*]{}, [*focusing*]{}, or [*defocusing GP hierarchies*]{}, depending on the type of the NLS governing the solutions obtained from factorized initial conditions.

In the present work, we investigate the Cauchy problem for the cubic and quintic GP hierarchy with focusing and defocusing interactions. Our results do not assume any factorization of the initial data. As a crucial ingredient of our arguments, we introduce Banach spaces $\cH_\xi^\alpha=\{ \, \Gamma\in\Gspace \,  | \, \| \, \Gamma \, \|_{\cH_\xi^\alpha} <\infty \, \}$ where \[bigG\] = { = ( \^[(k)]{}(x\_1,…,x\_k;x\_1’,…,x\_k’) )\_[k]{} | \^[(k)]{} &lt; } is the space of sequences of $k$-particle density matrices, and \_[\_\^]{} := \_[k]{} \^k \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} . The parameter $\xi>0$ is determined by the initial condition, and it sets the energy scale of a given Cauchy problem. If $\Gamma\in\cH_\xi^\alpha$, then $\xi^{-1}$ is the typical $H^\alpha$-energy per particle.

The parameter $\alpha$ determines the regularity of the solution, and our results hold for $\alpha\in\alphaset(d,p)$ where \[eq-alphaset-def-0\] (d,p) := {

[cc]{} (12,) & [if]{} d=1\
(d2-, ) & [if]{} d2 (d,p)(3,2)\
\[1,) & [if]{} (d,p)=(3,2) ,

. in dimensions $d\geq1$, and where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy. The parameter $\xi>0$ determines the energy scale of the problem.

The main results proven in this paper are:

1.  We prove local existence and uniqueness of solutions for the cubic and quintic GP hierarchy with focusing or defocusing interactions, in $\cH_\xi^\alpha$, for $\alpha\in \alphaset(d,p)$, which satisfy a spacetime bound $\|\opB\Gamma\|_{L^1_{t\in I}\cH^{\alpha}_{\xi}}<\infty$ for some $\xi>0$ (the operator $\opB$ is defined in Section \[sec-defandresults-1\] below). This spacetime bound has been conjectured by Klainerman and Machedon in [@klma]. It is of Strichartz-type, and is proven in Section \[sec-locwp-1\] using a Picard-type fixed point argument on the space $L^1_{t\in [0,T]}\cH_\xi^\alpha$; see inequality (\[eqn-BGamma-spacetime-bd-0\]) and Remark \[rem-Strichartz-1\] below.

    Accordingly, we conclude that a solution of the GP hierarchy in $\cH_\xi^\alpha$ is unique if and only if this spacetime bound holds.\

2.  We prove the global existence and uniqueness of solutions in $\cH_\xi^1$ satisfying the above noted spacetime bound, for the defocusing cubic GP hierarchy for $1\leq d\leq3$, and the defocusing quintic GP hierarchy for $1\leq d\leq 2$, provided that an a priori bound $\|\Gamma(t)\|_{\cH_\xi^1}<c$ holds for $\xi>0$ sufficiently small.\

3.  We indroduce generalized pseudoconformal transformations, and prove the invariance of the cubic GP hierarchy in $d=2$, and of the quintic GP hierarchy in $d=1$, under their application. Because the NLS obtained from factorized initial data in these cases are $L^2$-critical, we will, for brevity, refer to these GP hierarchies as being $L^2$-critical.\

4.  For the focusing cubic or quintic GP hierarchy, we prove lower bounds on the blowup rate in $\cH^\alpha_\xi$ and $\cL^r_\xi$, where both spaces are defined in Section \[sec-defandresults-1\] below.

An important ingredient of our proof of the local existence and uniqueness of solutions is the use of certain spacetime bounds for the non-interacting GP hierarchy established in [@klma] for the cubic GP hierarchy in $d=3$ (which were generalized to cubic in $d=2$ in [@kiscst], and to the quintic GP hierarchy in [@chpa]), and the “boardgame estimates" developed in [@klma] (and generalized to the quintic case in [@chpa]), which were motivated by the Feynman graph expansion techniques of [@esy1; @esy2]. For our discussion of blowup solutions of the focusing (cubic or quintic) GP hierarchy, we make extensive use of a quantity that controls the average $H^\alpha$-energy per particle, and, in a different form, the average $L^r$-norm per particle. It is introduced in Definition \[def-AvHLp-1\] below, and turns out to be the key observable for our discussion of blowup solutions.\

Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------

In Section \[sec-defandresults-1\], we introduce the cubic and quintic GP hierarchy, and state our main theorems. In Section \[sec-locwp-1\], we prove the local wellposedness of the Cauchy problem for the cubic and quintic GP hierarchy, for both focusing and defocusing interactions. In Section \[sec-gwp-GP-1\], the local wellposedness is enhanced to global wellposedness for the cubic and quintic defocusing GP hierarchies, using energy conservation. In Section \[sec-Proof-blowuprate-1\], we prove lower bounds on the blowup rate of blowup solutions in the spaces $\cH_\xi^\alpha$ and $\cL^r_\xi$ (see below for their definitions). In Section \[sec-confinv-1\], we prove the pseudoconformal invariance of the $L^2$-critical cubic (in $d=2$) and quintic (in $d=1$) GP hierarchies. In the Appendix, we reformulate the Klainerman-Machedon spacetime bounds in a form convenient for our work.

Definition of the model and statement of the main results {#sec-defandresults-1}
=========================================================

We introduce the space := \_[k=1]{}\^L\^2(\^[dk]{}\^[dk]{}) of sequences of density matrices := ( \^[(k)]{} )\_[k]{} where $\gamma^{(k)}\geq0$, $\tr\gamma^{(k)} =1$, and where every $\gamma^{(k)}(\ux_k,\ux_k')$ is symmetric in all components of $\ux_k$, and in all components of $\ux_k'$, respectively.

We call $\Gamma=(\gamma^{(k)})_{k\in\N}$ admissible if\
&& = dx\_[k+1]{} dx\_[k+p2]{} \^[(k+)]{}(\_[k]{},x\_[k+1]{},…,x\_[k+p2]{};\_k’,x\_[k+1]{},…,x\_[k+p2]{}) for all $k\in\N$.

Let $0<\xi<1$ and $r>1$. We define \_\^r := { | \_[\_\^r]{} &lt; } where \_[\_\^r]{} := \_[k=1]{}\^\^[ k]{} \^[(k)]{} \_[L\^r(\^[dk]{}\^[dk]{})]{} . Furthermore, we define \_\^ := { | \_[\_\^]{} &lt; } where \_[\_\^]{} = \_[k=1]{}\^\^[ k]{} \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} , with \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} = S\^[(k,)]{} \^[(k)]{} \_[L\^2(\^[dk]{}\^[dk]{})]{} , and $S^{(k,\alpha)}:=\prod_{j=1}^k\langle\nabla_{x_j}\rangle^\alpha\langle\nabla_{x_j'}\rangle^\alpha$.

Clearly, $\cL_\xi^r$, $\cH_\xi^\alpha$ are Banach spaces.

We note that Banach spaces of integral kernels of a similar type as those introduced above are, for instance, used for operator-theoretic renormalization group methods in the spectral analysis of quantum electrodynamics, [@bcfs].

Let $p\in\{2,4\}$. We consider the $p$-GP (Gross-Pitaevskii) hierarchy given by \[eq-def-b0-2\] i\_t \^[(k)]{} = \_[j=1]{}\^k \[-\_[x\_j]{},\^[(k)]{}\] + B\_[k+p2]{} \^[(k+p2)]{} in $d$ dimensions, for $k\in\N$. Here,\
&& := \_[j=1]{}\^k (B\_[j;k+1,…,k+p2]{}\^[(k+)]{})(t,x\_1,…,x\_k;x\_1’,…,x\_k’)\
&& := \_[j=1]{}\^kdx\_[k+1]{}dx\_[k+p2]{} dx\_[k+1]{}’dx\_[k+p2]{}’\
&&\
&& \^[(k+p2)]{}(t,x\_1,…,x\_[k+p2]{};x\_1’,…,x\_[k+p2]{}’) accounts for $\frac p2+1$-body interactions between the Bose particles.

For a factorized initial condition \[eq-initcond-fact-1\] \^[(k)]{}(0) = | \_0 \_0 |\^[k]{} with $\phi_0\in H^\alpha$, one obtains that \^[(k)]{}(t) = | (t) (t) |\^[k]{} is a solution of if $\phi_t$ satisfies the NLS \[eq-NLS-p-1\] i\_t \_t + \_x \_t - |\_t|\^p \_t = 0 with initial condition $\phi(0)=\phi_0$, where $\mu\in\{1,-1\}$. For $p=2$, this is the cubic NLS, and for $p=4$, this is the quintic NLS. The NLS is defocusing for $\mu=1$, and focusing for $\mu=-1$.

Accordingly, we refer to (\[eq-def-b0-2\]) as the [*cubic GP hierarchy*]{} if $p=2$, and as the [*quintic GP hierarchy*]{} if $p=4$. Moreover, for $\mu=1$ or $\mu=-1$ we refer to the GP hierarchies as being defocusing or focusing, respectively.

We recall the definition of the set $\alphaset(d,p)$, for $p=2,4$ and $d\geq1$, \[eq-alphaset-def-1\] (d,p) = {

[cc]{} (12,) & [if]{} d=1\
(d2-, ) & [if]{} d2 (d,p)(3,2)\
\[1,) & [if]{} (d,p)=(3,2)

. Our main result in this paper is the following theorem.

\[thm-main-0\] Let $0<\xi_2=\eta\xi_1\leq\xi_1<1$. Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$, and $0<\eta<1$ sufficiently small. Then, the following hold.

-   For every $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that there exists a unique solution $\Gamma(t)\in\cH_{\xi_2}^\alpha$ for $t\in[0,T]$ with $\|\opB\Gamma\|_{L^1_{t\in[0,T]}\cH_{\xi_2}^\alpha}<\infty$.\

-   Assume that given $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there are constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a solution $\Gamma(t)$ of the $p$-GP hierarchy (\[seclwp-pGP\]) in the space $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$.\
    \
    Then, the solution $\Gamma(t)\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ is [*unique*]{} if and only if $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ holds for some $\xi>0$.\
    \
    If the latter is satisfied, then in fact, the Strichartz-type bound \[eqn-BGamma-spacetime-bd-0\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} &&C(d,p,\_1,\_2) \_0\_[\_[\_1]{}\^]{} holds.\

\[rem-BBGKYlim-1\] An immediate implication of part (ii) of Theorem \[thm-main-0\] is that every solution $\Gamma(t)$ extracted by a diagonal argument from the $N\rightarrow\infty$ limit of the $N$-particle BBGKY hierarchies (with repulsive interactions) studied by Erdös-Schlein-Yau in [@esy1; @esy2], Kirkpatrick-Schlein-Staffilani in [@kiscst], and Chen-Pavlović in [@chpa], satisfies $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ for some $\xi>0$. This is true because uniqueness of those solutions has been established in these works with independent methods.

The role of the parameters $\xi_1,\xi_2$ is as follows: Given initial data $\Gamma_0=(\gamma^{(k)})_{k\in\N}$ with $\|\gamma^{(k)}\|_{H^\alpha(\R^{dk}\times\R^{dk})}<\infty$ for all $k$, we determine $\xi_1>0$ sufficiently small such that $\Gamma_0\in\cH_{\xi_1}^\alpha$. This means that the energy per particle in $\Gamma_0$ is bounded by $\xi_1^{-1}$. In cases of physical interest, $\xi_1>0$; the notion of an energy per particle will be quantified below. Then, we find a suitable $\xi_2=\eta\xi_1\ll\xi_1$ such that the Cauchy problem for the the GP hierarchy can be solved in a sufficiently large space $\cH^\alpha_{\xi_2}$. The requirement $\xi_2 \ll \xi_1$ is used to ensure that a solution $\Gamma(t)$ does not drift out of $\cH^\alpha_{\xi_2}$ for $t\in I=[0,T]$ with $T=T(\xi_2)>0$; we thereby impose the assumption that the energy per particle does not exceed $\xi_2^{-1}$ while $t\in I$, but once this assumption is violated, we may choose $0<\xi_2'<\xi_2$ to continue the solution to $T(\xi_2')>T(\xi_2)$.

In particular, there is no implication of the size of $\xi_2$ on the regularity accounted for by $\alpha$. For factorized initial data, the statement that the solution of the NLS remains in $H^\alpha$ for $t\in I$ is equivalent to the statement that the solution of the GP hierarchy remains in $\cH^\alpha_{\xi}$ for [*an arbitrary*]{} nonzero $\xi>0$.

\[rem-Strichartz-1\] We note that the estimate (\[eqn-BGamma-spacetime-bd-0\]), for the cubic GP hierarchy with $d=3$ and $\alpha=1$, proves the a priori spacetime bound conjectured in [@klma]. For factorized initial data $\Gamma=(|\phi_0\rangle\langle\phi_0|^{\otimes k})_{k\in\N}$ in the cubic case, so that $\Gamma=(|\phi(t)\rangle\langle\phi(t)|^{\otimes k})_{k\in\N}$ where $i\partial_t\phi+\Delta\phi-\mu|\phi|^2\phi=0$, it corresponds to the inequality ||\^2 \_[L\^1\_[tI]{}H\^]{}\^[13]{} C(T) \_0 \_[H\^]{} which is of Strichartz type. The example of factorized solutions with $\phi(t)\in H^1$, $t\in I$, is discussed in detail in [@klma].

We say that a solution $\Gamma(t)$ of the GP hierarchy blows up in finite time with respect to $H^\alpha$ if there exists $T^*<\infty$ such that for every $\xi>0$ there exists $T_{\xi,\Gamma}^*<T^*$ such that $\|\Gamma(t)\|_{\cH_{\xi}^\alpha}\rightarrow\infty$ as $t\nearrow T^{*}_{\xi,\Gamma}$, and $T_{\xi,\Gamma}^*\nearrow T^*$ as $\xi\rightarrow0$.

For the study of blowup solutions, it is convenient to introduce the following quantity.

\[def-AvHLp-1\] We refer to \_[H\^]{}() := \^[-1]{} , \_[L\^r]{}() := \^[-1]{} , respectively, as the typical (or average) $H^\alpha$-energy and the typical $L^r$-norm per particle.

We note that = ( | |\^[k]{} )\_[k]{} \_[H\^]{}() = \_[H\^]{}\^2 \_[L\^r]{}() = \_[L\^r]{}\^2 in the factorized case.

The fact that $\Gamma\in\cH_\xi^\alpha$ means that the typical energy per particle is bounded by $\Av_{H^\alpha}(\Gamma )<\xi^{-1}$. Therefore, the parameter $\xi$ determines the $H^\alpha$-energy scale in the problem. While solutions with a bounded $H^\alpha$-energy remain in the same $\cH_\xi^\alpha$, blowup solutions undergo transitions $\cH_{\xi_1}^\alpha\rightarrow\cH_{\xi_2}^\alpha\rightarrow\cH_{\xi_3}^\alpha\rightarrow\cdots$ where the sequence $\xi_1>\xi_2>\cdots$ converges to zero as $t\rightarrow T^*$.

It is easy to see that blowup in finite time of $\Gamma(t)$ with respect to $H^\alpha$ is equivalent to the statement that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^*$.

Clearly, $(\Av_{N}(\Gamma ))^{-1}$ is the convergence radius of $\|\Gamma\|_{\cN_\xi}$ as a power series in $\xi$, for the norms $N=H^\alpha,L^r$ and $\cN_\xi=\cH^\alpha_\xi ,\cL^r_\xi$, respectively.

\[thm-blowuprate-L2crit-1\] Assume that $\Gamma(t)$ is a solution of the (cubic $p=2$ or $p=4$ quintic) $p$-GP hierarchy with initial condition $\Gamma(t_0)=\Gamma_{0}\in\cH_{\xi}^\alpha$, for some $\xi>0$, which blows up in finite time. Then, the following lower bounds on the blowup rate hold:

1.  Assume that $\frac4d\leq p < \frac{4}{d-2\alpha}$. Then, ( \_[H\^]{}((t)) )\^[12]{} &gt; . Thus specifically, for the cubic GP hierarchy in $d=2$, and for the quintic GP hierarchy in $d=1$, ( \_[H\^1]{}((t )) )\^[12]{} , with respect to the Sobolev spaces $H^\alpha$, $\cH_\xi^\alpha$.

2.  ( \_[L\^r]{}((t )) )\^[12]{} , &lt; r.

We note that in the factorized case, the above lower bounds on the blow-up rate coincide with the known lower bounds on the blow-up rate for solutions to the NLS (see, for example, [@ca]).

The cubic GP hierarchy in $d=2$, and the quintic GP hierarchy in $d=1$ are distinguished by being invariant under a class of generalized pseudoconformal transformations, as presented below. Let us first recall pseudoconformal invariance on the level of the NLS (\[eq-NLS-p-1\]). If the NLS (\[eq-NLS-p-1\]) is $L^2$-critical, that is, $p=\frac 4d$, it is invariant under the pseudoconformal transformations \[eq-NLSpc-def-1\] [P]{}\_t (x) := e\^[-i]{} \_( ) , for $b\in\R\setminus\{0\}$. That is, i\_t[P]{}\_t + \_t - |[P]{}\_t|\^p [P]{}\_t = 0 ; see for instance [@ca]. There are two cases of $L^2$-critical NLS with $p\in\N$: The cubic ($p=2$) NLS in $d=2$, and the quintic ($p=4$) NLS in $d=1$.

For the GP hierarchy, one can likewise introduce pseudoconformal transformations, and as we prove in this paper, the GP hierarchy is pseudoconformally invariant when $p=2$ and $d=2$ (cubic), or $p=4$ and $d=1$ (quintic). This property is independent of whether the GP hierarchy is defocusing, $\mu=1$, or focusing, $\mu=-1$.

\[thm-pseudoconf-gamma-1\] For $d=2$ and $p=2$ (cubic), or $d=1$ and $p=4$ (quintic), the focusing or defocusing ($\mu\in\{1,-1\}$) GP hierarchy is invariant under the pseudoconformal transformations \[eq-pseudoconf-gamma-1\]\
&& := e\^[-i]{} \^[(k)]{}( , ; ) , for $b\in\R\setminus\{0\}$.

That is, \[eq-infhierarch-pseudoconf-1\] i\_t \^[(k)]{} + \_\^[(k)]{}\^[(k)]{} - B\_[k+p2]{} \^[(k+2)]{} = 0 , for all $k\geq1$.

The proof is given in Section \[sec-confinv-1\]. For a survey of related matters for the NLS, see for instance [@ca; @ra; @ta].

Of course, the following is immediately clear.

Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Moreover, assume that $\Gamma(t)\in\cH_{\xi_2}^\alpha$ solves the (cubic or quintic) focusing ($\mu=-1$) GP hierarchy with factorized initial condition $\Gamma_0=(|\phi_0\rangle\langle\phi_0|^{\otimes k})_{k\in\N}\in\cH_{\xi }^\alpha$ for some $\xi>0$, where $\phi_0\in H^\alpha $.\
\
Then, if there exists $T^*<\infty$ such that $\|\phi(t)\|_{H^\alpha}\rightarrow\infty$ as $t\nearrow T^*$, it follows that also $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^*$.

This follows from $\Av_{H^\alpha}(\Gamma(t))=\|\phi(t)\|_{H^\alpha}^2$ for product states.

For various scenarios in which blowup occurs for solutions of the cubic or quintic NLS, we refer to the literature; see for instance [@ca; @ra] for surveys.

Local existence and uniqueness of solutions for the focusing and defocusing GP hierarchy {#sec-locwp-1}
========================================================================================

In this section, we prove a local existence and uniqueness result for the cubic and quintic GP hierarchy for both focusing and defocusing interactions. We formulate all arguments for the cubic hierarchy ($p=2$). For the quintic hierarchy ($p=4$), the generalizations are straightforward, and will only be briefly described.

We introduce the notation \_\^[(k)]{} = \_[\_k]{} - \_[’\_k]{} with \_[\_k]{} = \_[j=1]{}\^k\_[x\_j]{} and \_[, x\_j]{} = \_[x\_j]{} - \_[x\_j’]{} . Moreover, we write \_ := ( \^[(k)]{}\_\^[(k)]{} )\_[k]{} and := ( B\_[k+1]{} \^[(k+1)]{} )\_[k]{} . Then, the $p$-GP hierarchy can be written as \[seclwp-pGP\] i\_t + \_ = . In integral formulation, it is formally given by \[int-pGP\] (t) = e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) . We will prove the local existence and uniqueness of such a solution in the following manner. We note that (\[int-pGP\]) can be formally written as a system of integral equations (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) \[int-pGP-sys-1\]\
(t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) , \[int-pGP-sys-2\] where we note that the second line is formally a self-consistent fixed point equation for $\opB\Gamma$.

Let $I:=[0,T]$. We introduce the product space \[def-Wspace-1\] := L\^\_[tI]{}\_\^ L\^1\_[tI]{}\_\^ , which we endow with the norm (, ) \_ := \_[L\^\_[tI]{}\_\^]{} + \_[L\^1\_[tI]{}\_\^]{} . Clearly, $( \Wspacexi , \| \, \cdot \, \|_{\Wspacexi} )$ is a Banach space. Then, we introduce the system (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) \[int-pGP-sys-Wspace-1\]\
(t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) , \[int-pGP-sys-Wspace-2\] which is formally equivalent to the system (\[int-pGP-sys-1\]), (\[int-pGP-sys-2\]).

Our main result in this section is the following existence and uniqueness theorem, which corresponds to part (i) in Theorem \[thm-main-0\].

\[thm-locwp-main-1\] Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Then, the following holds. For every $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a unique solution $(\Gamma(t),\Xi(t))$ of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]), in the space $\Wspace$. Moreover, the relation (t)=(t)L\^1\_[t I]{}\_[\_2]{}\^, holds for the solution, and the component $\Gamma(t)$ satisfies the $p$-GP hierarchy (\[seclwp-pGP\]), \[eq-GPhier-thm-1\] i\_t + \_ - = 0, with initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^\alpha$.

Moreover, there exists a constant $C(T,d,p,\xi_1,\xi_2)$ such that \_[L\^1\_[tI]{}\_[\_2]{}\^]{} C(T,d,p,\_1,\_2) \_0 \_[\_[\_1]{}\^]{} holds.

We note that our local existence and uniqueness result differs than the one proven in [@esy1; @esy2] in that performing the contraction principle on the space $\Wspace$ presumes finiteness of $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$. However, [^1] it is a priori conceivable that there exist solutions of the $p$-GP hierarchy $\Gamma(t)\in \cH_{\xi_2}^\alpha$ for which $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$ is not finite. We note that finiteness of $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$ is an essential element of this analysis, and has been assumed without proof in [@klma] as a key ingredient.

It was previously unknown whether this condition is generally satisfied for solutions of the GP hierarchy. Using Theorem \[thm-locwp-main-1\], we can prove that $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}<\infty$ is a sufficient and necessary condition for a solution $\Gamma(t)\in\cH_{\xi_2}^\alpha$ of (\[seclwp-pGP\]), with initial condition $\Gamma(0)\in\cH_{\xi_1}^\alpha$, to be unique. Accordingly, we arrive at the following result which corresponds to part (ii) in Theorem \[thm-main-0\].

\[thm-locwp-main-1-L1bd\] Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Assume moreover that for $\Gamma_0\in\cH_{{\xi_1}}^\alpha$ there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a solution $\Gamma(t)$ of the $p$-GP hierarchy (\[seclwp-pGP\]) in the space $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$.

Then, the solution $\Gamma(t)\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ is [*unique*]{} if and only if $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ holds for some $\xi>0$.

If the latter holds, then in fact, there exists a constant $C(T,d,p,\xi_1,\xi_2)$ such that \[eq-Strichartz-Wspace-1-1\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} C(T,d,p,\_1,\_2) \_0 \_[\_[\_1]{}\^]{} is satisfied.

We first prove Theorem \[thm-locwp-main-1\]. We make the key observation that the fixed point equation (\[int-pGP-sys-Wspace-2\]) determining the component $\Xi(t)$ of the desired solution $(\Gamma(t),\Xi(t))$ is self-contained, and independent of the component $\Gamma(t)$. Therefore, we can invoke the Picard fixed point principle on the space $L^1_{t\in I}\cH_{\xi_2}^\alpha$ (the second factor of the product space $\Wspace$) to first find a unique $\Xi\in L^1_{t\in I}\cH_{\xi_2}^\alpha$ solving (\[int-pGP-sys-Wspace-2\]).

We recall that the fixed point equation for $\Xi(t)$ is given by \[eq-opBGamma-fixedpt-1\] (t) = e\^[it\_]{} \_0 - i \_0\^t ds e\^[i(t-s)\_]{}(s), in the space $L^1\cH_{\xi_2}^\alpha$.

We define, for an admissible sequence of density matrices $\widetilde\Gamma=(\widetilde\gamma^{(k)})_{k\in\N}$, \[eq-Duh-j-def-1\]\
& := & (-i)\^j\_0\^t dt\_1 \_0\^[t\_[j-1]{}]{}dt\_j e\^[i(t-t\_1)\_\^[(k+1)]{}]{}B\_[k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& B\_[k+3]{} B\_[k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}(t\_j) . Then, any solution of (\[eq-opBGamma-fixedpt-1\]) satisfies the fixed point equation (obtained from iterating the Duhamel formula $k$ times for the $k$-th component of $\opB\Gamma$) ()\^[(k)]{}(t) = \_[j=1]{}\^[k-1]{}B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) + B\_[k+1]{}\_[k]{}()\^[(k+1)]{}(t) . To formulate a Picard-type fixed point argument, we define ()=( ()\^[(k)]{})\_[k]{} where the $k$-th component is given by \[eq-PhiGamma-fixed-pt\] ()\^[(k)]{}(t) = \_[j=1]{}\^[k-1]{}B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) + B\_[k+1]{}\_[k]{}()\^[(k+1)]{}(t) . Similarly as in [@chpa], we use different approaches when $d\geq2$ and when $d=1$. In dimension $d=1$, and for both the cubic and quintic GP hierarchy, we use a spatial a priori bound as in [@chpa] where we refer for details.

In dimensions $d\geq2$, we apply the Klainerman-Machedon spacetime bounds similarly to [@klma] and [@kiscst]. This is explained in detail in the Appendix.

We first consider the case $d\geq2$.

In the sequel, we will often abbreviate H\^\_k := H\^(\^[dk]{}\^[dk]{}) . We invoke Propositions \[prp-spacetime-bd-1\] and \[prp-spacetime-bd-2\] in the Appendix. They generalize the spacetime bounds and “board game" arguments developed in [@klma]. Proposition \[prp-spacetime-bd-2\] implies that for $\Gamma_0=(\gamma_0^{(k)})_{k\in\N}$,\
& &lt; & k C\^k \_[j=1]{}\^[k-1]{} (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\
& &lt; & k (C \_1\^[-1]{})\^[k]{} \_[j=1]{}\^[k-1]{} (c T \_1\^[-2]{})\^[2]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\
& &lt; & (c T \_1\^[-2]{}) k (C \_1\^[-1]{})\^[k]{} \_[j=1]{}\^[k-1]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{} . \[eq-Duhexp-aux-1\] Therefore,\
& &lt; & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_[j=1]{}\^[k-1]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\
& &lt; & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_[=1]{}\^[2k]{} \_1\^ \^[()]{}\_0\_[H\^\_]{}\
& &lt; & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_0\_[\_[\_1]{}\^]{}\
& &lt; & (c T \_1\^[-2]{}) \_0\_[\_[\_1]{}\^]{} \[eq-Duhexp-aux-2\] for $\xi_2=\eta\xi_1$, with $0<\eta\leq1$ sufficiently small.

This implies that, for $I=[0,T]$, and any $T>0$, ( \_[j=1]{}\^[k-1]{}B\_[k]{}\_j(\_0)\^[(k+1)]{}(t) )\_[k]{} L\^1\_[tI]{}\^\_[\_2]{} , provided that $\Gamma_0\in\cH_{\xi_1}^\alpha$, and $\xi_2=\eta\xi_1$ with $\eta>0$ sufficiently small.

Our next step is to prove that $\Phi$ is a contraction on $L^1_{t\in I}\cH^\alpha_{\xi_2}$. To this end, we use the bound (\_1)\^[(k)]{} -(\_2)\^[(k)]{} \_[L\_[tI]{}\^1H\^\_k]{} k (C T)\^ \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{} which is proven in Proposition \[prp-spacetime-bd-2\] in the Appendix.

We obtain \[eq-PhiGamma-contr-1\]\
& = & \_[k=1]{}\^\_2\^k (\_1)\^[(k)]{} -(\_2)\^[(k)]{} \_[L\_[tI]{}\^1H\^\_k]{}\
&& \_[k=1]{}\^k (C T \_2\^[-2]{})\^ \_2\^[2k]{} \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{}\
&& \_[k1]{}{ k (C T \_2\^[-2]{})\^ } \_[k=1]{}\^ \_2\^[2k]{} \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{}\
&& ( C T \_2\^[-2]{})\^[12]{} \_1 - \_2 \_[L\^1\_[tI]{}\_[\_2]{}\^]{}, for $T>0$ sufficiently small. This implies that $\Phi$ is a contraction on $L^1_{t\in I}\cH^\alpha_{\xi_2}$ provided that $T$ is sufficiently small.

Consequently, for a every initial condition $\Gamma_0\in\cH_{\xi_1}^\alpha$, there exist $T>0$ and $\xi_2=\eta\xi_1$ with $0<\eta\leq1$ such that there exists a unique solution $\Xi(t)$ of (\[eq-opBGamma-fixedpt-1\]) in $L^1_{t \in I}\cH^\alpha_{\xi_2}$,

Repeating the above arguments, we find that this solution satisfies \[eq-Strichartz-cH-1\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} &&(c T \_1\^[-2]{}) \_0\_[\_[\_1]{}\^]{} + ( C T \_2\^[-2]{} )\^[12]{} \_[L\^1\_[tI]{}\_[\_2]{}\^]{} . Hence, in particular, \[eqn-BGamma-spacetime-bd-1\] \_[L\^1\_[tI]{}\_[\_2]{}]{} && \_0\_[\_[\_1]{}\^]{} for sufficiently small $T>0$.

Given the solution $\Xi$ in $L^1_{t \in I}\cH^\alpha_{\xi_2}$ obtained from the contraction argument (\[eq-PhiGamma-contr-1\]), the right hand side of the integral equation (\[int-pGP-sys-Wspace-1\]) is completely determined by $\Gamma_0$ and $\Xi(s)$ where $s\in I$.

Accordingly, $\Gamma(t)$ is given by \[int-pGP-sys-Wspace-1-2\] (t) = e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) where the solution $\Xi(s)$ found in step [*(1)*]{} above has been substituted into the integral. Evidently, \[eq-Gamma-Duh-opBGamma-1\] (t) \_[ \_[\_2]{}\^]{} & & \_0 \_[\_[\_2]{}\^]{} + \_0\^t ds e\^[i(t-s)\_]{}(s)\_[\_[\_2]{}\^]{}\
& & \_0 \_[\_[\_2]{}\^]{} + \_0\^t ds (s)\_[\_[\_2]{}\^]{}\
& & \_0 \_[\_[\_2]{}\^]{} + (s) \_[L\^1\_[sI]{}\_[\_2]{}\^]{} , using the unitarity of $e^{i(t-s)\opDelta_\pm}$ with respect to $\cH_{\xi_2}^\alpha$. By (\[eqn-BGamma-spacetime-bd-1\]), we conclude that the last line is bounded, hence $\Gamma(t)$ determined by (\[int-pGP-sys-Wspace-1-2\]) indeed lies in $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$.

To prove uniqueness, we assume that $(\Gamma_1(t),\Xi_1(t))$ and $(\Gamma_2(t),\Xi_2(t))\in\Wspace$ correspond to solutions of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) to the same initial condition $\Gamma_1(0)=\Gamma_2(0)\in\cH_{\xi_1}^\alpha$, for $t\in[0,T]$. Then, linearity implies that $(\Gamma_1(t)-\Gamma_2(t),\Xi_1(t))-\Xi_2(t))\in\Wspace$ solves the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) with initial condition $\Gamma_1(0)-\Gamma_2(0)=(0)_{k\in\N}$. Hence, (\[eq-Gamma-Duh-opBGamma-1\]) implies that \_1(t)-\_2(t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_1(0)-\_2(0)\_[\_[\_1]{}\^]{}\
&=&0. This implies that $\Gamma_1(t)=\Gamma_2(t)$, for all $t<T$. This proves local existence and uniqueness of solutions for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]).

Next, we observe that applying $\opB$ to $\Gamma(t)$ produces the rhs of the fixed point equation (\[eq-opBGamma-fixedpt-1\]) for $\Xi(t)$. Therefore, we immediately find that \[eq-XiopBGammadiff-1\] = L\^1\_[tI]{}\_[\_2]{}\^. Finally, it is easy to verify for the unique solution $(\Gamma(t),\Xi(t))\in\Wspace$ of (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) that i\_t (t) + \_(t) = (t) . Therefore, using (\[eq-XiopBGammadiff-1\]), i\_t (t) + \_(t) - (t) \_[L\^1\_[tI]{}\_[\_2]{}\^]{} = 0 , which implies (\[eq-GPhier-thm-1\]).

For the quintic GP hierarchy, all steps of the above proof can be adopted with minor modifications. A key difference is the fact that (\[eq-Gamma-Duh-opBGamma-1\]) is replaced by \[eq-Gamma-Duh-opBGamma-quintic-1\] (t) \_[ \_[\_2]{}\^]{} & & \_0 \_[\_[\_2]{}\^]{} + \_[L\^1\_[tI]{}\_[\_2]{}\^]{}\
& & ( 1 + ) \_0\_[\_[\_1]{}\^]{} . This concluded the proof of Theorem \[thm-locwp-main-1\] for $d\geq2$. In the case $d=1$, we can straightforwardly adapt the proof given in [@chpa] of the uniqueness of solutions of the quintic GP hierarchy in $d=1$. The spacetime bounds of Proposition \[prp-spacetime-bd-1\] is not available in $d=1$ since it would produce divergent bounds. However, the spatial bounds in $d=1$ proven in [@chpa] apply for both the cubic and the quintic GP hierarchy, under the assumption that $\alpha>\frac12$.

The result is that we get a factor $t$ instead of $t^{\frac12}$, in all of the bounds found above for the cubic GP hierarchy that produced a factor $t^{\frac12}$. Accordingly, we find \[eq-Gamma-Duh-opBGamma-1D-1\] (t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} for the cubic GP hierarchy instead of (\[eq-Gamma-Duh-opBGamma-1\]), and \[eq-Gamma-Duh-opBGamma-quintic-1D-1\] (t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} for the quintic GP hierarchy instead of (\[eq-Gamma-Duh-opBGamma-quintic-1\]), respectively. Hence, we obtain local wellposedness for sufficiently small $T>0$.

In conclusion, we have proved the existence and uniqueness of solutions of the GP hierarchy (\[seclwp-pGP\]) in the space $\Wspace$ defined in (\[def-Wspace-1\]).

It is now straightforward to prove Theorem \[thm-locwp-main-1-L1bd\].

$"\Rightarrow"$. We first prove that uniqueness of $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ implies $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ for some $\xi>0$. Let $t'\in I_1=[0,T_1]$. Assume $\Gamma(t+t')$ is the unique solution in $\cH_{\xi_2}^1$ of the $p$-GP hierarchy for the initial condition $\Gamma(t)\in \cH_{\xi_1}^\alpha$, for $t'\in I_1$, and $0<\xi_1<1$ chosen appropriately.

We may also use $\Gamma(t)$ as an initial condition for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) in $\mathfrak{W}_{\xi_2}^\alpha(I_2)$. Thereby, we obtain a unique solution $( \Gamma'(t+t') , \Xi(t+t') $ in $\mathfrak{W}_{\xi_2}^\alpha(I_2)$, for $t' \in I_2 = [0,T_2]$, for $T_2>0$ and $0<\xi_2\leq\xi_1$ chosen small enough.

Let $I:=[0,\min\{T_1,T_2\}]$. From Theorem \[thm-locwp-main-1\], we infer that $\Gamma'(t+t')$ is also an element of $\cH_{\xi_2}^\alpha$, for $t\in I$, and solves the GP hierarchy equation with initial condition $\Gamma'(t)=\Gamma(t)$ for $t'=0$.

Since by assumption, $\Gamma(t+t')$ is the unique solution of the GP hierarchy in $\cH_{\xi_2}^\alpha$, solving the same GP hierarchy equation, and with the same initial condition $\Gamma(t)$ at $t'=0$. Thus, $\Gamma(t+t')=\Gamma'(t+t')$.

But since the solution $( \Gamma'(t+t') , \Xi(t+t') )$ in $\Wspace$ for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) has the property that $\| \opB\Gamma'(t + . ) \|_{L^1_{t'\in I} \cH^\alpha_{\xi_2}}
\leq C(T,d,p,\xi_1,\xi_2) \, \| \Gamma(t) \|_{\cH_{\xi_1}^\alpha}$ is bounded, we conclude that also, $\| \opB\Gamma(t + . ) \|_{L^1_{t'\in I} \cH^\alpha_{\xi_2}}
\leq C(T,d,p,\xi_1,\xi_2) \, \| \Gamma(t) \|_{\cH_{\xi_1}^\alpha}$.

$"\Leftarrow"$. Next, we prove that $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ for some $\xi>0$ implies uniqueness of $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$, for some $\xi_2$. We may set $\xi=\xi_2$ by choosing $\min\{\xi,\xi_2\}$.

We note that since $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ and $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$, it is clear that $(\Gamma,\opB\Gamma)\in\Wspace$. Accordingly, Theorem \[thm-locwp-main-1\] implies that $\Gamma$ is the unique solution in $\Wspace$ for the initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^\alpha$ for which $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ is satisfied.

Finally, Theorem \[thm-locwp-main-1\] implies that whenever $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ holds for some $\xi>0$, the stronger estimate (\[eq-Strichartz-Wspace-1-1\]) is satisfied.

$\;$\
\

On the global existence and uniqueness of solutions for defocusing GP hierarchies {#sec-gwp-GP-1}
=================================================================================

It is proved for the $d=2,3$ defocusing cubic case in [@esy1; @esy2; @kiscst], and for the $d=1,2$ defocusing quintic case in [@chpa], that whenever $\Gamma(t)=(\gamma^{(k)}(t))_{k\in\N}$ is obtained from the $N\rightarrow\infty$ limit of a bosonic $N$-particle BBGKY hierarchy with repulsive interactions, \[eq-GPsol-defoc-apriorien-1\] ( S\^[(k,1)]{} \^[(k)]{}\_0 ) &lt; C\^k for some constant $C$, then \[eq-GPsol-defoc-apriorien-2\] ( S\^[(k,1)]{} \^[(k)]{}(t) ) &lt; C\_0\^k with $C_0$ independent of $t\geq0$.

This follows from energy conservation in the $N$-particle system of which the GP hierarchy is the $N\rightarrow\infty$ limit. We first claim that this implies that $\Gamma(t)\in\cH_{\xi_1}^1$ for some $\xi_1>0$, and all $t\in\R$.

To see this, we consider a fixed $k$. Let $\gamma^{(k)}$ be non-negative, normalized trace class, $\tr(\gamma^{(k)})=1$, and hermitean. Then, we have that \^[(k)]{}(\_k;\_k’) = \_j \_j |\_j(\_k’)\_j(\_k)| for an orthogonal basis $\psi_j$ of $L^2(\R^{dk})$ with $\lambda_j\geq0$ and $\sum\lambda_j=1$. Then, \^[(k)]{} \_[H\^1]{} & = & \_[j,j’]{} \_j \_[j’]{} | \_[\_k]{}\_j | \_[\_k]{}\_[j’]{} |\^2\
&& (\_j \_j \_[\_k]{}\_j \^2 )\^2\
&=& ( ( S\^[(k,1)]{} \^[(k)]{} ) )\^2 . Thus, for a solution $\gamma^{(k)}(t)$ of the (cubic or quintic) GP hierarchy with initial condition satisfying (\[eq-GPsol-defoc-apriorien-1\]), we have that \^[(k)]{}(t) \_[H\^1]{} &lt; C\_0\^k with $C_0$ independent of $t$. Thus, for $\xi_1$ sufficiently small, we obtain the a priori bound (t)\_[\_[\_1]{}\^1]{} ( \_[k=1]{}\^(C\_0)\^[ 2k]{} )\^[12]{} &lt; , for all $t\in\R$.

This implies the existence of a solution $\Gamma(t)\in\cH_{\xi_1}^1$ globally in time. The key question is whether $\Gamma(t)\in \cH_{\xi_1}^1$ is the unique solution of the $p$-GP hierarchy with initial condition $\Gamma_0\in\cH_{\xi_1}^1$. To this end, we prove the following uniqueness result.

\[thm-defocGP-globalwp-1\] Assume that $1\leq d\leq3$ for $p=2$, and in $1\leq d\leq 2$ for $p=4$ such that $1 \in\alphaset(d,p)$. Assume that $\Gamma(t)\in\cH_{\xi_1}^1$, $t\in\R$, is a solution of the defocusing $p$-GP hierarchy for the initial condition $\Gamma(0)=\Gamma_0$, and satisfies the a priori bound $\|\Gamma(t)\|_{\cH_{\xi_1}^1}<c_0<\infty$ for $t\in\R$. Moreover, assume that for every sufficiently short open interval $I\in\R$, there exists a constant $\xi=\xi(I)$ with $0<\xi\leq \xi_1$ such that $\|\opB\Gamma\|_{L^1_{t\in I}\cH_{\xi}^1}<\infty$.

Then, there exists a constant $\xi_2=\xi_2(\xi_1)$ depending only on $\xi_1$, and a constant $T_0=T_0(d,p,\xi_1,\xi_2)$ such that the pair $(\Gamma,\opB\Gamma)$ is the unique solution of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) associated to the defocusing ($\mu=+1$) $p$-GP hierarchy with initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^1$, in the space := \_[I,|I|&lt;T\_0]{}ace , for $0<\xi_2\leq\xi_1$ sufficiently small. Hence in particular, \_[L\^1\_[tI]{}\_[\_2]{}\^1]{} C(T\_0,d,p,\_1,\_2) c\_0 for all $I\in\R$ with $|I|<T_0$.

Assume that $I'\subset\R$ is such that $\|\opB\Gamma\|_{L^1_{t\in I'}\cH_{\xi}^1}<\infty$. Let $I:=[\tau,T]\subset I'$ Then, Theorem \[thm-locwp-main-1-L1bd\] implies that in fact, there exists a constant $\xi_2\leq\xi_1$ such that \_[L\^1\_[tI ]{}\_[\_2]{}\^1]{} &lt; C\_0 () \_[\_[\_1]{}\^1]{} &lt; C\_0 c\_0 , for a constant $C_0=C_0(|I|,d,p,\xi_1,\xi_2)$.

This estimate holds independently of the value of $\tau\in\R$. Therefore, we can cover $\R$ with intervals $\R=\cup_j I_j$ where $I_j=[\tau_j,\tau_{j+1}]$, $j\in\Z$, and $|I_j|=|I|=: T_0$.

Assume that the initial time $t=0\in I_0$. Then, Theorem \[thm-locwp-main-1-L1bd\] implies that there exists a unique (forward and backward in time) solution $(\Gamma,\opB\Gamma)$ of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) in $\mathfrak{W}_{\xi_2}^1(I_0)$ for the initial condition initial condition $\Gamma_0$.

By iteration, we use $\Gamma(\tau_{1})$ as the initial condition for $t\in I_1$, and $\Gamma(\tau_0)$ as the terminal condition for $t\in I_{-1}$, and continue recursively, thereby covering $\R=\cup I_j$.

In conclusion, we obtain that there exists a unique global in time solution in $\Wspaceglob$ with initial condition $\Gamma_0\in\cH_{\xi_1}^1$, under the assumptions of the theorem.

As noted in Remark \[rem-BBGKYlim-1\], every solution $\Gamma(t)\in\cH_{\xi_1}^1$ extracted by a diagonal argument from the $N\rightarrow\infty$ limit of the $N$-particle BBGKY hierarchies (with repulsive interactions) studied by Erdös-Schlein-Yau in [@esy1; @esy2], Kirkpatrick-Schlein-Staffilani in [@kiscst], and Chen-Pavlović in [@chpa], satisfies $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ for some $\xi>0$, and all sufficiently short intervals $I\subset\R$. This follows a posteriori from the uniqueness of those solutions, which has been established in these works with independent methods, combined with Theorem \[thm-locwp-main-1-L1bd\] in the work at hand. Accordingly, $(\Gamma,\opB\Gamma)$ is automatically an element of $\Wspaceglob$ whenever $\Gamma(t)$ is obtained from an $N\rightarrow\infty$ limit of the above noted type.

$\;$\
\

Lower bound on the blowup rates {#sec-Proof-blowuprate-1}
===============================

In this section, we establish Theorem \[thm-blowuprate-L2crit-1\]. We adapt a standard proof given for $L^2$-critical focusing NLS to the GP hierarchy; see for instance [@ra]. Let $p \in \{2,4\}$. Similarly as in (\[eq-pseudoconf-gamma-1\]), one finds that the $p$-GP hierarchy is invariant under the rescaling \[eq-pseudoconf-gamma-2\]\
&& := \^[(k)]{}( t +((t))\^[-2]{} , ((t))\^[-1]{}\_k ; ((t))\^[-1]{}\_k’ ) If $\Gamma(t)=(\gamma^{(k)}(t))_{k\in\N}$ solves the $p$-GP hierarchy, then $\cS_{\lambda,t}\Gamma=(\cS_{\lambda,t}\gamma^{(k)} )_{k\in\N}$ is also a solution of the $p$-GP hierarchy. The proof can be straightforwardly adapted from the one given in Section \[sec-confinv-1\].

\
\
Blowup in finite time means that there exists $T^*<\infty$ such that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\rightarrow T^*$. To prove a lower bound on the blowup rate, we may assume that $1<\Av_{H^\alpha}(\Gamma(t))<\infty$ at a fixed time $t$, and choose (t) = (\_[H\^]{}((t)))\^ &gt; 1 . We note that $\frac{4}{d}<p<\frac{4}{d-2\alpha}$ implies that $2\alpha-d+\frac4p>0$. Let S\^[(k,)]{}\_[(t)]{} := \_[j=1]{}\^k ((t))\^[-1]{} \_[x\_j]{}\^ ((t))\^[-1]{} \_[x\_j’]{} \^ where $\langle b\nabla_{x}\rangle=\sqrt{1-b^2\Delta_{x}}$ for any $b\in\R$. Clearly, \[eq-Sdot-quasiblowup-L2-aux-1\]\
& = & ((t))\^[k(d-)]{} ( S\^[(k,)]{}\_[(t)]{} \^[(k)]{} ) (t+((t))\^[-2]{}) \_[L\^2\_[\_k,\_k’]{}]{} , and \[eq-scaleSkalpha-bd-1\] ((t))\^[-2k]{} S\^[(k,)]{} S\^[(k,)]{}\_[(t)]{} S\^[(k,)]{} since we are assuming that $\lambda(t)>1$.

We define \_&lt;(,t,) := ((t))\^[-4p+d-2]{} = (\_[H\^]{}((t)))\^[-1]{} and \[eq-xibigxi-id-1\] \_&gt;(,t,) := ((t))\^[-4p+d ]{} = (\_[H\^]{}((t)))\^[ ]{} . Clearly, (\[eq-Sdot-quasiblowup-L2-aux-1\]) and (\[eq-scaleSkalpha-bd-1\]) imply that \[eq-sandwich-bd-1\] (t+((t))\^[-2]{}) \_[\_[\_&lt;(,t,)]{}\^]{} & & \_[,t]{}() \_[ \^\_]{}\
&& (t+((t))\^[-2]{}) \_[\_[\_&gt;(,t,)]{}\^]{} . As a consequence of the definition of $\Av_{H^\alpha}(\Gamma(t))$, it follows that for $\tau=0$, 0 &lt; (t ) \_[\_[\_&lt;(,t,)]{}\^]{} &lt; c for any $0<\xi<1$.

To ensure that $\|  \Gamma(t+(\lambda(t))^{-2}\tau)  \|_{\cH_{\xi_>(\xi,t,\lambda)}^\alpha}<c$, we use the fact that according to (\[eq-xibigxi-id-1\]), $\xi_>(\xi,t,\lambda) $ can be made arbitrarily small by choosing $\xi$ small.

We note that our assumption $\frac{4}{d}<p<\frac{4}{d-2\alpha}$ implies that $2\alpha-d+\frac4p>0$ and $d-\frac4p>0$, so that the exponent on the rhs of (\[eq-xibigxi-id-1\]) is positive. If blowup occurs, such that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^*$, the above considerations necessitate the choice of values of $\xi$ (whose reciprocal determines the energy scale) tending to zero as $t\nearrow T^*$.

Thus, for $\xi_1>0$ sufficiently small, \_[,t]{}(0) \_[ \^\_[\_1]{}]{} (t ) \_[\_[\_&gt;(\_1,t,)]{}\^]{} Due to Theorem \[thm-locwp-main-1\], we may pick $0<\xi_2=\eta\xi_1\ll\xi_1<1$, such that there exists a solution $\cS_{\lambda,t}\gamma^{(k)} (\tau)\in\cH_{\xi_2}^\alpha$ if $\tau\in[0,\tau_{\max}]$, for $\tau_{\max}>0$ sufficiently small.

But this implies that (t+((t))\^[-2]{}) \_[\_[\_&lt;(\_2,t,)]{}\^]{} \_[,t]{}() \_[ \^\_[\_2]{}]{} &lt; for $\tau\in[0,\tau_{\max}]$ so that there is no blowup if $\tau$ lies in that interval. Therefore, the blowup time $T^*$ is bounded from below by T\^\* &gt; t+((t))\^[-2]{}\_ , and hence, ( \_[H\^]{}((t)) )\^[12]{} = (t)\^[(-+)]{} &gt; . This proves ($a$).

\
\
It is easy to see that \[eq-Lp-rescale\] \_[,t]{}\^[(k)]{} () \_[L\^r\_[\_k,\_k’]{}]{} = ((t))\^[-2k(2p-)]{} \^[(k)]{} (t+((t))\^[-2]{}) \_[L\^r\_[\_k,\_k’]{}]{} , which, in turn, implies that \_[,t]{}\^[(k)]{} (0) \_[\^r\_]{} &=& \_[k1]{}\^k \_[,t]{}\^[(k)]{} (0) \_[L\^r\_[\_k,\_k’]{}]{}\
& = & \_[k1]{} \^k ((t))\^[-2k(2p-)]{} \^[(k)]{} (t) \_[L\^r\_[\_k,\_k’]{}]{}\
& = & \_[k1]{} ( )\^[k]{} \^[(k)]{} (t) \_[L\^r\_[\_k,\_k’]{}]{} . \[bigLp-lambda\] However is bounded for every $\xi<1$, if we choose (t) = ( \_[L\^r]{}((t)) )\^ .

Now we argue as in the proof of the part ($a$) by using the local well-posedness Theorem \[thm-locwp-main-1\] to conclude that the $H^\alpha$ blowup time $T^*$ is bounded from below by T\^\* &gt; t+((t))\^[-2]{}\_ . Therefore ( \_[L\^r]{}((t)) )\^[12]{} = (t)\^[(-)]{} &gt; . Hence ($b$) is proved.

Proof of pseudoconformal invariance {#sec-confinv-1}
===================================

In this section, we prove Theorem \[thm-pseudoconf-gamma-1\]. We recall the pseudoconformal transformations \[eq-pseudoconf-gamma-1\]\
&& := e\^[-i]{} \^[(k)]{}( , ; ) , for any $b\in\R\setminus\{0\}$. Similarly as in the case of NLS, one can verify that\
&=& e\^[-i]{}\
&& ((i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{}) ( , ; ) . Now we shall prove the pseudoconformal invariance of the quintic GP hieararchy when $d=1$. In particular, we find that\
&=& e\^[-i]{} dx\_[k+1]{} dx\_[k+2]{} dx\_[k+1]{}’ dx\_[k+2]{}’\
&&(x\_j-x\_[k+1]{})(x\_j-x\_[k+1]{}’) (x\_j-x\_[k+2]{})(x\_j-x\_[k+2]{}’)\
&& \^[(k)]{}( , ; )\
&=& e\^[-i]{}\
&& \^[(k)]{}( , ; ) ,\
&=& e\^[-i]{}\
&& B\_[j;k+1,k+2]{}\^1\^[(k+2)]{} ( , ; ) with $B_{j;k+1,k+2}=B_{j;k+1,k+2}^1-B_{j;k+1,k+2}^2$; in $B_{j;k+1,k+2}^2$, the variable $x_j$ in $B_{j;k+1,k+2}^1$ is replaced by $x_j'$. Notably, we have used that e\^[-i]{}|\_[x\_[k+1]{}=x\_[k+2]{}=x\_[k+1]{}’=x\_[k+2]{}’=x\_j]{} = 1 . Thus, when $d=1$, we obtain\
&=& e\^[-i]{}\
&& ( (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1,k+2]{}\^[(k+2)]{} ) ( , ; )\
&=&0 . This proves pseudoconformal invariance of the quintic GP hierarchy in dimension $d=1$.

For the cubic GP hierarchy, the operators $B_{j;k+1,k+2}$ are replaced by operators $B_{j;k+1}$ which contract $x_j$, $x_j'$ only with $x_{k+1}$ and $x_{k+1}'$, (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1]{}\^[(k+1)]{} = 0 . The same considerations as above then produce\
&=& e\^[-i]{}\
&& ( (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1]{}\^[(k+1)]{} ) ( , ; )\
&=&0 if $d=2$. This proves Theorem \[thm-pseudoconf-gamma-1\].

The Klainerman-Machedon spacetime bounds {#sect-boundsfreeGP-1}
========================================

We present the Klainerman-Machedon spacetime bounds in dimensions $d\geq2$ in the form required for this paper, with $\alpha\in\alphaset(d,p)$; see (\[eq-alphaset-def-1\]). In the regime $\alpha>\frac d2-\frac{1}{2(p-1)}$, we present a simple argument to prove the result. In the endpoint case $(d,p)=(3,2)$ and $\alpha=1$, we invoke a result of [@klma].

\[prp-spacetime-bd-1\] Let $p=2,4$ account for the cubic and quintic GP hierarchy, respectively, and assume that $\alpha\in\alphaset(d,p)$. Let $\gamma^{(k+\frac p2)}$ be the solution of i\_t\^[(k+p2)]{}(t,\_[k+p2]{};\_[k+p2]{}’) + (\_[\_[k+p2]{}]{}-\_[\_[k+p2]{}’]{}) \^[(k+p2)]{}(t,\_[k+p2]{};\_[k+p2]{}’) = 0 with initial condition \^[(k+p2)]{}(0,) = \_0\^[(k+p2)]{}\^. Then, there exists a constant $C$ such that\
&& C S\^[(k+p2,)]{} \_0\^[(k+p2)]{} \_[L\^2 (\^[d(k+p2)]{}\^[d(k+p2)]{})]{} holds.

For notational convenience, we discuss the proof for the quintic GP hierarchy where $p=4$.

Let $(\tau,\uu_k,\uu_k')$, $\uq:=(q_1,q_2)$, and $\uq':=(q_1',q_2')$ denote the Fourier conjugate variables corresponding to $(t,\ux_k,\ux_k')$, $(x_{k+1},x_{k+2})$, and $(x_{k+1}',x_{k+2}')$, respectively.

Without any loss of generality, we may assume that $j=1$ in $B_{j;k+1,k+2}$. Then, abbreviating () := ( + (u\_1+q\_1+q\_2-q\_1’-q\_2’)\^2 + \_[j=2]{}\^k u\_j\^2 + ||\^2 - |\_k’|\^2 - |’|\^2 ) we find\
& = & \_dd\_k d\_k’ \_[j=1]{}\^ku\_j\^[2]{} u\_j’\^[2]{}\
&& ( dd’ () \^[(k+2)]{}(,u\_1+q\_1+q\_2-q\_1’-q\_2’,u\_2,…,u\_k,;\_k’,’) )\^2 . Using the Schwarz estimate, this is bounded by &&\_dd\_k d\_k’ I\_(,\_k,\_k’) dd’ ()\
&& u\_1+q\_1+q\_2-q\_1’-q\_2’ \^[2]{} q\_1\^[2]{} q\_2\^[2]{} q\_1’\^[2]{} q\_2’ \^[2]{} \_[j=2]{}\^ku\_j\^[2]{} \_[j’=1]{}\^ku\_[j’]{}’\^[2]{}\
&& | \^[(k+2)]{}(,u\_1+q\_1+q\_2-q\_1’-q\_2’,u\_2,…,u\_k,;\_k’,’) |\^2 where \[eq-Ialpha-def-1\]\
&& := d d’ . Similarly as in [@klma; @kiscst], we observe that $$\label{eq-u1-aux-bd-1}
          \bra u_1\ket^{2\alpha} 
                    \, \leq \, 
     C\Big[ \, \bra u_1+q_1+q_2-q_1'-q_2'\ket^{2\alpha} 
     + \bra q_1\ket^{2\alpha}
     + \bra q_2\ket^{2\alpha}
     + \bra q_1'\ket^{2\alpha} + \bra q_2'\ket^{2\alpha} \, \Big]
     \,,
     $$ so that I\_(,\_k,\_k’) \_[=1]{}\^5 J\_where $J_\ell$ is obtained from bounding the numerator of (\[eq-Ialpha-def-1\]) using (\[eq-u1-aux-bd-1\]), and from canceling the $\ell$-th term on the rhs of (\[eq-u1-aux-bd-1\]) with the corresponding term in the denominator of (\[eq-Ialpha-def-1\]). Thus, for instance, J\_1 &lt; d d’ , and each of the terms $J_\ell$ with $\ell=2,\dots,5$ can be brought into a similar form by appropriately translating one of the momenta $q_i$, $q_j'$.

Further following [@klma; @kiscst], we observe that the argument of the delta distribution equals + (u\_1+q\_1+q\_2-q\_1’)\^2 + \_[j=2]{}\^k u\_j\^2 + ||\^2 - |\_k’|\^2 - (q\_1’)\^2 - 2(u\_1+q\_1+q\_2-q\_1’)q\_2’ , and we integrate out the delta distribution using the component of $q_2'$ parallel to $(u_1+q_1+q_2-q_1')$. This leads to the bound \[eq-J1-bound-aux-1\] J\_1 & &lt; & C\_C dd q\_1’ where C\_ := \_ . Clearly, $C_\alpha$ is finite for any $\alpha>\frac12$. Moreover, it is clear that $J_1$ is monotonically decreasing in $\alpha$.

For the cubic GP hierarchy, the above arguments lead to the condition that instead of (\[eq-J1-bound-aux-1\]), the integral \[eq-J1-bound-aux-2\] d q\_1 must be bounded.

\
\
We first consider the case $p=4$ corresponding to the quintic GP hierarchy, and argue as follows. To bound (\[eq-J1-bound-aux-1\]), we pick a spherically symmetric function $h\geq0$ with rapid decay away from the unit ball in $\R^d$, such that $h^\vee(x)\geq0$ decays rapidly outside of the unit ball in $\R^d$, and \[eq-convuppbd-1\] &lt; h\*( \_[B\_1]{} + )(q) . (for example, $h(u)= c_1 e^{-c_2 u^2}$, for suitable constants $c_1,c_2$), where $\chi_{B_1} +\chi_{B_1}^c=1$ is a smooth partition of unity with $\chi_{B_1}$ supported on the unit ball, with $\chi_{B_1}(u)=1$ for $|u|\leq\frac12$, and $\chi_{B_1}(u)=0$ for $|u|>\frac12$. Clearly, $h* \chi_{B_1}$ and $h*\frac{\chi_{B_1}^c}{| \, \cdot \, |^{2\alpha}}$ are both in $L^\infty$, for any $\alpha>0$.

Then, assuming that $\alpha<\frac d2$, inserting this into (\[eq-J1-bound-aux-1\]), the most singular part is given by \[eq-J1-convFTprod-bd-1\]\
&=& C\_C dx ( )\^(x) ( (h\*)\^(x) )\^3\
&=& C\_C’ dx (h\^(x))\^3 ( ( (\_[B\_1]{}\^c)\^ ) (x) )\^3\
&&lt;& C\_C’ dx (h\^(x))\^3 ( )\^3 . For sufficiently large $C'$, this is an upper bound on all of the remaining terms that are obtained from substituting the bound (\[eq-convuppbd-1\]) into (\[eq-J1-bound-aux-1\]). We have here used that $(\chi_{B_1}^c)^\vee=1^\vee-\chi_{B_1}^\vee=\delta-\chi_{B_1}^\vee$, so that $|((\chi_{B_1}^c)^\vee*\frac{1}{| \, \cdot \, |^{d-2\alpha}})(x)| \leq C \, \frac{1}{|x|^{d-2\alpha}}$ holds for $\alpha<\frac d2$.

We conclude that (\[eq-J1-convFTprod-bd-1\]) is finite provided that the singularity at $x=0$ is integrable, since $h^\vee(x)$ falls off rapidly as $|x|\rightarrow\infty$. In dimension $d$, this is the case if the exponents in the denominator satisfy d-1 + 3d - 6 &lt; d , such that \[eq-alpha-lowbd-1\] &gt; d2-16 . This proves the claim for the quintic GP hierarchy, i.e., for $p=4$. In order to prove the lower bound (\[eq-alpha-lowbd-1\]) on $\alpha$, we have assumed that $\alpha<\frac d2$, which is consistent with it. Now, since $J_1$ is monotonically decreasing in $\alpha$, we arrive at the asserted result.

For the cubic GP hierarchy, the same considerations lead to the condition that $(\ref{eq-J1-bound-aux-1}) \,  < \, \infty$ if $\alpha \, > \, \frac d2-\frac12$. For a general $p$-GP hierarchy, one obtains the condition $\alpha>\frac d2-\frac{1}{2(p-1)}$.

\
\
In the situation $d=3$ and $p=2$ of the cubic GP hierarchy in 3 dimensions, we have the endpoint case $\frac d2-\frac1{2(p-1)}=1$. Klainerman and Machedon have proven in [@klma] that (\[eq-Ialpha-def-1\]) is bounded in this case.

Next, we prove the iterated spacetime estimates for the cubic GP hierarchy used in Section \[sec-locwp-1\], which involve the “boardgame estimates” of [@klma], which are motivated by the Feynman graph techniques in [@esy1; @esy2; @ey]. The corresponding results for the quintic GP hierarchy are obtained in an analogous manner, and we refer to [@chpa] for details.

\[prp-spacetime-bd-2\] Assume $\alpha$ as in Proposition \[prp-spacetime-bd-1\], for the cubic GP hierarchy ($p=2$). For brevity, let H\^\_k := H\^(\^[dk]{}\^[dk]{}) . Then, for $k\geq1$ and $j\leq k$, and $t\in I=[0,T]$, B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) \_[L\^1\_[tI]{}H\^\_k]{} &lt; k C\^k (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{} . Moreover, \[eq-Boardgame-rem-est-1\] B\_[k+1]{}\_[k]{}()\^[(k+1)]{} \_[L\^1\_[tI]{}H\^\_k]{} &lt; k C\^k (c T)\^[k2]{} ()\^[(2k)]{} \_[L\^1\_[t I]{}H\^\_[2k]{}]{} , where $\duh_j( \, \cdot \,)$ is defined in (\[eq-Duh-j-def-1\]).

Let $I=[0,T]$. Using an argument presented as a “board game", it is proven in [@klma] that the following holds.

Let ${\mathcal E}_{j,k+1}$ denote the space of sequences $\umu=(\mus(1),\dots,\mus(j))$ where $\mu(i)\in\{1,\dots,k+i\}$, where for every $i\in\{1,\dots,j\}$, one has $\mu(i)\geq\mu(i')$ for all $i'>i$. The elements of ${\mathcal E}_{j,k+1}$ parametrize $(k+j)\times j$ matrices in so-called “special upper echelon form" (see [@klma] for definitions). The cardinality of this set satisfies $|{\mathcal E}_{j,k+1}|\leq C^{j+k}$.

For every $\umu\in{\mathcal E}_{j,k+1}$, one associates the term\
&:=&\_[D]{} dt\_1 dt\_[j]{} e\^[i(t-t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{}e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 for a measurable subset $D\subset[0,t]^j$. Then, it is proven in [@klma] that \[eq-klma-boardgame-dec-1\]\
&& \_[\_[j,k+1]{}]{} B\_[k+1]{} ( \_j(\_0)\^[(k+1)]{}(t) )\_ \_[L\^2\_[tI]{}H\^]{} . For the proof of (\[eq-klma-boardgame-dec-1\]) in the case of the cubic GP hierarchy, we refer to [@klma]. For the case of the quintic GP hierarchy, we refer to [@chpa].

We have, under the given assumptions on $\alpha$, that for $I=[0,T]$ and $D\subset I^j$,\
&& \_[=1]{}\^k \_[D]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^1\_[tI]{}H\^\_k]{}\
&& k \_[I]{} dt \_D dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ H\^\_k]{}\
&& k \_[I\^[j+1]{}]{} dt dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{}\^[(k+j+1)]{}\_0 \_[ H\^\_k]{} . \[eq-BGamma-Duhj-aux-1\] By Cauchy-Schwarz with respect to the integral in $t$, this is bounded by && k T\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j\_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ L\^2\_t(I)H\^\_k]{}\
&& k T\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ L\^2\_t()H\^\_k]{} . Using Proposition \[prp-spacetime-bd-1\] and unitarity of $e^{-i t_1\Delta_\pm^{(k+1)}}$, this is bounded by && k (cT)\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ H\^\_[k+1]{}]{} \[ap-usingKM1\]\
&=& k (c T)\^[12]{} \_[I\^[j-1]{}]{} dt\_2 dt\_[j]{} B\_[(1),k+2]{} e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^1\_[t\_1]{}(I)H\^\_[k+1]{}]{} . \[ap-Hol\] Iterating the same steps as above, we find, after $j$ steps, & &\
&& k (c T)\^ \_[I]{} dt\_[j]{} B\_[(j-1),k+j]{} e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} \[ap-KMiterated\]\
&& B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^2\_[t\_[j-1]{}I]{}H\^\_[k+j-1]{}]{}\
&& k (c T)\^ \_[I]{} dt\_[j]{} B\_[(j),k+j+1]{} e\^[i t\_[j]{} \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[H\^\_[k+j]{}]{}\
& & k (c T)\^ \^[(k+j+1)]{}\_0 \_[H\^\_[k+j+1]{}]{} . \[A2final\] In the last step, we used Cauchy-Schwarz in $t_j$, and Proposition \[prp-spacetime-bd-1\].

Then, estimating by $C^{j+k}$ the number of terms in the sum over $\umu\in {\mathcal E}_{j,k+1}$, \[eq-BGamma-Duhj-combin-bd-1\] B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) \_[L\^1\_[tI]{}H\^]{} k C\^k (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^]{} , as claimed.

In the same manner, we prove (\[eq-Boardgame-rem-est-1\]). In this case, we have\
&& \_[=1]{}\^k \_[D]{} dt\_1 dt\_[k]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i t\_[k]{} \_\^[(2k)]{}]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\^1\_[tI]{}H\^\_k]{}\
&& k \_[I]{}dt \_[D]{} dt\_1 dt\_[k]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\
&& e\^[i t\_[k]{} \_\^[(2k)]{}]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[ H\^\_k]{} Applying the same arguments as above between (\[eq-BGamma-Duhj-aux-1\]) and (\[ap-KMiterated\]), with $j=k$, one finds the upper bound & &\
&& k (c T)\^ \_[I]{} dt\_[k]{} B\_[(k-1),2k ]{} e\^[i t\_[k]{} \_\^[(2k )]{}]{} \[ap-KMiterated-BGamma\]\
&& B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\^2\_[t\_[k-1]{}I]{}H\^\_[2k -1]{}]{}\
&& k (c T)\^ \_[I]{} dt\_[k]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[H\^\_[2k]{}]{}\
& = & k (c T)\^ B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\_[tI]{}\^1H\^\_[2k ]{}]{}\
& = & k (c T)\^ ()\^[(2k)]{}(t\_k) \_[L\_[tI]{}\^1H\^\_[2k ]{}]{} . \[A2final-BGamma\] Invoking the argument used to establish (\[eq-BGamma-Duhj-combin-bd-1\]), we arrive at (\[eq-Boardgame-rem-est-1\]).

For more details, we refer to [@chpa].

Acknowledgements {#acknowledgements .unnumbered}
----------------

We thank S. Klainerman and N. Tzirakis for inspiring discussions. We are deeply indebted to B. Schlein for extremely useful comments that helped us to significantly improve a previous version. The work of T.C. was supported by NSF grant DMS-0704031. The work of N.P. was supported NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.

[99]{}

R. Adami, G. Golse, A. Teta, [*Rigorous derivation of the cubic NLS in dimension one*]{}, J. Stat. Phys. [**127**]{}, no. 6, 1194–1220 (2007).

M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason [*Bose-Einstein Quantum Phase Transition in an Optical Lattice Model*]{}, Phys. Rev. A [**70**]{}, 023612 (2004).

V. Bach, T. Chen, J. Fröhlich and I. M. Sigal, [*Smooth Feshbach map and operator-theoretic renormalization group methods*]{}, J. Funct. Anal., [**203**]{} (1), 44-92 (2003).

T. Cazenave, [*Semilinear Schrödinger equations*]{}, Courant lecture notes [**10**]{}, Amer. Math. Soc. (2003).

T. Chen, N. Pavlović, [*The quintic NLS as the mean field limit of a Boson gas with three-body interactions*]{}, Preprint http://arxiv.org/abs/0812.2740.

A. Elgart, L. Erdös, B. Schlein, H.-T. Yau, [*Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons*]{}, Arch. Rat. Mech. Anal. [**179**]{}, no. 2, 265–283 (2006).

L. Erdös, B. Schlein, H.-T. Yau, [*Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate*]{}, Comm. Pure Appl. Math. [**59**]{} (12), 1659–1741 (2006).

L. Erdös, B. Schlein, H.-T. Yau, [*Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems*]{}, Invent. Math. [**167**]{} (2007), 515–614.

L. Erdös, H.-T. Yau, [*Derivation of the nonlinear Schrödinger equation from a many body Coulomb system*]{}, Adv. Theor. Math. Phys. [**5**]{}, no. 6, 1169–1205 (2001).

J. Fröhlich, S. Graffi, S. Schwarz, [*Mean-field- and classical limit of many-body Schrödinger dynamics for bosons*]{}, Comm. Math. Phys. [**271**]{}, no. 3, 681–697 (2007).

J. Fröhlich, A. Knowles, A. Pizzo, [*Atomism and quantization*]{}, J. Phys. A [**40**]{}, no. 12, 3033–3045 (2007).

J. Fröhlich, A. Knowles, S. Schwarz [*On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction*]{}, Preprint arXiv:0805.4299.

K. Hepp, [*The classical limit for quantum mechanical correlation functions*]{}, Comm. Math. Phys. [**35**]{}, 265–277 (1974).

S. Klainerman, M. Machedon, [*On the uniqueness of solutions to the Gross-Pitaevskii hierarchy*]{}, Commun. Math. Phys. [**279**]{}, no. 1, 169–185 (2008).

K. Kirkpatrick, B. Schlein, G. Staffilani, [*Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics*]{}, Preprint arXiv:0808.0505.

E.H. Lieb, R. Seiringer, [*Proof of Bose-Einstein condensation for dilute trapped gases*]{}, Phys. Rev. Lett. [**88**]{}, 170409 (2002).

E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, [*The mathematics of the Bose gas and its condensation*]{}, Birkhäuser (2005).

E.H. Lieb, R. Seiringer, J. Yngvason, [*A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas*]{}, Commun. Math. Phys. [**224**]{} (2001).

G. Perelman, [*On the formation of singularities in solutions of the critial nonlinear Schrödinger equation*]{}, Ann. H. Poinc., [**2**]{} (4), 605–673 (2001).

P. Raphael, [*Stability and blow up for the non linear Schrödinger Equation*]{}, Lecture notes for CMI workshop, Zürich (2008).

I. Rodnianski, B. Schlein, [*Quantum fluctuations and rate of convergence towards mean field dynamics*]{}, preprint arXiv:math-ph/0711.3087.

B. Schlein, [*Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics*]{}, Lecture notes for the minicourse held at the 2008 CMI Summer School in Zurich.

H. Spohn, [*Kinetic Equations from Hamiltonian Dynamics*]{}, Rev. Mod. Phys. [**52**]{}, no. 3, 569–615 (1980).

T. Tao, [*Nonlinear dispersive equations. Local and global analysis*]{}, CBMS 106, eds: AMS, 2006.

[^1]: We thank B. Schlein for calling our attention to this fact.
