text:
\section{Introduction}

The derivation of the nonlinear Schr\"odinger 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 \cite{esy1,esy2,ey,kiscst,klma,rosc} and the references therein,
and also \cite{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
\cite{ailisesoyn,lise,lisesoyn,liseyn} and the references therein.

The  procedure  developed in the landmark works of Erd\"os, Schlein, and Yau,
\cite{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 \cite{kiscst,sc}.
For factorized initial data, the
solutions of the GP hierarchy are governed by a cubic NLS for
systems with 2-body interactions, \cite{esy1,esy2,ey,kiscst},
and quintic NLS for systems with 3-body interactions, \cite{chpa}.
The proof of the uniqueness of solutions of the GP hierarchy
is the most difficult part of this analysis, and is obtained in \cite{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 \cite{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 \cite{kiscst} of Kirkpatrick, Schlein, and Staffilani,
the corresponding problem in $d=2$ is solved,
and the assumption made in \cite{klma} is replaced by a spatial a
priori bound which is proven in \cite{kiscst}.
Alternative methods to obtain dynamical mean field limits of interacting
Bose gases using operator-theoretic methods are developed by Fr\"ohlich et al in \cite{frgrsc,frknpi,frknsc}.

All of the above mentioned works discuss Bose gases with {\em 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 {\em attractive} interactions within this context.
Accordingly, we will refer to the corresponding GP hierarchies as   {\em cubic},
{\em quintic}, {\em focusing}, or {\em 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
${\mathcal H}_\xi^\alpha=\{ \, \Gamma\in{\mathfrak G} \,  | \, \| \, \Gamma \, \|_{{\mathcal H}_\xi^\alpha} <\infty \, \}$
where
\begin{eqnarray}\label{bigG}
	{\mathfrak G} \, = \, \{ \, \Gamma \, = \, ( \, \gamma^{(k)}(x_1,\dots,x_k;x_1',\dots,x_k') \, )_{k\in{\mathbb N}}
	\, | \,
	{\rm Tr} \gamma^{(k)} \, < \, \infty \, \}
\end{eqnarray}
is the space of sequences of $k$-particle density matrices, and
\begin{eqnarray}
	\| \, \Gamma \, \|_{{\mathcal H}_\xi^\alpha} \, := \,
	\sum_{k\in{\mathbb N}} \xi^k \, \| \, \gamma^{(k)} \, \|_{H^\alpha({\mathbb R}^{dk}\times{\mathbb R}^{dk})} \,.
\end{eqnarray}
The parameter $\xi>0$ is determined by the initial condition, and it sets the energy scale of
a given Cauchy problem.
If $\Gamma\in{\mathcal H}_\xi^\alpha$, 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  $\alpha\in{\mathfrak A}(d,p)$ where
\begin{eqnarray}\label{eq-alphaset-def-0}
	{\mathfrak A}(d,p) \, := \, \left\{
	\begin{array}{cc}
	(\frac12,\infty) & {\rm if} \; d=1 \\
	(\frac d2-\frac{1}{2(p-1)}, \infty) & {\rm if} \; d\geq2 \; {\rm and} \; (d,p)\neq(3,2)\\
	\big[1,\infty) & {\rm if} \; (d,p)=(3,2) \,,
	\end{array}
	\right.
\end{eqnarray}
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:
\begin{enumerate}
\item
We prove local existence and uniqueness of solutions
for the cubic and
quintic GP hierarchy with focusing or defocusing interactions,
in ${\mathcal H}_\xi^\alpha$, for $\alpha\in {\mathfrak A}(d,p)$,
which satisfy a spacetime bound $\|\widehat{B}\Gamma\|_{L^1_{t\in I}{\mathcal H}^{\alpha}_{\xi}}<\infty$
for some $\xi>0$ (the operator $\widehat{B}$ is defined in Section \ref{sec-defandresults-1}
below). This spacetime bound  has been
conjectured by Klainerman and Machedon in \cite{klma}.
It is of Strichartz-type, and is proven in Section \ref{sec-locwp-1}
using a Picard-type fixed point argument on the space $L^1_{t\in [0,T]}{\mathcal H}_\xi^\alpha$;
see inequality
(\ref{eqn-BGamma-spacetime-bd-0}) and Remark \ref{rem-Strichartz-1} below.

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

\item
We prove the global existence and uniqueness of solutions
in ${\mathcal H}_\xi^1$ 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 $\|\Gamma(t)\|_{{\mathcal H}_\xi^1}<c$ holds for $\xi>0$
sufficiently small.
\\

\item
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.
\\

\item
For the focusing cubic or quintic GP hierarchy,
we prove lower bounds on the blowup rate in ${\mathcal H}^\alpha_\xi$ and ${\mathcal L}^r_\xi$,
where both spaces are defined in Section \ref{sec-defandresults-1} below.
\end{enumerate}

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 \cite{klma} for the cubic GP hierarchy in $d=3$ (which were generalized
to cubic in $d=2$ in \cite{kiscst}, and to the quintic GP hierarchy in \cite{chpa}),
and the ``boardgame estimates" developed in \cite{klma} (and generalized to the quintic case in \cite{chpa}),
which were motivated by the Feynman graph expansion techniques of \cite{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 \ref{def-AvHLp-1} below,
and turns out to be the key observable for our discussion of blowup solutions.
\\

\subsection*{Organization of the paper}

In Section \ref{sec-defandresults-1}, we introduce the cubic and quintic GP hierarchy,
and state our main theorems.
In Section \ref{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 \ref{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 \ref{sec-Proof-blowuprate-1}, we prove lower bounds on the blowup rate
of blowup solutions in the spaces ${\mathcal H}_\xi^\alpha$ and ${\mathcal L}^r_\xi$ (see below for their
definitions).
In Section \ref{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.

\newpage

\section{Definition of the model and statement of the main results}
\label{sec-defandresults-1}

We introduce the space
\begin{eqnarray}
	{\mathfrak G} \, := \, \bigoplus_{k=1}^\infty L^2({\mathbb R}^{dk}\times{\mathbb R}^{dk})
\end{eqnarray}
of sequences of density matrices
\begin{eqnarray}
	\Gamma \, := \, (\, \gamma^{(k)} \, )_{k\in{\mathbb N}}
\end{eqnarray}
where $\gamma^{(k)}\geq0$, ${\rm Tr}\gamma^{(k)} =1$,
and where every $\gamma^{(k)}({\underline{x}}_k,{\underline{x}}_k')$ is symmetric in all components of ${\underline{x}}_k$,
and in all components of ${\underline{x}}_k'$, respectively.

We call $\Gamma=(\gamma^{(k)})_{k\in{\mathbb N}}$  admissible if
\begin{eqnarray}
	\lefteqn{
	\gamma^{(k)}({\underline{x}}_k;{\underline{x}}_k')
	}
	\\
	&&\, = \, \int dx_{k+1} \cdots dx_{k+\frac p2}
	\, \gamma^{(k+\frac{p}{2})}({\underline{x}}_{k},x_{k+1},\dots,x_{k+\frac p2};{\underline{x}}_k',x_{k+1},\dots,x_{k+\frac p2})
	\nonumber
\end{eqnarray}
for all $k\in{\mathbb N}$.

Let $0<\xi<1$ and $r>1$.
We define
\begin{eqnarray}
	{\mathcal L}_\xi^r \, := \, \Big\{ \, \Gamma \, \in \, {\mathfrak G} \, \Big| \, \|\Gamma\|_{{\mathcal L}_\xi^r} < \, \infty \, \Big\}
\end{eqnarray}
where
\begin{eqnarray}
	\|\Gamma\|_{{\mathcal L}_\xi^r} \, := \, \sum_{k=1}^\infty \xi^{ k}
	\| \, \gamma^{(k)} \, \|_{L^r({\mathbb R}^{dk}\times{\mathbb R}^{dk})} \,.
\end{eqnarray}
Furthermore, we define
\begin{eqnarray}
	{\mathcal H}_\xi^\alpha \, := \, \Big\{ \, \Gamma \, \in \, {\mathfrak G} \, \Big| \, \|\Gamma\|_{{\mathcal H}_\xi^\alpha} < \, \infty \, \Big\}
\end{eqnarray}
where
\begin{eqnarray}
	\|\Gamma\|_{{\mathcal H}_\xi^\alpha} \, = \, \sum_{k=1}^\infty \xi^{ k}
	\| \,  \gamma^{(k)} \, \|_{H^\alpha({\mathbb R}^{dk}\times{\mathbb R}^{dk})} \,,
\end{eqnarray}
with
\begin{eqnarray}
	\| \, \gamma^{(k)} \, \|_{H^\alpha({\mathbb R}^{dk}\times{\mathbb R}^{dk})} \, = \,
	\| \, S^{(k,\alpha)} \, \gamma^{(k)} \, \|_{L^2({\mathbb R}^{dk}\times{\mathbb R}^{dk})}  \,,
\end{eqnarray}
and $S^{(k,\alpha)}:=\prod_{j=1}^k\langle\nabla_{x_j}\rangle^\alpha\langle\nabla_{x_j'}\rangle^\alpha$.

Clearly, ${\mathcal L}_\xi^r$,  ${\mathcal H}_\xi^\alpha$ 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,
\cite{bcfs}.

Let $p\in\{2,4\}$.
We consider the $p$-GP (Gross-Pitaevskii) hierarchy given by
\begin{eqnarray} \label{eq-def-b0-2}
	i\partial_t \gamma^{(k)} \, = \, \sum_{j=1}^k [-\Delta_{x_j},\gamma^{(k)}]
	\, + \,  \mu B_{k+\frac p2} \gamma^{(k+\frac p2)}
\end{eqnarray}
in $d$ dimensions, for $k\in{\mathbb N}$. Here,
\begin{eqnarray}
	\lefteqn{
	\left(B_{k+\frac p2}\gamma^{(k+\frac{p}{2})}\right)(t,x_1,\dots,x_k;x_1',\dots,x_k')
	}
	\\
	&& := \,
	\sum_{j=1}^k \left(B_{j;k+1,\dots,k+\frac p2}\gamma^{(k+\frac{p}{2})}\right)(t,x_1,\dots,x_k;x_1',\dots,x_k')
	\nonumber\\
	&& := \, \sum_{j=1}^k\int dx_{k+1}\cdots dx_{k+\frac p2} dx_{k+1}'\cdots dx_{k+\frac p2}' \,
	\nonumber\\
	&&\quad\quad\quad\quad
	\big[ \, \prod_{\ell=k+1}^{k+\frac p2} \delta(x_j-x_{\ell})\cdots \delta(x_j-x_{\ell}' )
	- \prod_{\ell=k+1}^{k+\frac p2}\delta(x_j'-x_{\ell})\cdots\delta(x_j'-x_{\ell}') \, \big]
	\nonumber\\
	&& \quad\quad\quad\quad\quad\quad
	\gamma^{(k+\frac p2)}(t,x_1,\dots,x_{k+\frac p2};x_1',\dots,x_{k+\frac p2}') \,
	\nonumber\,
\end{eqnarray}
accounts for  $\frac p2+1$-body interactions between the Bose particles.

For a factorized initial condition
\begin{eqnarray}\label{eq-initcond-fact-1}
	\gamma^{(k)}(0) \, = \, | \, \phi_0 \, \rangle \langle \, \phi_0 \, |^{\otimes k}
\end{eqnarray}
with $\phi_0\in H^\alpha$, one obtains that
\begin{eqnarray}
	\gamma^{(k)}(t) \, = \, | \, \phi(t) \, \rangle \langle \, \phi(t) \, |^{\otimes k}
\end{eqnarray}
is a solution of \eqref{eq-def-b0-2} if $\phi_t$ satisfies the NLS
\begin{eqnarray}\label{eq-NLS-p-1}
	i\partial_t \phi_t \, + \, \Delta_x \phi_t \, - \, \mu \, |\phi_t|^p \, \phi_t \, = \, 0
\end{eqnarray}
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 (\ref{eq-def-b0-2}) as the {\em cubic GP hierarchy} if $p=2$,
and as the {\em 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 ${\mathfrak A}(d,p)$, for $p=2,4$ and $d\geq1$,
\begin{eqnarray}\label{eq-alphaset-def-1}
	{\mathfrak A}(d,p) \, = \, \left\{
	\begin{array}{cc}
	(\frac12,\infty) & {\rm if} \; d=1 \\
	(\frac d2-\frac{1}{2(p-1)}, \infty) & {\rm if} \; d\geq2 \; {\rm and} \; (d,p)\neq(3,2)\\
	\big[1,\infty) & {\rm if} \; (d,p)=(3,2)
	\end{array}
	\right.
\end{eqnarray}
Our main result in this paper is the following theorem.

\begin{theorem}\label{thm-main-0}
Let $0<\xi_2=\eta\xi_1\leq\xi_1<1$.
Assume that $\alpha\in{\mathfrak A}(d,p)$ where $d\geq1$ and $p\in\{2,4\}$, and $0<\eta<1$ sufficiently small.
Then, the following hold.
\begin{itemize}
\item[(i)]
For every $\Gamma_0\in{\mathcal H}_{{\xi_1}}^\alpha$, there
exist constants $T>0$ and $0<\xi_2\leq\xi_1$
such that there exists a unique
solution $\Gamma(t)\in{\mathcal H}_{\xi_2}^\alpha$ for $t\in[0,T]$ with
$\|\widehat{B}\Gamma\|_{L^1_{t\in[0,T]}{\mathcal H}_{\xi_2}^\alpha}<\infty$.
\\
\item[(ii)]
Assume that given $\Gamma_0\in{\mathcal H}_{{\xi_1}}^\alpha$, 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 (\ref{seclwp-pGP})
in the space $L^\infty_{t\in I}{\mathcal H}_{\xi_2}^\alpha$.
\\
\\
Then, the solution $\Gamma(t)\in L^\infty_{t\in I}{\mathcal H}_{\xi_2}^\alpha$
is {\em unique} if and only if
$\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi}^\alpha}<\infty$
holds for some $\xi>0$.
\\
\\
If the latter is satisfied, then in fact, the Strichartz-type bound
\begin{eqnarray}\label{eqn-BGamma-spacetime-bd-0}
	\|\widehat{B}\Gamma\|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}
	&\leq&C(d,p,\xi_1,\xi_2) \,
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha}
\end{eqnarray}
holds.
\\
\end{itemize}
\end{theorem}

\begin{remark}\label{rem-BBGKYlim-1}
An immediate implication of part (ii) of Theorem \ref{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 Erd\"os-Schlein-Yau in \cite{esy1,esy2},
Kirkpatrick-Schlein-Staffilani in
\cite{kiscst}, and Chen-Pavlovi\'c in \cite{chpa}, satisfies
$\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi}^\alpha}<\infty$
for some $\xi>0$.
This is true because uniqueness of those solutions has been established in these
works with independent methods.
\end{remark}

\begin{remark}
The role of the parameters $\xi_1,\xi_2$ is as follows:
Given initial data $\Gamma_0=(\gamma^{(k)})_{k\in{\mathbb N}}$ with
$\|\gamma^{(k)}\|_{H^\alpha({\mathbb R}^{dk}\times{\mathbb R}^{dk})}<\infty$ for all $k$,
we determine $\xi_1>0$ sufficiently small such that
$\Gamma_0\in{\mathcal H}_{\xi_1}^\alpha$.
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
${\mathcal H}^\alpha_{\xi_2}$.
The requirement $\xi_2 \ll \xi_1$ is used to ensure
that a solution $\Gamma(t)$ does not drift out of  ${\mathcal H}^\alpha_{\xi_2}$ 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  $T(\xi_2')>T(\xi_2)$.
\end{remark}

\begin{remark}
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 ${\mathcal H}^\alpha_{\xi}$ for {\em an arbitrary} nonzero $\xi>0$.
\end{remark}

\begin{remark}\label{rem-Strichartz-1}
We note that the estimate (\ref{eqn-BGamma-spacetime-bd-0}), for the cubic GP hierarchy
with $d=3$ and $\alpha=1$, proves the a priori spacetime bound conjectured in \cite{klma}.
For factorized initial data $\Gamma=(|\phi_0\rangle\langle\phi_0|^{\otimes k})_{k\in{\mathbb N}}$
in the cubic case, so that $\Gamma=(|\phi(t)\rangle\langle\phi(t)|^{\otimes k})_{k\in{\mathbb N}}$
where $i\partial_t\phi+\Delta\phi-\mu|\phi|^2\phi=0$, it corresponds to the inequality
\begin{eqnarray}
	\| \, |\phi|^2\phi \, \|_{L^1_{t\in I}H^\alpha}^{\frac13} \, \leq \, C(T) \, \| \, \phi_0 \, \|_{H^\alpha}
\end{eqnarray}
which is of Strichartz type.
The example of factorized solutions with $\phi(t)\in H^1$, $t\in I$, is discussed in
detail in \cite{klma}.
\end{remark}

\begin{definition}
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
$\|\Gamma(t)\|_{{\mathcal H}_{\xi}^\alpha}\rightarrow\infty$ as $t\nearrow T^{*}_{\xi,\Gamma}$,
and $T_{\xi,\Gamma}^*\nearrow T^*$ as $\xi\rightarrow0$.
\end{definition}

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

\begin{definition}\label{def-AvHLp-1}
We refer to
\begin{eqnarray}
	{\rm Av}_{H^\alpha}(\Gamma ) \, := \, \Big[ \, \sup\big\{ \, \xi >0 \, \big| \,
	\| \, \Gamma \, \|_{{\mathcal H}_\xi^\alpha} <\infty\, \big\} \, \Big]^{-1} \,,
\end{eqnarray}
\begin{eqnarray}
	{\rm Av}_{L^r}(\Gamma ) \, := \, \Big[ \, \sup\big\{ \, \xi >0 \, \big| \,
	\| \, \Gamma \, \|_{{\mathcal L}_\xi^r} <\infty\, \big\} \, \Big]^{-1} \,,
\end{eqnarray}
respectively, as the typical (or average) $H^\alpha$-energy and the typical $L^r$-norm per particle.
\end{definition}

We note that
\begin{eqnarray}
	\Gamma \, = \, ( \, | \, \phi \, \rangle \langle \, \phi \, |^{\otimes k} \, )_{k\in{\mathbb N}}
	\; \; \; \Rightarrow \; \; \;
 	{\rm Av}_{H^\alpha}(\Gamma ) \, = \, \|\phi\|_{H^\alpha}^2 \mbox{ and }
	{\rm Av}_{L^r}(\Gamma ) \, = \, \|\phi\|_{L^r}^2
\end{eqnarray}
in the factorized case.

The fact that $\Gamma\in{\mathcal H}_\xi^\alpha$ means that
the typical energy per particle is bounded by ${\rm Av}_{H^\alpha}(\Gamma )<\xi^{-1}$.
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 ${\mathcal H}_\xi^\alpha$,
blowup solutions undergo transitions
${\mathcal H}_{\xi_1}^\alpha\rightarrow{\mathcal H}_{\xi_2}^\alpha\rightarrow{\mathcal H}_{\xi_3}^\alpha\rightarrow\cdots$
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 ${\rm Av}_{H^\alpha}(\Gamma(t))\rightarrow\infty$
as $t\nearrow T^*$.

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

\begin{theorem}
\label{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 $\Gamma(t_0)=\Gamma_{0}\in{\mathcal H}_{\xi}^\alpha$,
for some $\xi>0$, which blows up in finite time. Then, the following
lower bounds on the blowup rate hold:
\begin{enumerate}
\item[($a$)]
Assume that $\frac4d\leq p < \frac{4}{d-2\alpha}$. Then,
\begin{eqnarray}
	( \, {\rm Av}_{H^\alpha}(\Gamma(t)) \, )^{\frac12}
	\, > \, \frac{C}{|T^*-t|^{(2\alpha-d+\frac4p)/4}} \,.
\end{eqnarray}
Thus specifically, for the cubic GP hierarchy in $d=2$, and for the quintic GP hierarchy in $d=1$,
\begin{eqnarray}
	( \, {\rm Av}_{H^1}(\Gamma(t )) \, )^{\frac12} \, \geq \, \frac{C}{|t -T^*|^{\frac12}} \,,
\end{eqnarray}
with respect to the Sobolev spaces $H^\alpha$, ${\mathcal H}_\xi^\alpha$.

\item[($b$)]
\begin{eqnarray}
	( \, {\rm Av}_{L^r}(\Gamma(t )) \, )^{\frac12} \, \geq \,
        \frac{C}{|t -T^*|^{\frac{1}{p}-\frac{d}{2r}}} \,, \mbox{ for } \frac{pd}{2} < r.
\end{eqnarray}
\end{enumerate}
\end{theorem}

\begin{remark}
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, \cite{ca}).
\end{remark}

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 (\ref{eq-NLS-p-1}).
If the NLS (\ref{eq-NLS-p-1}) is $L^2$-critical, that is, $p=\frac 4d$,
it is invariant under the pseudoconformal transformations
\begin{eqnarray}\label{eq-NLSpc-def-1}
	{\mathcal P}\phi_t (x)
	\, := \,
	\frac{1}{(1+bt)^{1/2}} \, e^{-i\frac{b x^2 }{1+bt}}
	\phi_{\frac{1}{1+bt}}\Big( \, \frac{x}{1+bt} \, \Big) \,,
\end{eqnarray}
for $b\in{\mathbb R}\setminus\{0\}$.
That is,
\begin{eqnarray}
	i\partial_t{\mathcal P}\phi_t \, + \, \Delta {\mathcal P}\phi_t
	\, - \, \mu \, |{\mathcal P}\phi_t|^p \, {\mathcal P}\phi_t \, = \, 0 \,;
\end{eqnarray}
see for instance \cite{ca}.
There are two cases of $L^2$-critical NLS with $p\in{\mathbb N}$: 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, $\mu=-1$.

\begin{theorem}
\label{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 \eqref{eq-def-b0-2} is invariant under the pseudoconformal
transformations
\begin{eqnarray}\label{eq-pseudoconf-gamma-1}
	\lefteqn{
	{\mathcal P}\gamma^{(k)} (t,{\underline{x}}_k;{\underline{x}}_k')
	}
	\nonumber\\
	&&
	\, := \,
	\frac{1}{(1+bt)^{dk}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\gamma^{(k)}\Big( \, \frac{1}{1+bt} \, , \, \frac{{\underline{x}}_k}{1+bt} \, ; \, \frac{{\underline{x}}_k'}{1+bt} \, \Big) \,,
\end{eqnarray}
for $b\in{\mathbb R}\setminus\{0\}$.

That is,
\begin{eqnarray}\label{eq-infhierarch-pseudoconf-1}
	i\partial_t {\mathcal P}\gamma^{(k)} \, + \,
	\Delta_\pm^{(k)}{\mathcal P}\gamma^{(k)}
	\, - \, \mu \,  B_{k+\frac p2} {\mathcal P}\gamma^{(k+2)} = 0 \,,
\end{eqnarray}
for all $k\geq1$.
\end{theorem}

The proof is given in Section \ref{sec-confinv-1}.
For a survey of related matters for the NLS, see for instance \cite{ca,ra,ta}.

Of course, the following is immediately clear.

\begin{theorem}
Assume that $\alpha\in{\mathfrak A}(d,p)$ where $d\geq1$ and $p\in\{2,4\}$.
Moreover, assume that $\Gamma(t)\in{\mathcal H}_{\xi_2}^\alpha$ solves the (cubic or quintic)
focusing ($\mu=-1$) GP hierarchy with factorized
initial condition $\Gamma_0=(|\phi_0\rangle\langle\phi_0|^{\otimes k})_{k\in{\mathbb N}}\in{\mathcal H}_{\xi }^\alpha$
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 ${\rm Av}_{H^\alpha}(\Gamma(t))\rightarrow\infty$  as $t\nearrow T^*$.
\end{theorem}

\begin{proof}
This follows from ${\rm Av}_{H^\alpha}(\Gamma(t))=\|\phi(t)\|_{H^\alpha}^2$ for product states.
\endprf

For various scenarios in which blowup occurs for solutions of the cubic
or quintic NLS, we refer to the literature; see for instance \cite{ca,ra} for
surveys.

\newpage

\section{Local existence and uniqueness of solutions for the focusing and defocusing GP hierarchy}
\label{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
\begin{eqnarray}
	\Delta_{\pm}^{(k)} \, = \, \Delta_{{\underline{x}}_k} - \Delta_{{\underline{x}}'_k}
\end{eqnarray}
with
\begin{eqnarray}
	\Delta_{{\underline{x}}_k} \, = \, \sum_{j=1}^k\Delta_{x_j}
\end{eqnarray}
and
\begin{eqnarray}
	\Delta_{\pm, x_j} \, = \, \Delta_{x_j} - \Delta_{x_j'}  \,.
\end{eqnarray}
Moreover, we write
\begin{eqnarray}
	\widehat{\Delta}_\pm \Gamma \, := \, ( \, \Delta^{(k)}_\pm \gamma^{(k)} \, )_{k\in{\mathbb N}}
\end{eqnarray}
and
\begin{eqnarray}
	\widehat{B} \Gamma \, := \, ( \, B_{k+1} \gamma^{(k+1)} \, )_{k\in{\mathbb N}} \,.
\end{eqnarray}
Then, the $p$-GP hierarchy \eqref{eq-def-b0-2} can be written as
\begin{eqnarray} \label{seclwp-pGP}
        i\partial_t \Gamma \, + \, \widehat{\Delta}_\pm \Gamma \, = \, \mu \widehat{B} \Gamma \,.
\end{eqnarray}
In integral formulation, it is formally given by
\begin{eqnarray} \label{int-pGP}
	\Gamma(t) \, = \, e^{it\widehat{\Delta}_\pm}\Gamma_0 \, - \, i \mu
        \int_0^t ds \, e^{i(t-s)\widehat{\Delta}_\pm} \widehat{B}\Gamma(s) \,.
\end{eqnarray}
We will prove the local existence and uniqueness of such a solution in the following manner.
We note that (\ref{int-pGP}) can be formally written as a system of integral equations
\begin{eqnarray}
		\Gamma(t) & = & e^{it\widehat{\Delta}_\pm}\Gamma_0 \, - \, i \mu
        \int_0^t ds \, e^{i(t-s)\widehat{\Delta}_\pm} \widehat{B}\Gamma(s)
        \label{int-pGP-sys-1}\\
        \widehat{B}\Gamma(t) & = & \widehat{B} \, e^{it\widehat{\Delta}_\pm}\Gamma_0 \, - \, i \mu
        \int_0^t ds \, \widehat{B} \, e^{i(t-s)\widehat{\Delta}_\pm} \widehat{B}\Gamma(s) \,,
        \label{int-pGP-sys-2}
\end{eqnarray}
where we note that the second line is formally a self-consistent fixed point equation for $\widehat{B}\Gamma$.

Let $I:=[0,T]$. We introduce the product space
\begin{eqnarray}\label{def-Wspace-1}
	{\mathfrak{W}_{\xi}^\alpha(I)} \, := \,
	L^\infty_{t\in I}{\mathcal H}_\xi^\alpha \, \times \,
	L^1_{t\in I}{\mathcal H}_\xi^\alpha \,  ,
\end{eqnarray}
which we endow with the norm
\begin{eqnarray}
	\| \, (\Gamma, \Xi) \, \|_{{\mathfrak{W}_{\xi}^\alpha(I)}} \, := \,
	\| \, \Gamma \, \|_{L^\infty_{t\in I}{\mathcal H}_\xi^\alpha}
	\, + \,
	\| \, \Xi \, \|_{L^1_{t\in I}{\mathcal H}_\xi^\alpha} \,.
\end{eqnarray}
Clearly, $( {\mathfrak{W}_{\xi}^\alpha(I)} , \| \, \cdot \, \|_{{\mathfrak{W}_{\xi}^\alpha(I)}} )$ is a Banach space.
Then, we introduce the system
\begin{eqnarray}
		\Gamma(t) & = & e^{it\widehat{\Delta}_\pm}\Gamma_0 \, - \, i \mu
        \int_0^t ds \, e^{i(t-s)\widehat{\Delta}_\pm} \Xi(s)
        \label{int-pGP-sys-Wspace-1} \\
        \Xi(t) & = & \widehat{B} \, e^{it\widehat{\Delta}_\pm}\Gamma_0 \, - \, i \mu
        \int_0^t ds \, \widehat{B} \, e^{i(t-s)\widehat{\Delta}_\pm} \Xi(s) \,,
        \label{int-pGP-sys-Wspace-2}
\end{eqnarray}
which is formally equivalent to the system (\ref{int-pGP-sys-1}), (\ref{int-pGP-sys-2}).

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

\begin{theorem}\label{thm-locwp-main-1}
Assume that $\alpha\in{\mathfrak A}(d,p)$ where $d\geq1$ and $p\in\{2,4\}$.
Then, the following holds. For every $\Gamma_0\in{\mathcal H}_{{\xi_1}}^\alpha$, 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 (\ref{int-pGP-sys-Wspace-1}), (\ref{int-pGP-sys-Wspace-2}), in the space ${\mathfrak{W}_{\xi_2}^\alpha(I)}$.
Moreover, the relation
\begin{eqnarray}
	\Xi(t)=\widehat{B}\Gamma(t)\in L^1_{t \in I}{\mathcal H}_{\xi_2}^\alpha \,,
\end{eqnarray}
holds for the solution, and the component $\Gamma(t)$ satisfies the
$p$-GP hierarchy (\ref{seclwp-pGP}),
\begin{eqnarray}\label{eq-GPhier-thm-1}
	i\partial_t\Gamma \, + \, \widehat{\Delta}_\pm\Gamma \, - \,
	\mu \, \widehat{B} \Gamma \, = \, 0\,,
\end{eqnarray}
with initial condition $\Gamma(0)=\Gamma_0\in{\mathcal H}_{\xi_1}^\alpha$.

Moreover, there exists a constant $C(T,d,p,\xi_1,\xi_2)$ such that
\begin{eqnarray}
	\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha} \, \leq \,
	C(T,d,p,\xi_1,\xi_2) \, \| \, \Gamma_0 \, \|_{{\mathcal H}_{\xi_1}^\alpha}
\end{eqnarray}
holds.
\end{theorem}

We note that our local existence and uniqueness result differs
than the one proven in \cite{esy1,esy2} in that performing the contraction principle on the space
${\mathfrak{W}_{\xi_2}^\alpha(I)}$ presumes finiteness of $\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}$.
However,
\footnote{We thank B. Schlein for calling our attention to this fact.}
it is a priori conceivable that there exist solutions of the $p$-GP hierarchy
$\Gamma(t)\in {\mathcal H}_{\xi_2}^\alpha$ for which $\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}$
is not finite.
We note that finiteness of $\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}$ is an
essential element of this analysis, and has been assumed without proof in \cite{klma} as a key
ingredient.

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

\begin{theorem}\label{thm-locwp-main-1-L1bd}
Assume that $\alpha\in{\mathfrak A}(d,p)$ where $d\geq1$ and $p\in\{2,4\}$.
Assume moreover that for $\Gamma_0\in{\mathcal H}_{{\xi_1}}^\alpha$ 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 (\ref{seclwp-pGP})
in the space $L^\infty_{t\in I}{\mathcal H}_{\xi_2}^\alpha$.

Then, the solution $\Gamma(t)\in L^\infty_{t\in I}{\mathcal H}_{\xi_2}^\alpha$
is {\em unique} if and only if
$\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi}^\alpha}<\infty$
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
\begin{eqnarray}\label{eq-Strichartz-Wspace-1-1}
	\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha} \, \leq \,
	C(T,d,p,\xi_1,\xi_2) \, \| \, \Gamma_0 \, \|_{{\mathcal H}_{\xi_1}^\alpha}
\end{eqnarray}
is satisfied.
\end{theorem}

We first prove Theorem \ref{thm-locwp-main-1}.
\begin{proof}
We make the key observation that the fixed point equation
(\ref{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 $L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha$ (the second factor of the product space
${\mathfrak{W}_{\xi_2}^\alpha(I)}$) to first
find a unique $\Xi\in L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha$ solving (\ref{int-pGP-sys-Wspace-2}).

\noindent\underline{\em (1) Existence and uniqueness of the component $\Xi(t)$}

We recall that the fixed point equation for $\Xi(t)$ is given by
\begin{eqnarray}\label{eq-opBGamma-fixedpt-1}
	\Xi(t) \, = \, \widehat{B} e^{it\widehat{\Delta}_\pm} \Gamma_0 \, - \, i
	\mu \, \int_0^t \, ds \, \widehat{B} e^{i(t-s)\widehat{\Delta}_\pm}\Xi(s)\,,
\end{eqnarray}
in the space $L^1{\mathcal H}_{\xi_2}^\alpha$.

We define, for an admissible sequence of density matrices
$\widetilde\Gamma=(\widetilde\gamma^{(k)})_{k\in{\mathbb N}}$,
\begin{eqnarray}\label{eq-Duh-j-def-1}
	\lefteqn{
	{\rm Duh}_j(\widetilde\Gamma)^{(k+1)}(t)
	}
	\nonumber\\
	& := & (-i\mu)^j\int_0^t dt_1 \cdots \int_0^{t_{j-1}}dt_j
	e^{i(t-t_1)\Delta_\pm^{(k+1)}}B_{k+2}e^{i(t_1-t_2)\Delta_\pm^{(k+2)}}
	\nonumber\\
	&&\quad\quad\quad\quad\quad\quad
	B_{k+3} \cdots
	\cdots B_{k+j+1} e^{i t_j \Delta_\pm^{(k+j+1)}} \widetilde\gamma^{(k+j+1)}(t_j) \,.
\end{eqnarray}
Then, any solution of (\ref{eq-opBGamma-fixedpt-1}) satisfies the fixed point equation
(obtained from iterating the Duhamel formula $k$ times for the $k$-th component of $\widehat{B}\Gamma$)
\begin{eqnarray}
	(\widehat{B}\Gamma)^{(k)}(t) \, = \, \sum_{j=1}^{k-1}B_{k+1}{\rm Duh}_j(\Gamma_0)^{(k+1)}(t)
	\, + \, B_{k+1}{\rm Duh}_{k}(\widehat{B}\Gamma)^{(k+1)}(t) \,.
\end{eqnarray}
To formulate a Picard-type fixed point argument, we define
\begin{eqnarray}
	\Phi(\Xi)=( \Phi(\Xi)^{(k)})_{k\in{\mathbb N}}
\end{eqnarray}
where the $k$-th component is given by
\begin{eqnarray}\label{eq-PhiGamma-fixed-pt}
	\Phi(\Gamma)^{(k)}(t) \, = \, \sum_{j=1}^{k-1}B_{k+1}{\rm Duh}_j(\Gamma_0)^{(k+1)}(t)
	\, + \, B_{k+1}{\rm Duh}_{k}(\Xi)^{(k+1)}(t) \,.
\end{eqnarray}
Similarly as in \cite{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 \cite{chpa}
where we refer for details.

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

We first consider the case $d\geq2$.

In the sequel, we will often abbreviate
\begin{eqnarray}
	H^\alpha_k \, := \, H^\alpha({\mathbb R}^{dk}\times {\mathbb R}^{dk}) \,.
\end{eqnarray}
We invoke Propositions \ref{prp-spacetime-bd-1} and
\ref{prp-spacetime-bd-2} in the Appendix.
They generalize the  spacetime bounds and ``board game"
arguments developed in \cite{klma}.
Proposition \ref{prp-spacetime-bd-2}  implies that for $\Gamma_0=(\gamma_0^{(k)})_{k\in{\mathbb N}}$,
\begin{eqnarray}
	\lefteqn{
	\| \, \sum_{j=1}^{k-1}B_{k+1}{\rm Duh}_j(\Gamma_0)^{(k+1)}(t) \, \|_{L^1_{t\in I}H^\alpha_k}
	}
	\nonumber\\
	& < &  k C^k \sum_{j=1}^{k-1} (c T)^{\frac {(j+1)}2} \|\gamma^{(k+j+1)}_0\|_{H^\alpha_{k+j+1}}
	\\
	& < & k  (C \xi_1^{-1})^{k} \sum_{j=1}^{k-1} (c T \xi_1^{-2})^{\frac {(j+1)}2}
	\xi_1^{k+j+1} \|\gamma^{(k+j+1)}_0\|_{H^\alpha_{k+j+1}}
	\\
	& < & (c T \xi_1^{-2})  k  (C \xi_1^{-1})^{k} \sum_{j=1}^{k-1}  \xi_1^{k+j+1}
	\|\gamma^{(k+j+1)}_0\|_{H^\alpha_{k+j+1}} \,.
	\label{eq-Duhexp-aux-1}
\end{eqnarray}
Therefore,
\begin{eqnarray}
	\lefteqn{
	\sum_{k=1}^\infty \xi_2^{k}\| \, \sum_{j=1}^{k-1}B_{k+1}{\rm Duh}_j(\Gamma_0)^{(k+1)}(t) \,
	\|_{L^1_{t\in I}H^\alpha_{k}}
	}
	\nonumber\\
	& < & (c T \xi_1^{-2})
	\sum_{k=1}^\infty k  \Big(C \frac{\xi_2}{\xi_1}\Big)^k \sum_{j=1}^{k-1}  \xi_1^{k+j+1}
	\|\gamma^{(k+j+1)}_0\|_{H^\alpha_{k+j+1}}
	\\
	& < & (c T \xi_1^{-2})
	\sum_{k=1}^\infty k \Big(C \frac{\xi_2}{\xi_1}\Big)^k
	\sum_{\ell=1}^{2k}  \xi_1^{\ell} \|\gamma^{(\ell)}_0\|_{H^\alpha_\ell}
	\\
	& < & (c T \xi_1^{-2})
	\sum_{k=1}^\infty k \big(C \eta \big)^k
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha}
	\\
	& < & (c T \xi_1^{-2})
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha}
	\label{eq-Duhexp-aux-2}
\end{eqnarray}
for $\xi_2=\eta\xi_1$, with $0<\eta\leq1$ sufficiently small.

This implies that, for $I=[0,T]$, and any $T>0$,
\begin{eqnarray}
	\Big( \, \sum_{j=1}^{k-1}B_{k}{\rm Duh}_j(\Gamma_0)^{(k+1)}(t) \, \Big)_{k\in{\mathbb N}}
	\, \in \, L^1_{t\in I}{\mathcal H}^\alpha_{\xi_2} \,,
\end{eqnarray}
provided that $\Gamma_0\in{\mathcal H}_{\xi_1}^\alpha$, and $\xi_2=\eta\xi_1$ with $\eta>0$ sufficiently small.

Our next step is to prove that $\Phi$ is a contraction on $L^1_{t\in I}{\mathcal H}^\alpha_{\xi_2}$.
To this end, we use the bound
\begin{eqnarray}
	\|\Phi(\Xi_1)^{(k)} -\Phi(\Xi_2)^{(k)} \|_{L_{t\in I}^1H^\alpha_k}
	\, \leq \, k \, (C T)^{\frac{k}{2}} \| \Xi_1^{(2k)} - \Xi_2^{(2k)} \|_{L^1_{t\in I}H^\alpha_{2k}}
\end{eqnarray}
which is proven in Proposition \ref{prp-spacetime-bd-2} in the Appendix.

We obtain
\begin{eqnarray}\label{eq-PhiGamma-contr-1}
	\lefteqn{
	\|\Phi(\Xi_1)-\Phi(\Xi_2)\|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}
	}
	\nonumber\\
	& =  &
	\sum_{k=1}^\infty\xi_2^k
	\|\Phi(\Xi_1)^{(k)} -\Phi(\Xi_2)^{(k)} \|_{L_{t\in I}^1H^\alpha_k}
	\nonumber\\
	&\leq&
	\sum_{k=1}^\infty k \, (C T \xi_2^{-2})^{\frac{k}{2}}
	\xi_2^{2k} \, \| \Xi_1^{(2k)} - \Xi_2^{(2k)} \|_{L^1_{t\in I}H^\alpha_{2k}}
	\nonumber\\
	&\leq&
	\sup_{k\geq1}\{ \, k \, (C T \xi_2^{-2})^{\frac{k}{2}} \, \} \, \sum_{k=1}^\infty
	\xi_2^{2k} \, \| \Xi_1^{(2k)} - \Xi_2^{(2k)} \|_{L^1_{t\in I}H^\alpha_{2k}}
	\nonumber\\
	&\leq&
	( C T \xi_2^{-2})^{\frac12} \,
	\| \Xi_1 - \Xi_2 \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}\,,
\end{eqnarray}
for $T>0$ sufficiently small.
This implies that $\Phi$ is a contraction on $L^1_{t\in I}{\mathcal H}^\alpha_{\xi_2}$
provided that $T$ is sufficiently small.

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

Repeating the above arguments, we find that this solution satisfies
\begin{eqnarray}\label{eq-Strichartz-cH-1}
	\| \, \Xi  \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}
	&\leq&(c T \xi_1^{-2})
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha}
	\, + \, ( C T \xi_2^{-2} )^{\frac12} \,
	\| \, \Xi \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha} \,.
\end{eqnarray}
Hence, in particular,
\begin{eqnarray}\label{eqn-BGamma-spacetime-bd-1}
	\| \, \Xi \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}}
	&\leq&\frac{c T \xi_1^{-2} }{1-(c T \xi_2^{-2} )^{\frac 12}}
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha} \,
\end{eqnarray}
for sufficiently small $T>0$.

\noindent\underline{\em (2) Determination and uniqueness of the component $\Gamma(t)$}

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

Accordingly,  $\Gamma(t)$ is given by
\begin{eqnarray}\label{int-pGP-sys-Wspace-1-2}
	\Gamma(t) \, = \, e^{it\widehat{\Delta}_\pm}\Gamma_0 \, - \, i \mu \int_0^t ds \, e^{i(t-s)\widehat{\Delta}_\pm}
	\Xi(s) \,
\end{eqnarray}
where the solution $\Xi(s)$ found in step {\em (1)} above has been substituted into the integral. Evidently,
\begin{eqnarray}\label{eq-Gamma-Duh-opBGamma-1}
	\| \Gamma(t) \|_{ {\mathcal H}_{\xi_2}^\alpha}
	& \leq & \| \Gamma_0 \|_{{\mathcal H}_{\xi_2}^\alpha}
	\, + \, \|\int_0^t  ds \,   e^{i(t-s)\widehat{\Delta}_\pm}\Xi (s)\|_{{\mathcal H}_{\xi_2}^\alpha}
	\nonumber\\
	& \leq & \| \Gamma_0 \|_{{\mathcal H}_{\xi_2}^\alpha}
	\, + \, \int_0^t  ds \, \| \Xi (s)\|_{{\mathcal H}_{\xi_2}^\alpha}
	\nonumber\\
	& \leq & \| \Gamma_0 \|_{{\mathcal H}_{\xi_2}^\alpha}
	\, + \, \| \Xi(s) \|_{L^1_{s\in I}{\mathcal H}_{\xi_2}^\alpha} \,,
\end{eqnarray}
using the unitarity of $e^{i(t-s)\widehat{\Delta}_\pm}$ with respect to ${\mathcal H}_{\xi_2}^\alpha$.
By (\ref{eqn-BGamma-spacetime-bd-1}), we conclude that the last line is bounded,
hence $\Gamma(t)$ determined by (\ref{int-pGP-sys-Wspace-1-2})
indeed lies in $L^\infty_{t\in I}{\mathcal H}_{\xi_2}^\alpha$.

To prove uniqueness, we assume that $(\Gamma_1(t),\Xi_1(t))$ and $(\Gamma_2(t),\Xi_2(t))\in{\mathfrak{W}_{\xi_2}^\alpha(I)}$
correspond to solutions of the system (\ref{int-pGP-sys-Wspace-1}), (\ref{int-pGP-sys-Wspace-2}) to
the same initial condition $\Gamma_1(0)=\Gamma_2(0)\in{\mathcal H}_{\xi_1}^\alpha$, for $t\in[0,T]$.
Then, linearity implies that
$(\Gamma_1(t)-\Gamma_2(t),\Xi_1(t))-\Xi_2(t))\in{\mathfrak{W}_{\xi_2}^\alpha(I)}$
solves the system (\ref{int-pGP-sys-Wspace-1}), (\ref{int-pGP-sys-Wspace-2})
with initial condition $\Gamma_1(0)-\Gamma_2(0)=(0)_{k\in{\mathbb N}}$.
Hence, (\ref{eq-Gamma-Duh-opBGamma-1}) implies that
\begin{eqnarray}
	\| \Gamma_1(t)-\Gamma_2(t) \|_{ {\mathcal H}_{\xi_2}^\alpha}
	& \leq & \Big( \, 1 \,+ \, \frac{c T \xi_1^{-2}}{1-(c T \xi_2^{-2})^{\frac 12}} \, \Big) \,
	\|\Gamma_1(0)-\Gamma_2(0)\|_{{\mathcal H}_{\xi_1}^\alpha}
	\nonumber\\
	&=&0\,.
\end{eqnarray}
This implies that $\Gamma_1(t)=\Gamma_2(t)$, for all $t<T$.
This proves local existence and uniqueness of solutions for the
system (\ref{int-pGP-sys-Wspace-1}), (\ref{int-pGP-sys-Wspace-2}).

Next, we observe that applying $\widehat{B}$ to $\Gamma(t)$ produces the rhs of
the fixed point equation (\ref{eq-opBGamma-fixedpt-1}) for $\Xi(t)$. Therefore,
we immediately find that
\begin{eqnarray}\label{eq-XiopBGammadiff-1}
	\Xi \, = \,  \widehat{B} \Gamma  \; \; \; \in \, L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha \,.
\end{eqnarray}
Finally, it is easy to verify for the unique solution
$(\Gamma(t),\Xi(t))\in{\mathfrak{W}_{\xi_2}^\alpha(I)}$ of  (\ref{int-pGP-sys-Wspace-1}), (\ref{int-pGP-sys-Wspace-2}) that
\begin{eqnarray}
	i\partial_t \Gamma(t) \, + \, \widehat{\Delta}_\pm \Gamma(t) \, = \, \mu \, \Xi(t) \,.
\end{eqnarray}
Therefore, using (\ref{eq-XiopBGammadiff-1}),
\begin{eqnarray}
	\| \, i\partial_t \Gamma(t) \, + \, \widehat{\Delta}_\pm \Gamma(t)
	\, - \, \mu \, \widehat{B} \Gamma(t) \, \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha} \, = \, 0 \,,
\end{eqnarray}
which implies (\ref{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 (\ref{eq-Gamma-Duh-opBGamma-1}) is replaced by
\begin{eqnarray}\label{eq-Gamma-Duh-opBGamma-quintic-1}
	\| \Gamma(t) \|_{ {\mathcal H}_{\xi_2}^\alpha} & \leq & \| \Gamma_0 \|_{{\mathcal H}_{\xi_2}^\alpha}
	\, + \,  \| \widehat{B}\Gamma \|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^\alpha}
	\nonumber\\
	 & \leq & \Big( \, 1 \,+ \, \frac{c T \xi_1^{-4}}{1-(c T \xi_2^{-4})^{\frac 12}} \, \Big) \,
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha} \,.
\end{eqnarray}
This concluded the proof of Theorem \ref{thm-locwp-main-1} for $d\geq2$.

In the case $d=1$, we can straightforwardly adapt the proof given in \cite{chpa}
of the uniqueness of solutions of the quintic GP hierarchy
in $d=1$. The spacetime bounds of Proposition \ref{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 \cite{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
\begin{eqnarray}\label{eq-Gamma-Duh-opBGamma-1D-1}
	\| \Gamma(t) \|_{ {\mathcal H}_{\xi_2}^\alpha} & \leq &
	\Big( \, 1 \,+ \, \frac{c T^2 \xi_1^{-2}}{1-c T \xi_2^{-1} } \, \Big) \,
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha}
\end{eqnarray}
for the cubic GP hierarchy instead of (\ref{eq-Gamma-Duh-opBGamma-1}), and
\begin{eqnarray}\label{eq-Gamma-Duh-opBGamma-quintic-1D-1}
	\| \Gamma(t) \|_{ {\mathcal H}_{\xi_2}^\alpha}
	 & \leq & \Big( \, 1 \,+ \, \frac{c T^2 \xi_1^{-4}}{1- c T  \xi_2^{-2} } \, \Big) \,
	\|\Gamma_0\|_{{\mathcal H}_{\xi_1}^\alpha}
\end{eqnarray}
for the quintic GP hierarchy instead of (\ref{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 (\ref{seclwp-pGP}) in
the space ${\mathfrak{W}_{\xi_2}^\alpha(I)}$ defined in (\ref{def-Wspace-1}).
\endprf

It is now straightforward to prove Theorem \ref{thm-locwp-main-1-L1bd}.

\begin{proof}
$"\Rightarrow"$. We first prove that uniqueness of $\Gamma\in L^\infty_{t\in I}{\mathcal H}_{\xi_2}^\alpha$
implies  $\| \widehat{B}\Gamma \|_{L^1_{t \in I} {\mathcal H}^\alpha_{\xi}}<\infty$ for
some $\xi>0$.
Let $t'\in I_1=[0,T_1]$.
Assume $\Gamma(t+t')$ is the unique solution in ${\mathcal H}_{\xi_2}^1$
of the $p$-GP hierarchy for the initial condition $\Gamma(t)\in {\mathcal H}_{\xi_1}^\alpha$,
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
(\ref{int-pGP-sys-Wspace-1}), (\ref{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  \ref{thm-locwp-main-1}, we infer that $\Gamma'(t+t')$ is also an
element of ${\mathcal H}_{\xi_2}^\alpha$, 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 ${\mathcal H}_{\xi_2}^\alpha$, 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 ${\mathfrak{W}_{\xi_2}^\alpha(I)}$  for the system
(\ref{int-pGP-sys-Wspace-1}), (\ref{int-pGP-sys-Wspace-2}) has the property that
$\| \widehat{B}\Gamma'(t + . ) \|_{L^1_{t'\in I} {\mathcal H}^\alpha_{\xi_2}}
\leq C(T,d,p,\xi_1,\xi_2) \, \| \Gamma(t) \|_{{\mathcal H}_{\xi_1}^\alpha}$
is bounded, we conclude that also,
$\| \widehat{B}\Gamma(t + . ) \|_{L^1_{t'\in I} {\mathcal H}^\alpha_{\xi_2}}
\leq C(T,d,p,\xi_1,\xi_2) \, \| \Gamma(t) \|_{{\mathcal H}_{\xi_1}^\alpha}$.

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

We note that since
$\Gamma\in L^\infty_{t\in I}{\mathcal H}_{\xi_2}^\alpha$ and
$\| \widehat{B}\Gamma \|_{L^1_{t \in I} {\mathcal H}^\alpha_{\xi}}<\infty$, it is clear that
$(\Gamma,\widehat{B}\Gamma)\in{\mathfrak{W}_{\xi_2}^\alpha(I)}$. Accordingly, Theorem \ref{thm-locwp-main-1}
implies that $\Gamma$ is the unique solution in ${\mathfrak{W}_{\xi_2}^\alpha(I)}$ for the
initial condition $\Gamma(0)=\Gamma_0\in{\mathcal H}_{\xi_1}^\alpha$ for which
$\| \widehat{B}\Gamma \|_{L^1_{t \in I} {\mathcal H}^\alpha_{\xi}}<\infty$ is satisfied.

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

$\;$\\ \\

\section{On the global existence and uniqueness of solutions for defocusing GP hierarchies}
\label{sec-gwp-GP-1}

It is proved for the $d=2,3$ defocusing cubic case in \cite{esy1,esy2,kiscst},
and for the $d=1,2$ defocusing quintic
case in \cite{chpa}, that whenever $\Gamma(t)=(\gamma^{(k)}(t))_{k\in{\mathbb N}}$ is  obtained from the
$N\rightarrow\infty$ limit of a bosonic $N$-particle BBGKY hierarchy with
repulsive interactions,
\begin{eqnarray}\label{eq-GPsol-defoc-apriorien-1}
	{\rm Tr} \big( \, S^{(k,1)} \, \gamma^{(k)}_0 \, \big) \, < \, C^k
\end{eqnarray}
for some constant $C$, then
\begin{eqnarray}\label{eq-GPsol-defoc-apriorien-2}
	{\rm Tr} \big( \, S^{(k,1)} \, \gamma^{(k)}(t) \, \big) \, < \, C_0^k
\end{eqnarray}
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 $\Gamma(t)\in{\mathcal H}_{\xi_1}^1$ for some $\xi_1>0$,
and all $t\in{\mathbb R}$.

To see this, we consider a fixed $k$. Let $\gamma^{(k)}$ be non-negative,
normalized trace class,
${\rm Tr}(\gamma^{(k)})=1$, and hermitean.
Then, we have that
\begin{eqnarray}
	\gamma^{(k)}({\underline{x}}_k;{\underline{x}}_k') \, = \, \sum_j \lambda_j \,
	|\psi_j({\underline{x}}_k')\rangle\langle\psi_j({\underline{x}}_k)|
\end{eqnarray}
for an orthogonal basis $\psi_j$ of $L^2({\mathbb R}^{dk})$ with $\lambda_j\geq0$ and $\sum\lambda_j=1$.
Then,
\begin{eqnarray}
	\| \, \gamma^{(k)} \, \|_{H^1}
	& = & \sum_{j,j'} \lambda_j \, \lambda_{j'}
	\big|\big\langle \, \langle\nabla_{{\underline{x}}_k}\rangle\psi_j \,
	\big| \,  \langle\nabla_{{\underline{x}}_k}\rangle\psi_{j'} \, \big\rangle \, \big|^2
	\\
	&\leq&
	\Big(\sum_j \lambda_j \,
	\big\| \, \langle\nabla_{{\underline{x}}_k}\rangle\psi_j \, \big\|^2 \, \Big)^2
	\\
	&=&
	\Big( \, {\rm Tr} \big( \, S^{(k,1)} \, \gamma^{(k)} \, \big) \, \Big)^2 \,.
\end{eqnarray}
Thus, for a solution $\gamma^{(k)}(t)$ of the (cubic or quintic) GP hierarchy
with initial condition satisfying (\ref{eq-GPsol-defoc-apriorien-1}), we have that
\begin{eqnarray}
	\| \, \gamma^{(k)}(t) \, \|_{H^1}
	\, < \, C_0^k
\end{eqnarray}
with $C_0$ independent of $t$.
Thus, for $\xi_1$ sufficiently small, we obtain the a priori bound
\begin{eqnarray}
	\|\Gamma(t)\|_{{\mathcal H}_{\xi_1}^1}
	\, \leq \, \Big( \, \sum_{k=1}^\infty (C_0\xi)^{ 2k} \, \Big)^{\frac12} \, < \, \infty \, ,
\end{eqnarray}
for all $t\in{\mathbb R}$.

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

\begin{theorem}\label{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 $1 \in{\mathfrak A}(d,p)$.
Assume that $\Gamma(t)\in{\mathcal H}_{\xi_1}^1$,  $t\in{\mathbb R}$,
is a solution of the defocusing $p$-GP hierarchy for the initial condition
$\Gamma(0)=\Gamma_0$, and
satisfies the a priori bound $\|\Gamma(t)\|_{{\mathcal H}_{\xi_1}^1}<c_0<\infty$
for $t\in{\mathbb R}$.
Moreover, assume that  for every
sufficiently short open interval $I\in{\mathbb R}$,
there exists a constant $\xi=\xi(I)$ with $0<\xi\leq \xi_1$ such that
$\|\widehat{B}\Gamma\|_{L^1_{t\in I}{\mathcal H}_{\xi}^1}<\infty$.

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 $(\Gamma,\widehat{B}\Gamma)$ is the unique solution
of the system  (\ref{int-pGP-sys-Wspace-1}), (\ref{int-pGP-sys-Wspace-2})
associated to the defocusing ($\mu=+1$) $p$-GP hierarchy with initial condition
$\Gamma(0)=\Gamma_0\in{\mathcal H}_{\xi_1}^1$, in the space
\begin{eqnarray}
	{\widetilde{\mathfrak{W}}_{\xi_2,T_0}^1} \, := \, \bigcup_{I\subset {\mathbb R},|I|<T_0}{\mathfrak{W}_{\xi_2}^1(I)} \,,
\end{eqnarray}
for $0<\xi_2\leq\xi_1$ sufficiently small.
Hence in particular,
\begin{eqnarray}
	\|\widehat{B}\Gamma\|_{L^1_{t\in I}{\mathcal H}_{\xi_2}^1} \, \leq \, C(T_0,d,p,\xi_1,\xi_2) \, c_0
\end{eqnarray}
for all $I\in{\mathbb R}$ with $|I|<T_0$.
\end{theorem}

\begin{proof}
Assume that $I'\subset{\mathbb R}$ is such that $\|\widehat{B}\Gamma\|_{L^1_{t\in I'}{\mathcal H}_{\xi}^1}<\infty$.
Let $I:=[\tau,T]\subset I'$
Then,  Theorem \ref{thm-locwp-main-1-L1bd} implies that in fact, there exists a constant $\xi_2\leq\xi_1$
such that
\begin{eqnarray}
	\|\widehat{B}\Gamma\|_{L^1_{t\in I }{\mathcal H}_{\xi_2}^1} \, < \, C_0 \, \| \, \Gamma(\tau) \, \|_{{\mathcal H}_{\xi_1}^1}
	\, < \, C_0 \, c_0 \,,
\end{eqnarray}
for a constant $C_0=C_0(|I|,d,p,\xi_1,\xi_2)$.

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

Assume that the initial time $t=0\in I_0$. Then,  Theorem \ref{thm-locwp-main-1-L1bd} implies that
there exists a unique (forward and backward in time) solution  $(\Gamma,\widehat{B}\Gamma)$
of the system  (\ref{int-pGP-sys-Wspace-1}), (\ref{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 ${\mathbb R}=\cup I_j$.

In conclusion, we obtain that there exists a unique global in time solution
in  ${\widetilde{\mathfrak{W}}_{\xi_2,T_0}^1}$ with initial condition $\Gamma_0\in{\mathcal H}_{\xi_1}^1$,
under the assumptions of the theorem.
\endprf

\begin{remark}
As noted in Remark \ref{rem-BBGKYlim-1},
every solution $\Gamma(t)\in{\mathcal H}_{\xi_1}^1$ extracted by a diagonal argument
from the $N\rightarrow\infty$ limit of the $N$-particle BBGKY hierarchies
(with repulsive interactions)
studied by Erd\"os-Schlein-Yau in \cite{esy1,esy2}, Kirkpatrick-Schlein-Staffilani in
\cite{kiscst}, and Chen-Pavlovi\'c in \cite{chpa}, satisfies
$\| \, \widehat{B} \Gamma \, \|_{L^1_{t\in I}{\mathcal H}_{\xi}^\alpha}<\infty$
for some $\xi>0$, and all sufficiently short intervals $I\subset{\mathbb R}$.
This follows a posteriori from the uniqueness of those solutions,
which has been established in these
works with independent methods, combined with Theorem \ref{thm-locwp-main-1-L1bd} in the work at hand.
Accordingly, $(\Gamma,\widehat{B}\Gamma)$
is automatically an element of ${\widetilde{\mathfrak{W}}_{\xi_2,T_0}^1}$ whenever  $\Gamma(t)$ is obtained from an $N\rightarrow\infty$
limit of the above noted type.
\end{remark}

$\;$ \\ \\

\section{Lower bound on the blowup rates}
\label{sec-Proof-blowuprate-1}

In this section, we establish Theorem \ref{thm-blowuprate-L2crit-1}.
We adapt a standard proof given for $L^2$-critical focusing NLS to
the GP hierarchy; see for instance \cite{ra}.
Let $p \in \{2,4\}$.
Similarly as in (\ref{eq-pseudoconf-gamma-1}), one finds that
the $p$-GP hierarchy is invariant under the rescaling
\begin{eqnarray}\label{eq-pseudoconf-gamma-2}
	\lefteqn{
	{\mathcal R}_{\lambda,t}\gamma^{(k)} (\tau,{\underline{x}}_k;{\underline{x}}_k')
	}
	\nonumber\\
	&&
	\, := \,
	\frac{1}{(\lambda(t))^{4k/p}} \,
	\gamma^{(k)}\Big( \, t +(\lambda(t))^{-2}\tau \, , \, (\lambda(t))^{-1}{\underline{x}}_k \, ; \,
	(\lambda(t))^{-1}{\underline{x}}_k' \, \Big) \,
\end{eqnarray}
If $\Gamma(t)=(\gamma^{(k)}(t))_{k\in{\mathbb N}}$
solves the $p$-GP hierarchy, then
${\mathcal R}_{\lambda,t}\Gamma=({\mathcal R}_{\lambda,t}\gamma^{(k)} )_{k\in{\mathbb N}}$
is also a solution of the $p$-GP hierarchy.
The proof can be straightforwardly adapted from the one given in Section \ref{sec-confinv-1}.

\noindent\underline{\em Proof of statement (a).}
\\
\\
Blowup in finite time means that there exists $T^*<\infty$ such that
${\rm Av}_{H^\alpha}(\Gamma(t))\rightarrow\infty$
as $t\rightarrow T^*$.
To prove a lower bound on the blowup rate, we may assume that
$1<{\rm Av}_{H^\alpha}(\Gamma(t))<\infty$ at a fixed time $t$, and choose
\begin{eqnarray}
	\lambda(t) \, = \, ({\rm Av}_{H^\alpha}(\Gamma(t)))^{\frac{1}{2\alpha-d+\frac4p}} \, > \, 1 \,.
\end{eqnarray}
We note that $\frac{4}{d}<p<\frac{4}{d-2\alpha}$  implies that $2\alpha-d+\frac4p>0$.
Let
\begin{eqnarray}
	S^{(k,\alpha)}_{\lambda(t)}
	\, := \, \prod_{j=1}^k\langle \, (\lambda(t))^{-1} \, \nabla_{x_j}\rangle^\alpha \,
	\langle \, (\lambda(t))^{-1} \, \nabla_{x_j'} \, \rangle^\alpha
\end{eqnarray}
where $\langle b\nabla_{x}\rangle=\sqrt{1-b^2\Delta_{x}}$ for any $b\in{\mathbb R}$.
Clearly,
\begin{eqnarray}\label{eq-Sdot-quasiblowup-L2-aux-1}
	\lefteqn{
	\Big\| \, S^{(k,\alpha)} {\mathcal R}_{\lambda,t}\gamma^{(k)} (\tau) \, \Big\|_{L^2_{{\underline{x}}_k,{\underline{x}}_k'}}
	}
	\nonumber\\
	& = &  (\lambda(t))^{k(d-\frac{4}{p})} \, \| \, (\, S^{(k,\alpha)}_{\lambda(t)}
	\, \gamma^{(k)} \, ) (t+(\lambda(t))^{-2}\tau)\, \|_{L^2_{{\underline{x}}_k,{\underline{x}}_k'}} \,,
\end{eqnarray}
and
\begin{eqnarray}\label{eq-scaleSkalpha-bd-1}
	(\lambda(t))^{-2\alpha k} S^{(k,\alpha)} \, \leq \,
	S^{(k,\alpha)}_{\lambda(t)}
	\, \leq \, S^{(k,\alpha)}
\end{eqnarray}
since we are assuming that $\lambda(t)>1$.

We define
\begin{eqnarray}
	\xi_<(\xi,t,\lambda) \, := \, \xi \, (\lambda(t))^{-\frac4p+d-2\alpha }
	\, = \, \xi \, ({\rm Av}_{H^\alpha}(\Gamma(t)))^{-1}
\end{eqnarray}
and
\begin{eqnarray}\label{eq-xibigxi-id-1}
	\xi_>(\xi,t,\lambda) \, := \, \xi (\lambda(t))^{-\frac4p+d }
	\,  = \, \xi  ({\rm Av}_{H^\alpha}(\Gamma(t)))^{\frac{d-\frac4p}{2\alpha-d+\frac4p} }  \,.
\end{eqnarray}
Clearly, (\ref{eq-Sdot-quasiblowup-L2-aux-1}) and (\ref{eq-scaleSkalpha-bd-1}) imply that
\begin{eqnarray}\label{eq-sandwich-bd-1}

	\| \, \Gamma(t+(\lambda(t))^{-2}\tau) \, \|_{{\mathcal H}_{\xi_<(\xi,t,\lambda)}^\alpha}

	& \leq &
	\|\, {\mathcal R}_{\lambda,t}\Gamma (\tau) \, \|_{ {\mathcal H}^\alpha_{\xi}}

	 \\
	&\leq&

	\| \, \Gamma(t+(\lambda(t))^{-2}\tau) \, \|_{{\mathcal H}_{\xi_>(\xi,t,\lambda)}^\alpha} \,.
	\nonumber
\end{eqnarray}
As a consequence of the definition of ${\rm Av}_{H^\alpha}(\Gamma(t))$,
it follows that for $\tau=0$,
\begin{eqnarray}
	0 \, < \, \| \, \Gamma(t  ) \, \|_{{\mathcal H}_{\xi_<(\xi,t,\lambda)}^\alpha} \, < \, c
\end{eqnarray}
for any $0<\xi<1$.

To ensure that $\|  \Gamma(t+(\lambda(t))^{-2}\tau)  \|_{{\mathcal H}_{\xi_>(\xi,t,\lambda)}^\alpha}<c$,
we use the fact that according to (\ref{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 (\ref{eq-xibigxi-id-1}) is positive.
If blowup occurs, such that ${\rm Av}_{H^\alpha}(\Gamma(t))\rightarrow\infty$ 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,
\begin{eqnarray}
	\|\, {\mathcal R}_{\lambda,t}\Gamma (0) \, \|_{ {\mathcal H}^\alpha_{\xi_1}}
	\, \leq \,
	\| \, \Gamma(t ) \, \|_{{\mathcal H}_{\xi_>(\xi_1,t,\lambda)}^\alpha}
\end{eqnarray}
Due to Theorem  \ref{thm-locwp-main-1}, we may   pick $0<\xi_2=\eta\xi_1\ll\xi_1<1$,
such that there exists a solution
${\mathcal R}_{\lambda,t}\gamma^{(k)} (\tau)\in{\mathcal H}_{\xi_2}^\alpha$ if
$\tau\in[0,\tau_{\max}]$, for $\tau_{\max}>0$ sufficiently small.

But this implies that
\begin{eqnarray}
	\| \, \Gamma(t+(\lambda(t))^{-2}\tau) \, \|_{{\mathcal H}_{\xi_<(\xi_2,t,\lambda)}^\alpha}
	\, \leq \,
	\|\, {\mathcal R}_{\lambda,t}\Gamma (\tau) \, \|_{ {\mathcal H}^\alpha_{\xi_2}} \, < \, \infty
\end{eqnarray}
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
\begin{eqnarray}
	T^* \, > \, t+(\lambda(t))^{-2}\tau_{\max} \,,
\end{eqnarray}
and hence,
\begin{eqnarray}
	( \, {\rm Av}_{H^\alpha}(\Gamma(t)) \, )^{\frac12} \, = \,
        \lambda(t)^{(\alpha-\frac{d}{2}+\frac{2}{p})}
	\, > \, \frac{C}{|T^*-t|^{(2\alpha-d+\frac4p )/4}} \,.
\end{eqnarray}
This proves ($a$).

\noindent\underline{\em Proof of statement (b).}
\\
\\
It is easy to see that
\begin{eqnarray}\label{eq-Lp-rescale}
	\Big\| \, {\mathcal R}_{\lambda,t}\gamma^{(k)} (\tau) \, \Big\|_{L^r_{{\underline{x}}_k,{\underline{x}}_k'}}
	\, = \, (\lambda(t))^{-2k(\frac2p-\frac{d}{r})} \,
	\| \,  \gamma^{(k)} (t+(\lambda(t))^{-2}\tau)\, \|_{L^r_{{\underline{x}}_k,{\underline{x}}_k'}} \,,
\end{eqnarray}
which, in turn, implies that
\begin{eqnarray}
	\|\, {\mathcal R}_{\lambda,t}\gamma^{(k)} (0) \, \|_{{\mathcal L}^r_\xi}
	&=&
	\sum_{k\geq1}\xi^k\Big\| \, {\mathcal R}_{\lambda,t}\gamma^{(k)} (0) \, \Big\|_{L^r_{{\underline{x}}_k,{\underline{x}}_k'}}
	\nonumber\\
	& = & \sum_{k\geq1} \xi^k \, (\lambda(t))^{-2k(\frac2p-\frac{d}{r})} \,
	\| \,  \gamma^{(k)} (t)\, \|_{L^r_{{\underline{x}}_k,{\underline{x}}_k'}}
	\nonumber\\
	& = & \sum_{k\geq1} \left( \, \frac{\xi}{(\lambda(t))^{(\frac4p-\frac{2d}{r})}} \, \right)^{k} \,
	\| \,  \gamma^{(k)} (t)\, \|_{L^r_{{\underline{x}}_k,{\underline{x}}_k'}} \,. \label{bigLp-lambda}
\end{eqnarray}
However \eqref{bigLp-lambda} is bounded for every $\xi<1$, if we choose
\begin{eqnarray}
	\lambda(t) \, = \, ( \, {\rm Av}_{L^r}(\Gamma(t)) \, )^{\frac{1}{\frac{4}{p}-\frac{2d}{r}}} \,.
\end{eqnarray}

Now we argue as in the proof of the part ($a$) by using
the local well-posedness Theorem \ref{thm-locwp-main-1}
to conclude that the $H^\alpha$ blowup time $T^*$ is bounded from below by
\begin{eqnarray}
	T^* \, > \, t+(\lambda(t))^{-2}\tau_{\max} \,.
\end{eqnarray}
Therefore
\begin{eqnarray}
	( \, {\rm Av}_{L^r}(\Gamma(t)) \, )^{\frac12} \, = \,
        \lambda(t)^{(\frac{2}{p}-\frac{d}{r})}
	\, > \, \frac{C}{|T^*-t|^{\frac{1}{p} - \frac{d}{2r}}} \,.
\end{eqnarray}
Hence ($b$) is proved.
\qed

\section{Proof of pseudoconformal invariance}
\label{sec-confinv-1}

In this section, we prove Theorem \ref{thm-pseudoconf-gamma-1}.
We recall the pseudoconformal transformations
\begin{eqnarray}\label{eq-pseudoconf-gamma-1}
	\lefteqn{
	{\mathcal P}\gamma^{(k)} (t,{\underline{x}}_k;{\underline{x}}_k')
	}
	\nonumber\\
	&&
	\, := \,
	\frac{1}{(1+bt)^{dk}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\gamma^{(k)}\Big( \, \frac{1}{1+bt} \, , \, \frac{{\underline{x}}_k}{1+bt} \, ; \, \frac{{\underline{x}}_k'}{1+bt} \, \Big) \,,
\end{eqnarray}
for any $b\in{\mathbb R}\setminus\{0\}$.
Similarly as in the case of NLS, one can verify that
\begin{eqnarray}
	\lefteqn{
	(i\partial_t+ \Delta_{{\underline{x}}_k}-\Delta_{{\underline{x}}_k'}) {\mathcal P}\gamma^{(k)} (t,{\underline{x}}_k;{\underline{x}}_k')
	}
	\\
	&=&\frac{1}{(1+bt)^2}\frac{1}{(1+bt)^{dk}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\nonumber\\
	&&\quad\quad\quad
	((i\partial_t+ \Delta_{{\underline{x}}_k}-\Delta_{{\underline{x}}_k'})\gamma^{(k)})
	\Big( \, \frac{1}{1+bt} \, , \, \frac{{\underline{x}}_k}{1+bt} \, ; \, \frac{{\underline{x}}_k'}{1+bt} \, \Big) \,.
	\nonumber
\end{eqnarray}
Now we shall prove the pseudoconformal invariance of the quintic GP hieararchy when $d=1$.
In particular, we find that
\begin{eqnarray}
	\lefteqn{
	B_{j;k+1,k+2}^1{\mathcal P}\gamma^{(k+2)} (t,{\underline{x}}_k;{\underline{x}}_k')
	}
	\nonumber\\
	&=&
	\frac{1}{(1+bt)^{d(k+2)}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\int dx_{k+1} dx_{k+2} dx_{k+1}' dx_{k+2}'
	\nonumber\\
	&&\quad \delta(x_j-x_{k+1})\delta(x_j-x_{k+1}')
	\delta(x_j-x_{k+2})\delta(x_j-x_{k+2}')
	\nonumber\\
	&&\quad\quad
	\gamma^{(k)}\Big( \, \frac{1}{1+bt} \, , \, \frac{({\underline{x}}_k,x_{k+1},x_{k+2})}{1+bt} \, ;
	\, \frac{({\underline{x}}_k',x_{k+1}',x_{k+2}')}{1+bt} \, \Big)
	\\
	&=&
	\frac{1}{(1+bt)^{2d}}\frac{1}{(1+bt)^{dk}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\nonumber\\
	&&\quad\quad
	\gamma^{(k)}\Big( \, \frac{1}{1+bt} \, , \, \frac{({\underline{x}}_k,x_{j},x_{j})}{1+bt} \, ;
	\, \frac{({\underline{x}}_k',x_{j},x_{j})}{1+bt} \, \Big) \,,
	\\
	&=&
	\frac{1}{(1+bt)^{2d}}\frac{1}{(1+bt)^{dk}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\nonumber\\
	&&\quad\quad
	B_{j;k+1,k+2}^1\gamma^{(k+2)} \Big( \, \frac{1}{1+bt} \, , \, \frac{{\underline{x}}_k}{1+bt} \, ;
	\, \frac{{\underline{x}}_k'}{1+bt} \, \Big)

\end{eqnarray}
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
\begin{eqnarray}
	e^{-i\frac{b((x_{k+1}^2+x_{k+2}^2)-({\underline{x}}_{k+1}'^2+x_{k+2}'^2))}{1+bt}}\Big|_{x_{k+1}=x_{k+2}=x_{k+1}'=x_{k+2}'=x_j}
	\, = \, 1 \,.
\end{eqnarray}
Thus, when $d=1$, we obtain
\begin{eqnarray}
	\lefteqn{
	\Big( \, (  i\partial_t+ \Delta_{{\underline{x}}_k}-\Delta_{{\underline{x}}_k'})
	{\mathcal P}\gamma^{(k)} - \mu \sum_{j=1}^k B_{j;k+1,k+2}
	{\mathcal P}\gamma^{(k+2)} \, \Big) (t,{\underline{x}}_k;{\underline{x}}_k')
	}
	\nonumber\\
	&=&\frac{1}{(1+bt)^2}\frac{1}{(1+bt)^{dk}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\nonumber\\
	&&
	\Big(  (i\partial_t+ \Delta_{{\underline{x}}_k}-\Delta_{{\underline{x}}_k'})\gamma^{(k)}
	- \mu \sum_{j=1}^k B_{j;k+1,k+2}\gamma^{(k+2)} \, \Big)
	\Big( \, \frac{1}{1+bt} \, , \, \frac{{\underline{x}}_k}{1+bt} \, ; \, \frac{{\underline{x}}_k'}{1+bt}   \Big)
	\nonumber\\
	&=&0 \,.
\end{eqnarray}
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 $x_{k+1}'$,
\begin{eqnarray}
	(i\partial_t+ \Delta_{{\underline{x}}_k}-\Delta_{{\underline{x}}_k'})\gamma^{(k)}
	- \mu\sum_{j=1}^k B_{j;k+1}\gamma^{(k+1)} \, = \, 0 \,.
\end{eqnarray}
The same considerations as above then produce
\begin{eqnarray}
	\lefteqn{
	\Big( \, (  i\partial_t+ \Delta_{{\underline{x}}_k}-\Delta_{{\underline{x}}_k'})
	{\mathcal P}\gamma^{(k)} -  \mu \sum_{j=1}^k B_{j;k+1}
	{\mathcal P}\gamma^{(k+1)} \, \Big) (t,{\underline{x}}_k;{\underline{x}}_k')
	}
	\nonumber\\
	&=&\frac{1}{(1+bt)^2}\frac{1}{(1+bt)^{dk}} \, e^{-i\frac{b(|{\underline{x}}_k|^2-|{\underline{x}}_k'|^2)}{1+bt}}
	\nonumber\\
	&&
	\Big( \, (i\partial_t+ \Delta_{{\underline{x}}_k}-\Delta_{{\underline{x}}_k'})\gamma^{(k)}
	-  \mu \sum_{j=1}^k B_{j;k+1}\gamma^{(k+1)} \, \Big)
	\Big( \, \frac{1}{1+bt} \, , \, \frac{{\underline{x}}_k}{1+bt} \, ; \, \frac{{\underline{x}}_k'}{1+bt} \, \Big) \,
	\nonumber\\
	&=&0
\end{eqnarray}
if $d=2$. This proves Theorem \ref{thm-pseudoconf-gamma-1}.
\qed

\newpage

redpajama_set_name: RedPajamaArXiv


text:
Werner Schurrmann (* 10. Februar 1926), ein ehemaliger deutscher Fußballspieler, der in Brieske-Senftenberg von 1949 bis 1956 Erstligafußball bestritten hatte.

Sportliche Laufbahn
Die Betriebssportgemeinschaft (BSG) "Franz Mehring" Marga wurde 1949 Vizemeister der  Fußball-Landesklasse Brandenburg. Zum Spieleraufgebot gehörte auch der 23-jährige Stürmer Werner Schurrmann. Als Brandenburger Vizemeister nahm die Mannschaft mit Schurrmann an der Fußball-Ostzonenmeisterschaft 1949 teil, schied aber bereits nach zwei Spielen aus. Gleichzeitig qualifizierte man sich für die Teilnahme an der Fußball-Liga des Deutschen Sportausschusses, die erstmals 1949/50 als höchste ostdeutsche Spielklasse ausgetragen wurde.

In den 26 ausgetragenen Spielen wurde Werner Schurrmann 20-mal eingesetzt und wurde hauptsächlich im Mittelfeld aufgeboten. Zur Saison 1950/51 nannte sich die BSG in Aktivist Brieske-Ost nach dem beheimateten Ortsteil von Senftenberg Brieske um, und nach der Gründung der DDR wurde die DS-Liga in DDR-Oberliga umbenannt. Schurrmann war in dieser und den folgenden beiden Spielzeiten als Mittelfeldspieler gesetzt und verpasste von den 102 ausgetragenen Spielen lediglich vier Begegnungen. War er 1949/50 noch torlos gewesen, war er nun in allen drei Spielzeiten mit insgesamt zehn Toren erfolgreich. Danach erlitt Schurmanns Karriere einen Bruch. 1953/54 kam er nur in der Rückrunde auf sieben Einsätze. Auch 1954/55, als die BSG in den Sportclub Aktivist Brieske-Senftenberg umgewandelt wurde, spielte er ebenfalls nur siebenmal in der Oberliga und kam nur unregelmäßig zum Einsatz. Im Herbst 1955 wurde der DDR-Fußball nach sowjetischem Vorbild  auf die Kalenderjahr-Saison umgestellt, dazu wurden in der Oberliga zwischen August und Dezember 13 Spiele ausgetragen. Dort fand Schurrmann seine alte Form zurück, bestritt zwölf Begegnungen und erzielte drei Tore. Auch 1956 schien es zunächst, als könnte Schurrmann wieder seinen Platz im Mittelfeld behaupten, doch nach zehn Spieltagen fiel er erneut aus und kam danach nur sporadisch zu weiteren drei Einsätzen. Danach gab er im Alter von 30 Jahren seine Laufbahn als Leistungsfußballer auf. Innerhalb von acht Spielzeiten in der DS/Oberliga war er auf 157 Einsätze und 14 Tore gekommen.

Literatur
 Deutsches Sportecho: Jahrgänge 1949–1956.
 Hanns Leske: Enzyklopädie des DDR-Fußballs. Verlag Die Werkstatt, Göttingen 2007, ISBN 978-3-89533-556-3, S. 459.
 Andreas Baingo, Michael Horn: Die Geschichte der DDR-Oberliga. Verlag Die Werkstatt, Göttingen 2004, ISBN 3-89533-428-6, S. 301.

Weblinks

Fußballspieler (DDR)
Fußballspieler (SC Aktivist Brieske-Senftenberg)
Deutscher
DDR-Bürger
Geboren 1926
Mann

redpajama_set_name: RedPajamaWikipedia


He seemed knowledgeable but his body language was very negative and he lacked bedside manners. Not once did he smile or introduce Himself in the beginning. It was an awkward appointment.
الدكتور واضح شاطر .. هذه اول تجربية لي خلاني اشرح حالتي بعدها حاول يوضح لي كل شي ويعطيني معلومات اكثر .. عن حالتي .. كل الشكر ..


As All Hallow's Eve approaches I am extremely excited about Halloween decorations! But man, they get pricey! So I did what comes natural to a crafter – found a way to decorate on budget. I will be showing you how I made this awesomely spooky skull candle holder. With some inexpensive supplies this do it yourself project is quick and easy to put together. Let's get started!
Terracotta plates – price will depend on the size you buy. Mine were less than $1 each because I got the smaller sized ones.
Now that you have your supplies together, take the skulls and figure out how you would like them arranged. As you can see, I had the middle skull face a different direction than the other two skulls.
Once you've figured out how you'd like to stack them, take one of the skulls and a terracotta plate and glue them together. (Please note: in the photo I had the terracotta plate right side up, I ended up pulling it off and flipping it over but I failed to take a new photo. The finished product photos show this change).
Proceed to glue each skull together – followed by the second terracotta plate being glued right side up on top of the last skull. At this point, the project probably looks a bit odd, but don't worry! The paint will make the project complete!
Arrange the pieces how you'd like them and then glue everything together!
One the glue is dry you can paint your project! I went outside with my project and a cardboard box I had been using for other painting projects and got to work. Because I was using spray paint I sat the project inside of the box and sprayed away! I would spray the side I could see and then back off for five or so minutes while it dried.
I walked away between sprays to get away from the paint fumes. It took a about 20-30 minutes to get the entire thing coated evenly with paint.
Once the paint is dried, you now have a spooky candle holder for your Halloween decor! Don't have or use candles? No worries! I currently have a few cute trinkets sitting in mine! I just wouldn't recommend using anything super heavy – a small candle or a few small, light trinkets would be best!
Once you've finished the project I'd love to see how yours turned out! Tag me on Instagram @the_littlest_liar or on Twitter @thedancingdish! Don't forget to use #LittlestLiarDIY and #IGGPPC!
Sara (AKA thelittlestliar) is a lovely princess (at least in her head she is) who lives in South Carolina, USA. Her blog updates usually consist of Fandom or Geeky DIY projects or super yummy baking recipes. She has many passions which include crafts, DIY projects, baking, reading, drawing, and pretty nails. She dedicately watches Pretty Little Liars, The Fosters, and Scream Queens. Sara can be found on instagram @the_littlest_liar and on Twitter @thedancingdish.


text:
Q: Play tags$audio file until the end in Shiny App R I have a shiny app that is kind of like a Piano keyboard. I press a button and it plays a sound. It looks something like this:
if (input$down[1] == 65){
tags$audio(src = "Low_C.wav", type = "audio/wav", autoplay = NA, controls = FALSE)}
else if(input$down[1] == 87){
tags$audio(src = "Low_C_Sharp.wav", type = "audio/wav", autoplay = NA, controls = NA)}
Every time I press 65 (letter a on my keyboard), it plays low c. But when I press 87 (letter w) it stops playing low c and plays low c sharp.
I want to be able to press 65, it play low c until the end of "Low_C.wav" even if I press 87 before then end, it should play them together.

A: Okay, so I figured it out and thought I'd share.
Instead of tags$audio, I used
tags$script('new Audio ("Low_C.wav").play()')
and
tags$script('new Audio ("Low_C_Sharp.wav").play()')
It can play multiple sounds at once this way.

redpajama_set_name: RedPajamaStackExchange


text:
Network: Bbc Two
Full Episodes: 4
Last Episode, 06 March 2000: Season 1, Episode 4:
Nature Boy Episodes
14 February 2000 Season 1, Episode 1:
06 March 2000 Season 1, Episode 4:
Andrew Marrs History Of The World
El C.I.D.
John Bishop's Big Year

redpajama_set_name: RedPajamaCommonCrawl


text:
50 high school football players suspended for bullying and more
The following story was written by Gregg Doyel for CBSSports.com. You can read the rest of the story here.
The month of October is National Bullying Prevention Month. We will be featuring some bullying stories with positive endings like the following……
Photo by Daniel X. O'Neil
Matt Labrum doesn't want this story to be about him, so that's his second outlandish demand in a week. The first one, maybe you heard about: Labrum is the football coach at Union High in Roosevelt, Utah, and after some discipline issues on the team he demanded their jerseys. He was suspending the Cougars.
Labrum and his staff handed each player a piece of paper after their game last Friday night. It was a prepared statement that said a lot of things — harsh things, honest things, tough things — but the sentence that got the most attention was this one:
TURN IN YOUR JERSEYS NOW!
Whole team — suspended. Ever heard of such a thing? No? Neither had anyone around that part of Utah, which is why the suspension of the Union Cougars became a story in the Deseret News. And once it was there, it was everywhere. The networks wanted to talk to Labrum. National websites. USA Today. All of them, all of us, wanted to talk to the wacky football coach with the notion that football is nothing without character and respect.
By the time I found him, Matt Labrum was tired of talking about it. And he'd never wanted to talk about himself in the first place.
"The last thing I want is for this to be about me," he emailed me. "I'll talk to you — if you make it about the players."
Um. What? Not sure that's possible. The star of this story, the hero, is the football coach. The players? The ones who were acting out in class, skipping school, making poor grades? The ones who started cyber-bullying another student, ganging up on the kid on a social media site — a textbook example of the strong and numerous picking on the weak and isolated? I didn't want to make this story about the players.
But then I called a few of them. Talked to some of their parents. Then a few more. Found out exactly how the Union Cougars have been spending their time since being suspended.
And now I'm thinking Matt Labrum might be onto something. Sure, I could make this about the coach.
But I'd rather make it about the players.
Karter Rook shoveled pig crap.
That's how Rook paid his penance — well, that's how he started to pay his penance — for being part of the Union High football team. Understand, Karter Rook isn't necessarily a bad kid, and he seems actually like quite a nice one. Lots of these kids are no doubt good and decent people, but what was happening at Union High was too pervasive for the coach to pick and choose who to punish. Besides which, Matt Labrum didn't know exactly who had been behaving the worst.
He knew which kids were acting up in class, who had skipped some school, whose grades were slipping. Labrum teaches gym at Union, so if any of his player have issues at school, he's going to hear about it.
It was the bullying that drove Labrum to act. It was done anonymously, on a social networking site called Ask.fm, but the victim went to a Union High guidance counselor and laid out the evidence that suggested the tormenters were members of the football team.
Enough, Labrum decided. Enough. He met with his coaching staff a few days before Union's game last Friday against Judge Memorial Catholic High, and deep into the night they kicked around disciplinary ideas until finally settling on this one:
Everyone's off the team.
Everyone.
There were 50 kids on the varsity — the freshman team wasn't affected, Labrum says — and as they cooled down in the locker room after Union's loss to Judge Memorial, each of them was handed a piece of paper that started like this:
"Gentlemen, we are not pleased with how our football brothers are representing OUR family, school, community, alumni, and yourselves."
The letter went on to say, "We want change — and are going to make changes now."
First, the jerseys. Turn them in. To get them back — to be reinstated to the team — players would have to perform service for their family. And they would have to perform service for their community. They also would have to report for a two-hour study hall after school, "and we better have specific items to work on," the letter said.
All of that went down late Friday night. Sophomore offensive tackle Karter Rook went home after the game, went to bed, woke up the next morning and grabbed a shovel. His family has a pigpen. And it was filthy.
"Cleaning the pigpen was something I was going to do," said his mother, Jennifer Rook, "but Karter just realized, 'I need to contribute more.' Now we have the best looking pigpen in the county."
That was Saturday. And that's just one kid. There are 49 others on the team, and each spent the day working around his house — and documenting it, with pictorial proof — to show Labrum.
Read the rest of the story here.
bullyingcyberbullying


Elected officials and advocates from across New Jersey lauded Governor Phil Murphy's Fiscal Year 2020 budget, delivered today. The Governor's proposed budget lays out a plan to put New Jersey on a responsible path forward through over $1.1 billion in sustainable savings and support for our middle class through targeted investments in NJ TRANSIT, school funding, social services, and property tax relief.
"New Jersey deserves a budget built upon the values that can deliver a more secure and growing middle class – sustainable savings, responsible fiscal management, smart and targeted investment, and fairness," said Governor Murphy.


text:
Q: Manually pass list of strings to pipe If a command returns a list of strings I can pipe it to another command.
For instance:
command1 | command2

Lets say command1 returns 2 lines.  Is there a way to do the same thing on the command line without a command that generates the lines?
I know this below doesn't work but maybe explains better what I mean:
("string 1", "string 2) | command2

I want to manually specify 2 lines to send into the pipe instead of using a command to generate them.
Is that possible?

A: You can use echo commands:
{ echo "string 1"; echo "string 2"; } | command2

or printf to do it with one command:
printf 'string 1\nstring 2\n' | command2

But if it's many lines, a here-doc is usually easier:
command2 <<EOF
string 1
string 2
EOF


text:
QUANTUM SECURE MESSAGE – DON'T LET YOUR PRICELESS THING BE HACKED
Kabilan s, 12 months ago 4 min read 16
The field of quantum information science seeks to engineer real-world devices that can store and process quantum states of information. They believed that computers operating according to such principles are capable of solving problems exponentially faster than existing computers, while quantum networks have provable security guarantees.
The same concept can be applied by making more precise sensors and measurement devices. Constructing such systems is a significant challenge because quantum effects are typically confined to the atomic scale. However, through careful engineering, several physical platforms have been identified for quantum computing, including superconducting circuits, laser-cooled atoms, ions, electron spins in semiconductors.
Already we saw the basic concepts of quantum technology and its fascinating applications of quantum technology in various fields. If you haven't visited that page click the following link: https://engineer29.com/quantum-physics-and-its-main-theories/ . Today we are going to see about the new application of quantum technology.
Here we go, What do you know about messages? Nowadays, most of the important information is sent through messages. But you doubt if the message is conveyed correctly, without hacking or stealing the data and information? Definitely not, most of our data can be hacked or stolen by a third party easily. To prevent such types of cons, They have introduced new technology through which Messages are sent using quantum technology.
Quantum computing concept. Abstract glowing electronic circuit.
What is Quantum secure messages?
As I told you already, The information, data, Chats, Instructions conveyed (like Message from our bank, that we have withdrawn 1000) can be hacked easily. To avoid such issues, quantum messages have been introduced. Basically, the message will be encrypted with a secure key, which is a long mixture of numbers and characters. In the 1980s researchers developed a theoretical way or method for generating secure keys with the help of quantum mechanics. Usually, the messages are sent using electromagnetic waves to travel, but in this type, messages are sent through quantum waves.
security insurance concept
Today's achievements:
08.02.2022 that was a great day, The Indian space research organization (ISRO) has successfully demonstrated free-space quantum communication and sent messages over a range of 300 m. Also, they have developed a NAVIC receiver for reducing the time between transmitter and receiver. This demonstration included a live stream video conference using quantum key encrypted signals. which is the latest technology that is introduced in quantum technology.
Our world is developing towards securely encrypted information which is very difficult to hack and we can trust today's technology without any doubt. Researchers at the university of science and technology in China researched these quantum keys and worked out in 2018 how to share your quantum keys secretly. It is a wonderful technology and in the future, it will be executed on a mass scale.
AIM correct to achieve your goal
Quantum Tunnel:
Quantum tunneling is a quantum mechanical phenomenon in which the wave function will propagate through the potential barrier. The wave function will disappear and reappear on the other side. Quantum tunneling occurs with a barrier thickness of 1-3nm or smaller than that. Some researchers said if the wave function transmits into the barrier and without reappearing on the other side, is called as tunneling effect.
This is not predicted by classical mechanics. It plays a crucial role in physical phenomena, like nuclear fusion. It is used in quantum computing, tunnel diode, and in scanning tunneling microscopes. Tunneling may be explained in terms of Heisenberg's uncertainty principle in which quantum objects can be defined as a wave or particle. It can be one of the mechanisms of proton decay.
It is mainly used in the implications of the functioning of nanotechnology. Tunneling is a fundamental technique that is used to program the floating gate of memory. It is also used in the cold emission, Tunnel junction, quantum-dot cellular automation, tunnel diode, tunnel field-effect transistors, nuclear fusion, radioactive decay, astrochemistry in interstellar clouds, quantum biology, quantum conductivity, scanning tunneling microscope, kinetic isotope effect are some of the important applications of quantum tunneling in the following fields.
Artificial DNA molecule, the concept of artificial intelligence, on a dark background
Click here to know about the metaverse marriage
Tags #cold emission #Encrypted #Hacking #ISRO #messages #nuclear fusion #Physics research center #quantum computing #quantum dot #quantum physics #Quantum tunneling #radioactive decay #tunnel diode #tunnel junction #Wave function
KNOW A THING ABOUT MOBILE COMMS - PCM MADE EASY
MAKE WONDERS WITH GRAPH MODELS
Kabilan s
KNOW A THING ABOUT MOBILE COMMS – PCM MADE EASY
ECE, Uncategorized
Simple and ultimate analysis of the microwave
Kabilan s, 9 months ago 3 min read
Kabilaan K
Haa quantum physics 🌚


Post-doctoral fellowship training is integral to the academic life of the Center. Four senior faculty hold K24 Mentoring Awards to protect time to work with post-docs and junior faculty in order to foster their work and progress to independence. Fellows work on their own independent research projects in a mentored environment with research coordination, project management, regulatory, and statistical support.
We are pleased to announce that the K12 Career Development Program in Substance Use and Addiction Medicine, based at the Massachusetts General Hospital (MGH) and funded by the National Institute on Drug Abuse of the National Institutes of Health, is recruiting one post-doctoral clinician- scientist to start on or after July 1, 2019. The award aims to help the Scholar establish an independent research program in patient-oriented substance use and addiction research.
The program provides salary, training and project support for a 2- to 3-year period of intensive mentored research training and career development. Funding includes an annual stipend of $80,000 for 75-100% effort and up to $35,000 annually in allowable research and training costs that may be applied to travel to scientific meetings, research project costs, and tuition expenses with approval of the mentor and program leaders.
A limited number of individually-tailored, post-doctoral fellowships are available to highly motivated individuals whose research interests coincide with or further the goals of the Center. Former fellows have gone onto independent academic research careers and to important leadership positions at NIH. Doctoral level candidates in diverse fields are eligible to apply. Please see the staff page for current fellows and their backgrounds. Send a cover letter and CV to Dr. Evins to be considered.
ACGME approved fellowship in addiction psychiatry available as a one-year fellowship or as a two-year half-time fellowship combined with sponsored research or private practice. Psychiatrists who complete the fellowship are board eligible in addiction psychiatry. Previous clinical fellows have gone on to successful academic careers at Harvard and Yale, to direct private organizations that provide treatment services for people with psychiatric and substance use disorders in the community, and to private practice.
For more information on how to apply for any of these fellowships, go to our How to Apply page.


text:
Tom Carson won two National Magazine Awards during his stint as Esquire's "Screen" columnist and has been nominated twice more as GQ's movie reviewer. Formerly a staff writer at LA Weekly and The Village Voice, he is the author of Gilligan's Wake (a New York Times Notable Book of the Year for 2003) and Daisy Buchanan's Daughter.
Pope Francis's "Cardinal" Rules
Thinking about Otto Preminger's film 50 years later in the context of Pope Francis.
Tom CarsonJan 10, 2014
AP Photo/Alessandra Tarantino
What to make of last year's onslaught of Francismania? Like the sucker for pop-culture phenomenons I am, I haven't enjoyed anything so much since the Harry Potter books took off. As a veteran secular humanist, I can't help feeling some simpatico with the killjoys striving to remind us that the first Pope in memory to rate an affectionate New Yorker cover still presides over an essentially reactionary organization whose core doctrines haven't changed. But all the same, screw it: they're messing with everybody else's good time . Those of us without a dogma in this hunt just dig waiting for Pope Frank's next Bob Newhart-ish "He said that??" surprise, even as we relish the consternation he's provoked in everybody from Rush Limbaugh—"pure Marxism," the great man flatulated—to Home Depot founder Ken Langone, who fretted that Francis doesn't understand how good rich Americans are. Cardinal Timothy Dolan had to reassure Langone that the latest Pontiff does indeed love his flock's...
Peter O'Toole, Always in Character
The actor, who died earlier this week at the age of 81, turned every role into a feat of concentrated charm and wit.
Tom CarsonDec 18, 2013
In a fantasy sequence in the 1967 movie Casino Royale—not a straight Bond flick, like the 2006 version, but a famously messy spoof—Peter Sellers, playing one of several rival 007s, sees a haggard-looking Scots bagpiper emerge from soundstage fog. "Are you Richard Burton?" the piper asks, and Sellers retorts, "No, I'm Peter O'Toole!" At which the piper—played by O'Toole himself—is overcome by emotion. "Then you're the finest man who ever breathed," he says, and vanishes into the mist. What's nice is that O'Toole has the wit to say it in character—that is, in character as a demented Scots piper who reveres Peter O'Toole, with nary a hint of a wink to the audience. It's not just that he obviously knew it would be funnier that way, and we can only imagine how Burton's peculiarly preening self-contempt would have soured the cameo. No matter how silly the job at hand was, O'Toole just couldn't help acting. But he also seldom made it look like a job, which was a...
"Spring Breakers" Was the Best Movie of the Year. Seriously.
It was imbecilic but gorgeous—just like America.
AP Images/A24 Films
First off, some disclaimers: I haven't seen Joel and Ethan Coen's Inside Llewyn Davis yet. I thought 12 Years A Slave and Gravity were both just swell, and I had a great time at Ron Howard's unexpectedly lively—and worldly— Rush. In spite of groaning at Woody Allen's bald-faced hijacking of A Streetcar Named Desire and his Rip Van Winkle unfamiliarity with the current century, I even got semi-down with Blue Jasmine, and so on. But that said, as we gear up to dispatch 2013 to the rear-view mirror, there's no contest when it comes to naming my favorite American movie of the year. That would be Spring Breakers, cult director Harmony Korine's sleaze-addicted, druggily impressionistic—and wondrously beautiful—tale of vacuous college girls gone wild in St. Pete. It features bare breasts galore, surreally goofy robberies and mayhem, and an all but unrecognizable James Franco wearing what look like brass knuckles on his teeth as a white rapper/wannabe gangsta named...
Sondheim Looks Back
The composer reflects on his storied Broadway career in a new HBO documentary.
AP Images/Henny Ray Abrams
Stephen Sondheim is such an engaging and peppery talker about his own work that any run-of-the-mill documentarian—one unafflicted by genuinely terrible bad breath, anyway—could probably get gold just by turning on the camera. But Six by Sondheim, which airs on HBO on Monday, isn't run-of-the-mill at all. Directed by the composer's longtime stage collaborator, James Lapine, this openly celebratory doc is elaborately built around a half dozen capstone songs in Sondheim's career, from Company' s "Being Alive" to (duh) "Send in The Clowns" to Sunday in The Park With George 's climactic ode. Several of the numbers get full-on restaging—Todd Haynes guest-directs a stone-brilliant version of Follies' "I'm Still Here"—and the interview segments are stitched together from TV Q&A's stretching over five decades along with Lapine's own. This kind of homage can easily turn precious...
Another Note for the Bettie Page Files
A new documentary, Bettie Page Reveals All, provides not only a bonanza of images of the proto-hipster pinup girl in her prime, but her later-in-life musings as well.
Tom CarsonNov 22, 2013
It's fun to think of Bettie Page as not only Jackie Kennedy's secret 20th-century sister but as Hugh Hefner's female rival. Six decades ago, Playboy 's incredibly dull founder created an empire that turned women into interchangeable sex objects in the guise of liberating Americans from their Puritan hangups. All Page had to counter him with was a body that couldn't be mistaken for anyone else's and an extraordinarily expressive face. Hef made millions while Page posed for chump change for amateur shutterbugs, at least until softcore fetish king Irving Klaw and then pinup photographer Bunny Yeager recognized her uniqueness and she wound up in Playboy itself. And yet she won; she beat him. Hefner and Playboy are both still with us, but so is Ovaltine—and when was the last time you had any? Page, who died in 2008, is not only the most iconic sexual image of midcentury's louche subcultures but a 21st-century heroine to hipsters and feminists alike. Kids who don't give two hoots...


The Good The Cambridge SoundWorks PlayDock i comes with a great wireless remote that lets you navigate within the iPod's menus. The stereo separation feature works well, and many songs sound rich, open, and crystal clear. The unit can run off of batteries, and line-in and AV-out cables are included in the package.
The Bad The PlayDock i is too big and heavy to be fully portable, and certain songs played on the speaker unit suffer from an unpleasant vibration.
The Bottom Line Cambridge SoundWorks' PlayDock i has desirable features, decent sound quality, and a semi-portable design--you could do much worse for an iPod-ready tabletop speaker unit.
Creative Labs bought Cambridge SoundWorks back in 1998 (ancient times in tech years) and has continued to produce speakers under the brand. The latest product to come out of the company is the iPod-ready PlayDock i, a $200 tabletop speaker unit with a fantastic remote and decent sound quality. Though the system suffers a bit from its plastic housing, it beats out competitors with desirable features and a semi-portable design.
You probably aren't going to choose the Playdock i based on style. It looks like pretty much every other iPod-ready speaker set on the market: not ugly, but not striking. It has the white, piano-gloss finish, black fabric speaker grille, and semi-rectangular shape, though the PlayDock does have some nice curvature to its backside. The iPod dock protrudes from the front of the speaker a touch, making it less streamlined than speakers like the Altec Lansing iM7, but more secure than all those sets where the iPod juts out from the top. The top of the PlayDock houses the power and stereo separation buttons and a volume knob that clicks satisfyingly when you turn it. On the back, you'll find a built-in handle, a bass adjustment knob, and various ports: power in, auxiliary line-in, A/V out, and headphone out.
The PlayDock i even accepts eight C batteries via a compartment on the bottom, though at roughly 13.5x5.5x6-inches, I wouldn't consider the unit terribly portable. However, it would fit nicely on a counter, shelf, desktop, or nightstand, and it's not a stretch for picnics and camping. Creative doesn't include the batteries, but you do get a line-in cable for connecting non-iPod MP3 players, six iPod dock adapters, and an A/V cable for porting videos and photos from the iPod to your TV while the player's docked in the speaker unit. The PlayDock also charges the iPod while it's plugged in.
As with any decent tabletop speaker system for the iPod, the Playdock i also comes with a wireless remote. It operates via IR--rather than RF--so you'll need direct line-of-site with the speaker to use it. The remote worked fine up to about 20 feet or so away, which is plenty for most applications. Along with playback, power, and volume controls, the remote includes some handy extra buttons I haven't seen on many others. There's a back key for stepping up through the menus, up/down keys for navigating the menus, and an OK button for making menu selections. This is great, because it means you never have to get up and use the iPod itself for navigating.
There are also two extra buttons on the remote that are "not used" according to the manual. One is marked FM/MP3, presumably for switching between FM and MP3 modes. As neither the iPod nor the PlayDock have FM tuners, you can immediately see why this is useless. Another slightly more interesting key has tiny icons representing video and photo on it. My guess is that this would be used for toggling into TV out mode for viewing photos and videos on a connected TV. There's probably something in the iPod's firmware that prevents this from working at this point (or maybe Creative just couldn't figure it out), but it's compelling in that maybe it's a preview of things to come.
When put to the test, the PlayDock i proved to be an adequate performer. The stereo separation function worked well, opening up the sound noticeably, and I preferred to listen with this feature turned on. Overall sound quality was very good in some cases, and not so great in others. Red Hot Chili Peppers' "Scar Tissue" was rich and encompassing, the detail and understated bass in Les Brown's "Sentimental Journey" shined, and the vocals in Olive's "Safer Hands" were open and haunting. However, DJ Cam's mellow hip-hop track "Voodoo Chile" suffered from an unpleasant vibration for every snare hit, and The Mars Volta's "Televators" offered the same irksome hum during the crescendo of the chorus. This seems to be a recurring problem with plastic-housed speaker units, so you may be willing to live with it if you must have an iPod-specific speaker (rather than connecting the unit directly to a higher-end home system).