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\markboth{\hfil Homogeneous Boltzmann equation \hfil EJDE--2003/Mon.~04}
{EJDE--2003/Mon.~04\hfil M. Escobedo, S. Mischler, \& M. A. Valle \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Monogrpah 04, 2003. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Homogeneous Boltzmann equation in quantum relativistic kinetic theory
 %
\thanks{ {\em Mathematics Subject Classifications:} 82B40, 82C40, 83-02.
\hfil\break\indent
{\em Key words:}  Boltzmann equation, relativistic particles, 
entropy maximization,  \hfil\break\indent
Bose distribution, Fermi distribution, Compton scattering, Kompaneets equation.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted  November 29, 2002. Published January 20, 2003.} }
\date{}
%
\author{Miguel Escobedo, St\'ephane Mischler, \& Manuel A. Valle}
\maketitle

\begin{abstract} 
\noindent
 We consider some mathematical questions about Boltzmann 
 equations for quantum  particles, relativistic or non 
 relativistic.  Relevant particular cases such as Bose, 
 Bose-Fermi, and  photon-electron gases are studied. 
 We also consider some simplifications such as the
 isotropy of the distribution functions and the asymptotic 
 limits (systems where one of the species is at equilibrium). 
 This gives rise to interesting mathematical questions from a 
 physical point of view. New results are presented about the 
 existence  and long time behaviour of the solutions to some 
 of these problems. 
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}

\tableofcontents



\section{Introduction} %1

When quantum methods are applied to molecular encounters, some
divergence from the classical results appear. It is then necessary in some
cases to modify the classical theory in order to account for the quantum
effects which are present in the collision processes;
see  \cite[Sec. 17]{CC}, where the domain of applicability of the classical
kinetic theory is discussed in detail.
In spite of their formal similarity, the equations for classical and quantum
kinetic theory display very different features. Surprisingly,
the appropriate Boltzmann equations, which
account for quantum effects, have received scarce attention in the
mathematical literature.

In this work, we consider some mathematical questions about Boltzmann
equations for quantum particles,  relativistic and  not relativistic.
The general interest in different models involving that kind of equations has
increased recently. This is so because they are supposedly reliable for 
computing
non equilibrium properties of Bose-Einstein condensates
on sufficiently large times and distance scales;
see for example \cite{JP,ST1,ST2} and references therein. We study some
relevant particular cases (Bose, Bose-Fermi, photon-electron gases),
simplifications such as the isotropy of the distribution functions,
and asymptotic limits (systems where one of the species is at
equilibrium) which are important from a physical point of view and give
rise to interesting mathematical questions.

Since quantum and classic or relativistic
particles are involved, we are lead to consider such a
general type of equations.
We first consider the
homogeneous Boltzmann equation for a quantum gas constituted by a single
specie of particles, bosons or fermions. We solve the
entropy maximization  problem under the moments constraint
in the general quantum relativistic case.
The question of the well posedness, i.e. existence, uniqueness, stability of
solutions and of the long time behavior of the solutions is also treated in some
relevant particular cases. One could also consider other qualitative
properties such as regularity, positivity, eternal solution in a purely
kinetic perspective or  study the relation between the
Boltzmann equation and the underlying quantum field theory,
 or a more phenomenological description, such as
the based on hydrodynamics, but we do not go further in these directions.

\subsection{The Boltzmann equations} %1.1

To begin with, we focus our attention on a gas composed of
identical and indiscernible particles.
When two particles with respective momentum $p$ and $p_*$ in
$\mathbb{R}^3$ encounter each other, they collide and we denote $p'$ and
$p'_*$ their new momenta  after the collision. We assume that the
collision is elastic, which means that the total momentum and the total
energy of the system  constituted by this pair of particles are conserved.
More precisely, denoting by $\mathcal{E}(p)$ the energy of one particle
with momentum $p$, we assume that
$$
\begin{gathered}
p' + p'_* = p + p_* \\
\mathcal{E}(p') + \mathcal{E}(p'_*) = \mathcal{E}(p) + \mathcal{E}(p_*).
\end{gathered} \eqno(1.1)
$$
We denote $\mathcal{C}$ the set of all $4$-tuplets of particles
$(p,p_*,p',p'_*) \in \mathbb{R}^{12}$ satisfying (1.1).
The expression of the energy $\mathcal{E}(p)$ of a particle in function of its
momentum $p$ depends on the type of the particle;
$$
\begin{aligned}
&\mathcal{E}(p) = \mathcal{E}_{nr}(p) = {|p|^2 \over 2 \, m} \quad \hbox{for a non
relativistic particle,} \\
&\mathcal{E}(p) = \mathcal{E}_r(p) = \gamma mc^2; \,\gamma = {\sqrt{1+(|p|^2/c^2m^2)}}
\quad \hbox{for a relativistic particle, }\\
&\mathcal{E}(p) = \mathcal{E}_{ph}(p) = c|p| \quad
\hbox{for massless particle such as a photon or neutrino.}
\end{aligned} \eqno(1.2)
$$
Here, $m$ stands for the mass of the particle and $c$ for the velocity of
light. The velocity $v = v(p)$ of a particle with momentum $p$ is
defined by
$v(p) = \nabla_p \mathcal{E}(p)$, and therefore
$$
\begin{aligned}
v(p) &= v_{nr}(p) = {p \over m} \quad \hbox{for a non relativistic
particle,} \\
v(p) &= v_r(p) = {p \over m \gamma}  \quad \hbox{for a relativistic
particle,}\\
v(p) &= v_{ph}(p) = c {p \over |p|} \quad \hbox{for a photon .}
\end{aligned}  \eqno(1.3)
$$

Now we consider a gas constituted by a very large number (of order the
Avogadro number $A \sim 10^{23}/\hbox{mol}$) of a
single specie of identical and indiscernible particles. The very large
number of particles makes  impossible
(or irrelevant) the knowledge of the position
and momentum $(x,p)$ (with
$x$ in a domain $\Omega\subset \mathbb{R}^3$ and $p \in \mathbb{R}^3$) of every 
particle of
the gas. Then, we introduce $f = f(t,x,p) \ge 0$,
the gas density distribution of particles which at
time $t \ge 0$ have position $x \in
\mathbb{R}^3$ and momentum $p \in \mathbb{R}^3$. Under the hypothesis of 
molecular chaos and
of low density of the gas, so that particles collide by pairs  (no collision 
between three or more particles
occurs),  Boltzmann \cite{B} established that
the evolution of a classic (i.e. no quantum nor relativistic) gas  density
$f$ satisfies
$$
\begin{gathered}
{\partial f \over \partial t}  + v(p) \cdot \nabla_x f = Q(f(t,x,.))(p) \\
f(0,.) = f_{\rm in},
\end{gathered} \eqno(1.4)
$$
where $f_{\rm in} \ge 0$ is the initial gas distribution and $Q(f)$ is the
so-called Boltzmann collision
kernel. It describes the change of the momentum of the particles due
to the collisions.

A similar equation was proposed by  Nordheim \cite{N} in 1928  and by
Uehling \&  Uhlenbeck \cite{UU} in 1933 for the description of a quantum
gas, where only the collision term $Q(f)$ had to be changed to take into account 
the quantum
degeneracy of the particles.
The relativistic generalization of the Boltzmann equation including the
effects of collisions was given by  Lichnerowicz and
Marrot \cite{LM} in 1940.


Although this is by no means a review article we may nevertheless give some 
references for the
interested readers. For the classical Boltzmann equation we refer to  Villani's 
recent review
\cite{V} and the rather complete bibliography therein.
Concerning the relativistic kinetic theory, we refer to the monograph \cite{St} 
by J. M. Stewart
and the classical expository text \cite{GLW} by  Groot, Van Leeuwen and  Van 
Weert. A
mathematical  point of view, may be found in the books by Glassey \cite{G}
and Cercignani and  Kremer \cite{CK}. In \cite{J},  J\" uttner gave the
relativistic equilibrium distribution. Then,  Ehlers
in \cite{E}, Tayler \&  Weinberg in \cite{TW} and  Chernikov in \cite{Ch} proved  
the H-theorem for the
relativistic Boltzmann equation. The existence of global classical solutions for 
data close to equilibrium
is shown by R. Glassey and W. Strauss in \cite{GS1}.
The asymptotic stability of the equilibria is studied in
\cite{GS1}, and \cite{GS2}.  For these questions see also the book \cite{G}.
The global existence of renormalized solutions
is proved by Dudynsky and  Ekiel Jezewska in \cite{DE}.
The asymptotic behaviour of the global solutions is
also considered in \cite{An}.

In all the following we make the assumption that the density $f$ only
depends on the momentum. The collision term $Q(f)$ may then be expressed in all 
the
cases described above as
$$
\begin{gathered}
Q(f)(p)=\iiint_{\mathbb{R} ^9} W(p, p_{*}, p', p'_{*})q(f)\, dp_{*}dp'dp'_{*}
\\
q(f)\equiv q(f)(p, p_{*}, p', p'_{*})=[f'f'_{*}(1+\tau f)(1+\tau f_{*})
-ff_{*}(1+\tau f')(1+\tau f'_{*})]\\
\tau \in \{-1, 0, 1\},
\end{gathered} \eqno{(1.5)}
$$
where as usual, we denote:
$$ f=f(p),\quad  f_{*}=f(p_{*}),\quad f'=f(p'),\quad f'_{*}=f(p'_{*}),
$$
and $W$ is a non negative measure called transition rate, which may be written 
in
general as:
$$
W(p, p_{*}, p', p'_{*})=w(p, p_{*}, p', p'_{*})\delta
(p+p_{*}-p'-p'_{*})\delta (\mathcal{E}(p)+\mathcal{E}(p_{*})-\mathcal{E}(p')-
\mathcal{E}(p'_{*}))\eqno{(1.6)}
$$
where $\delta $ represents the Dirac measure. The quantity $Wdp'dp_*'$
is the probability for the initial
state $|p, p_{*}\rangle$ to scatter and become a final state of two particles
whose momenta lie in a small region $dp'\, dp_{*}'$.


The character relativistic or not, of the particles  is taken into
account in the expression of the energy of the particle $\mathcal{E}(p)$
given by (1.2).  The effects due to quantum degeneracy are included in the term 
$q(f)$ when $\tau \not
=0$, and depend on the bosonic or fermionic character of the involved particles. 
These are associated
with the fact that, in quantum mechanics, identical particles cannot be 
distinguished,
not even in principle. For dense gases at low temperature, this kind of terms 
are crucial. However, for non
relativistic dilute gases, quantum degeneracy plays no role and can be safely 
ignored ($\tau =0$).


The function $w$ is directly related to the differential cross section $\sigma $ 
(see (5.11)), a quantity
that is intrinsic to the colliding particles and the kind of interaction between 
them. The calculation of
$\sigma $ from the underlying interaction potential is a central problem in non 
relativistic quantum
mechanics, and  there are a few examples of isotropic interactions (the Coulomb 
potential, the delta
shell,\dots) which have an exact solution. However, in a complete relativistic 
setting or when many-body
effects due to collective dynamics lead to the screening of interactions, the 
description of these in terms
of a potential is impossible. Then, the complete framework of quantum field 
theory (relativistic or not)
must be used in order to perform perturbative computations of the involved 
scattering cross section in $w$.
We give some explicit examples in the Appendix \ref{A3} but let us only mention 
here the case
$w=1$ which corresponds to a non relativistic short range interaction
(see Appendix \ref{A3}). Since the particles
are indiscernible, the collisions are reversible and the two interacting 
particles form a closed
physical system.  We have then:
$$
\begin{aligned}
&W(p, p_{*}, {p'}, {p'_{*}})=W(p_{*}, p, {p'}, {p'_{*}})=W({p'}, {p'_{*}}, p, p_{*})\\
&+\quad   \hbox{Galilean invariance (in the non relativistic case)}\\
&+\quad   \hbox{Lorentz invariance (in the relativistic case).}
\end{aligned} \eqno{(1.7)}
$$
To give a sense to the expression (1.5) under general assumptions on the 
distribution $f$ is not a
simple question in general. Let us only  remark here that
$Q(f)$ is well defined as a measure when $f$ and $w$ are assumed to be 
continuous. But we will
see below that this is not always a reasonable assumption. It is one
of the purposes of this work to clarify this question in part.

The Boltzmann equation reads then very similar, formally at least, in all
the different contexts: classic, quantum and relativistic. In particular
some of the fundamental physically relevant properties of the solutions
$f$ may be formally established in all the cases in the same way:
conservation of the total number of particles, mean impulse and total energy; 
existence of an
``entropy function'' which increases along the trajectory (Boltzmann's
H-Theorem).  For any $\psi = \psi(p)$, the symmetries (1.7), imply the
fundamental and elementary identity
$$
\int_{\mathbb{R}^3} Q(f) \psi \, dp = \frac 14 \iiint_
{\mathbb{R}^{12}} W(p, p_{*}, p', p'_{*}) \,
 \,  q(f)\bigl(\psi + \psi_* - \psi' - \psi'_* \bigr) \, dp dp_* dp' dp'_*.
  \eqno(1.8)
$$
Taking $\psi(p) =1$, $\psi (p)= p_i$ and $\psi (p)=\mathcal{E}(p)$ and using the
definition of $\mathcal{C}$, we obtain that the particle number,
the momentum and the energy of a solution $f$ of the Boltzmann equation (1.5) 
are
conserved along the trajectoires, i.e.
$$
{d \over dt} \int_{\mathbb{R}^3} f(t,p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp =
\int_{\mathbb{R}^3} Q (f) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) \end{pmatrix} 
\, dp = 0,
\eqno(1.9)
$$
so that
$$
\int_{\mathbb{R}^3} f(t,p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp =
\int_{\mathbb{R}^3} f_{\rm in}(p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp.
\eqno(1.10)
$$
The entropy functional is defined by
$$
H(f) := \int_{R^3} h(f(p)) \, dp,
\quad h(f) = \tau^{-1} (1 + \tau f) \ln
(1 + \tau f) - f \ln f. \eqno(1.11)
$$
Taking in (1.8) $\psi = h'(f) = \ln (1 + \tau f) - \ln f$, we get
$$
\int_{\mathbb{R}^3} Q(f) \, h'(f) \, dp = \frac14  \, D (f)
\eqno(1.12)
$$
with
$$
\begin{aligned}
D(f) &= \iiiint_{\mathbb{R}^{12}} W \, e(f) \, dp dp_* dp' dp'_* \\
e(f) &= j\bigl( f f_* (1 + \tau \,f') (1 + \tau \,f'_*) , f' f'_* (1 + \tau \,f) 
(1 + \tau\,f_*) \\
j(s,t) &= (t-s) (\ln \, t - \ln \, s) \ge 0.
\end{aligned} \eqno(1.13)
$$
We deduce from the equation that the entropy is increasing along trajectories,
i.e.
$$
{d \over dt} H (f(t,.))  = \frac14  D(f) \ge 0.\eqno(1.14)
$$

The main qualitative characteristics of $f$ are described by these two
properties: conservation (1.9) and
increasing entropy (1.14). It is therefore natural to expect that as $t$
tends to $\infty$ the function $f$ converges to a function
$f_\infty$ which
 realizes the maximum of the entropy
$H(f)$ under the moments constraint (1.10).
A first simple and heuristic remark is that if $f_\infty$ solves the
entropy maximization problem with
constraints (1.10), there exist Lagrange multipliers $\mu \in \mathbb{R}$,
$\beta^0 \in \mathbb{R}$ and $\beta \in
\mathbb{R}^3$ such that
$$
\langle \nabla H(f_\infty), \varphi\rangle
=  \int_{\mathbb{R}^3} h'(f_\infty) \varphi \, dp  =
\langle \beta^0 \mathcal{E}(p) - \beta \cdot p - \mu,\varphi\rangle \quad
 \forall \varphi,
$$
which implies
$$
\ln (1 + \tau f_\infty) - \ln f_\infty = \beta^0 \mathcal{E}(p) - \beta
\cdot p - \mu
$$
and therefore
$$
f_\infty (p) = {1 \over e^{\nu(p)} - \tau}\quad
\hbox{with }\nu(p):=\beta^0 \mathcal{E}(p) - \beta \cdot p - \mu \,.
\eqno{(1.15)}
$$
The function $f_{\infty }$ is called a Maxwellian when $\tau = 0$, a
Bose-Einstein distribution when
$\tau > 0$ and a Fermi-Dirac distribution when $\tau < 0$.

\subsection{The classical case} %1.2

Let us consider for a moment the case $\tau =0$, i.e. the classic
Boltzmann equation, which has been widely studied.
It is known that for any initial data $f_{\rm in}$  there exists
a unique distribution $f_\infty$ of the form (1.15) such that
$$
\int_{\mathbb{R}^3} f_{\infty }(p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp =
\int_{\mathbb{R}^3} f_{\rm in}(p) \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) 
\end{pmatrix} \, dp.
\eqno(1.16)
$$

We may briefly recall the main results about the Cauchy problem and
the long time behaviour of the solutions which are
known up to now. We refer to \cite{C,L},
for a more detailed exposition and their proofs.

\begin{theorem}[Stationary solutions] \label{thm1}
For any measurable function $f \ge 0$ such that
$$
\int_{\mathbb{R}^3} f (1, p, {|p|^2 \over 2} ) \, dp = (N,P,E)
\eqno(1.17)
$$
for some $N,E > 0$, $P \in \mathbb{R}^3$, the following four assertions are 
equivalent:
\begin{itemize}
\item[(i)] $f$ is the Maxwellian
$${\mathcal{M}_{N,P,E} = \mathcal{M}[\rho,u,\Theta] = {\rho \over (2 \pi 
\Theta)^{3/2}}\exp \bigl( - {|p-u|^2 \over 2 \Theta} \bigr) }
$$
where $(\rho,u,\Theta)$ is  uniquely determined by
$N = \rho$, $P = \rho u$, and $E = {\rho \over 2} (|u|^2 +3 \Theta)$;

\item[(ii)] $f$ is the solution of the maximization problem
$$
H(f) = \max \{ H(g), \ g \hbox{ satisfies the moments equation } (1.10) \},
$$
where ${ H(g) = - \int_{\mathbb{R}^3} g \log g \, dp}$ stands for
the classical entropy;
\item[(iii)] $Q(f) = 0$;
\item[(iv)] $D(f) = 0$.
\end{itemize}
\end{theorem}

Concerning the evolution problem one can prove.

\begin{theorem} \label{thm2}
Assume that $w = 1$ (for simplicity). For any
initial data $f_{\rm in} \ge 0$ with finite
number of particles, energy and entropy, there exists a unique global solution 
$f \in
C([0,\infty);L^1(\mathbb{R}^3))$ which
conserves the particle number, energy and momentum.
Moreover, when $t \to \infty$, $f(t,.)$ converges to the Maxwellian $M$
with same particle number, momentum an
energy (defined by Theorem \ref{thm1}) and more precisely, for any $m > 0$ there
exists $C_m = C_m(f_0)$ explicitly
computable such that
$$
\| f - M \|_{L^1} \le {C_m \over (1+t)^m}.\eqno{(1.18)}
$$
\end{theorem}

We refer to \cite{Ar1,E,MW,Lu} for existence, conservations and
uniqueness and to \cite{Ar2,V,W,C,TV} for convergence to the equilibrium.
Also note that Theorem \ref{thm2} can be extended (sometimes only partially) to
a large class of cross-section $W$
we refer to \cite{V} for details and references.

\begin{remark} \label{rmk1.1} \rm
The proof of the equivalence (i) - (ii) only involves the entropy $H(f)$
and not the collision integral $Q(f)$ itself.
\end{remark}
\begin{remark} \label{rmk1.2} \rm
To show that (i), (iii) and (iv) are equivalent one
has first to define the quantities $Q(f)$ and $D(f)$ for the functions $f$
belonging to the physical functional space. The first difficulty is to
define precisely the collision integral $Q(f)$, (see Section 3.2).
\end{remark}

\subsection{Quantum and/or relativistic gases} %1.3

The Boltzmann equation looks formally
very similar in the different contexts: classic, quantum and
relativistic, but it actually presents some very different features in each of 
these different contexts.
 The two following remarks give some insight on these differences.

The natural spaces for the density $f$ are the
spaces of distributions $f \ge 0$ such that the ``physical" quantities are
bounded:
$$
\int_{\mathbb{R}^3} f (1 + \mathcal{E}(p)) \, dp < \infty
\quad\hbox{ and }\quad
H(f) < \infty,\eqno{(1.19)}
$$
where $H$ is given by (1.11). This provides the following  different conditions:
$$
\begin{aligned}
&f\in L^1_s\cap {L\log L}\quad \hbox{in the non quantum case, relativistic or
not}\\
&f\in L^1_s\cap L^{\infty }\quad \hbox{in the Fermi case, relativistic or not}\\
&f\in L^1_s \quad \hbox{in the Bose case, relativistic or not,}
\end{aligned}
$$
where
$$
L^1_s=\{f\in L^1(\mathbb{R}^3); \int_{\mathbb{R}^3} \,(1 + |p|^s)\,d|f|(p) 
<\infty\}
\eqno{(1.20)}$$
and $s = 2$ in the non relativistic case, $s = 1$ in the relativistic case.


On the other hand, remember that the density entropy $h$ given by (1.11) is:
$$
h(f) = \tau^{-1} (1 + \tau f)  \ln  (1 + \tau  f) - f  \ln  f.
$$
In the Fermi case we have $\tau =-1$ and then  $h(f) = +\infty$ whenever
$f \notin [0,1]$. Therefore the estimate $H(f)<\infty $ provides a strong
$L^\infty$ bound on $f$. But, in the Bose case,  $\tau =1$. A simple calculus 
argument then
shows that  $h(f)\sim \ln f$ as $f\to \infty $. Therefore  the entropy estimate 
$H(f)<\infty $ does not gives
any additional bound on $f$.

Moreover, and still concerning the Bose case, the following is shown in
\cite{CL}, in the context of the
Kompaneets equation (cf. Section 5). Let $a\in \mathbb{R}^3$ be any fixed vector 
and
$(\varphi _n)_{n\in \mathbb{N}}$  an approximation of the identity:
$$
(\varphi _n)_{n\in \mathbb{N}};\quad \varphi _n\to \delta _a.
$$
Then for any $f\in L^1_2$, the quantity
$H(f+\varphi _n)$ is well defined by (1.11) for all $n\in
\mathbb{N}$ and moreover,
$$N(f+\alpha \varphi _n)\to N(f)+\alpha ,\quad \hbox{and}\quad H(f+\varphi 
_n)\to
H(f)\quad \hbox{as}\quad n\to \infty.\eqno{(1.21)}
$$
See Section 2 for the details. This indicates that the expression of $H$ given 
in (1.11) may be extended to
nonnegative measures and that, moreover, the singular part of the measure does 
not contributes to the
entropy. More precisely, for any non negative measure
$F$ of the form $F=gdp+G$, where
$g\ge 0$ is an integrable function and
$G\ge 0$ is singular with respect to the Lebesgue measure $dp$, we define the 
Bose-Einstein entropy of $F$ by
$$
H(F) := H(g)= \int_{\mathbb{R}^3} \bigl[ (1+g) \ln (1+g) - g \ln g
\bigr] \, dp.\eqno{(1.22)}
$$

The discussion above shows how different is the quantum from the non quantum 
case, and even the Bose from the
Fermi case. Concerning the Fermi gases, the Cauchy
problem has been studied by  Dolbeault \cite{D} and Lions \cite{L}, under the 
hypothesis (H1) which includes the hard sphere
case
$w=1$. As it is indicated by the remark above, the estimates at our disposal in 
this case are even better than
in the classical case. In particular the collision term $Q(f)$ may be defined in 
the same way as in the
classical case. But as far as we know,
no analogue of Theorem \ref{thm1} was known for Fermi gases. The problem for 
Bose gases is
essentially open as we shall see below. Partial results for radially symmetric 
$L^1$
distributions have been obtained by Lu \cite{Lu} under strong cut off 
assumptions on the
function
$w$.

\subsubsection{Equilibrium states, Entropy} %1.3.1


As it is formally indicated by the identity (1.9), the particle number, momentum 
and energy of the
solutions to the Boltzmann equation are conserved along the trajectories. It is 
then very
natural to consider the following entropy maximization problem: given $N>0$, 
$P\in \mathbb{R}^3$ and $E\in \mathbb{R}$, find a
distribution $f$ which maximises the entropy $H$ and whose moments are $(N, P, 
E)$. The solution of this
problem is well known in the non quantum non relativistic case ( and is recalled 
in
Theorem \ref{thm1} above). In \cite{J},
 J\" uttner in [Ju] gave the relativistic Maxwellians.
 The question is also treated by  Chernikov in \cite{Ch}.
For the complete resolution of the moments equation in the relativistic non 
quantum case we refer to
 Glassey \cite{Gl} and  Glassey \& W. Strauss [GS].
 We solve the quantum relativistic case  in \cite{EMV}.
The general result may be stated as follows.

\begin{theorem} \label{thm3}
For every possible choice of $(N, P, E)$ such that the set $K$ defined by
$$
g\in K\quad \hbox{if and only if}\quad  \int_{\mathbb{R}^3} g (1, p, {|p|^2 
\over 2} ) \, dp = (N,P,E),
$$
is non empty, there exists a unique solution
$\overline{f}$ to the entropy maximization problem
$$
\overline{f}\in K,\quad H(\overline{f})=\max\{H(g); \, g\in K\}.
$$
Moreover,
$\overline{f}=f_{\infty }$ given by (1.15) for the nonquantum and Fermi case,
while for the Bose case $\overline{f}=f_{\infty }+\alpha
\delta _{\overline{p}}$ for some $\overline{p}\in \mathbb{R}^3$.
\end{theorem}

It was already observed by Bose and Einstein \cite{B, E1, E2} that for systems 
of bosons in thermal equilibrium  a careful
analysis of the statistical physics of the problem leads to enlarge the class of 
steady distributions to include also the
solutions containing a Dirac mass. On the other hand, the strong uniform bound 
introduced by the Fermi
entropy over the Fermi distributions leads to include in the family of Fermi 
steady states the so called degenerate
states. We present in Section 2 the detailed mathematical results of these two 
facts both for relativistic
and non relativistic particles. The interested reader may find the detailed
proofs in \cite{EMV}.

\subsubsection{Collision kernel, Entropy dissipation, Cauchy Problem} % 1.3.2

Theorem \ref{thm3} is the natural extension to quantum particles of the results 
for  non quantum particles, i.e.
points (i) and (ii) of Theorem \ref{thm1}. The extension of the points (iii) and 
(iv), even for the non relativistic
case, is more delicate. In the Fermi case it is possible to define the collision 
integral $Q(f)$ and the
entropy dissipation $D(f)$ and to solve the problem under some additional
conditions (see Dolbeault \cite{D} and Lions \cite{L}).
We consider this problem and related questions in Section 3.2.


In the equation for bosons, the first difficulty is to define the collision 
integral $Q(f)$
and the entropy dissipation
$D(f)$ in a sufficiently general setting. This question was treated by Lu in 
\cite{Lu} and solved under the
following additional assumptions:
\begin{itemize}
\item[(i)] $f\in L^1$ is radially symmetric.

\item[(ii)] Strong truncation on $w$.
\end{itemize}

These two conditions are introduced in order to give a sense to the collision 
integral.
Unfortunately, the second one is
not satisfied by the main physical examples such as $w=1$ (see Appendix 
\ref{A3}).
Moreover Theorem \ref{thm2.3} shows that the
natural framework to study the quantum Boltzmann equation, relativistic or
not, for Bose gases is the space of non negative measures. This is an
additional difficulty with respect to the non quantum or Fermi
cases. We partly extend the study of Lu to the case where $f$ is a
non negative radially symmetric measure.

\subsection{Two species gases, the Compton-Boltzmann equation} %1.4

Gases composed of two different species of particles,for example bosons and 
fermions, are interesting
by themselves for physical reasons and have thus been considered in the physical 
literature (see the
references below). On the other hand, from a mathematical point of vue, they 
provide simplified
but still interesting versions of Boltzmann equations for quantum particles. 
Their study may
be then a first natural step to understand the behaviour of this type of 
equations.


Let us then call
$F(t,p) \ge 0$ the density of Bose particles and $f(t,p) \ge 0$ that
of Fermi particles. Under the low density assumption, the evolution of the gas
is now given by the following system of Boltzmann equations (see \cite{CC}):
$$
\begin{gathered}
{\partial F \over \partial t} = Q_{1,1}(F,F) + Q_{1,2}(F,f) \quad F(0,.)
= F_{\rm in} \\
{\partial f \over \partial t} = Q_{2,1}(f,F) + Q_{2,2}(f,f) \quad f(0,.)
= f_{\rm in}.
\end{gathered} \eqno(1.23)
$$
The collision terms $Q_{1,1}(F,F)$ and $Q_{2,2}(f,f)$ stand for collisions
between particles of the same specie and are given by (1.2).
The collision terms $Q_{1,2}(F,f)$ and $Q_{2,1}(f,F)$ stands for collisions
between particles of  two different species:
$$
\begin{gathered}
Q_{1,2}(F,f) =  \int_{\mathbb{R}^3}  \int_{\mathbb{R}^3}
 \int_{\mathbb{R}^3} W_{1,2} q_{1,2}\, dp_* dp' dp'_*,\\
q_{1,2}= F' f'_* (1+F) (1 - \tau  \,f_*)
-  F f_* (1 + F') (1 - \tau  f'_*),
\end{gathered}\eqno(1.24)
$$
and $Q_{2,1}(f,F)$ is given by a similar expression. Note that the measure
$W_{1,2}=W_{1,2}(p,p_*,p',p'_*)$ satisfies
the micro-reversibility hypothesis
$$
W_{1,2}(p',p'_*,p,p_*) = W_{1,2}(p,p_*,p',p'_*),
\eqno(1.25)
$$
but not the indiscernibility hypothesis $W_{1,2}(p_*,p,p',p'_*) =
W_{1,2}(p,p_*,p',p'_*)$ as in (1.7) since the two colliding  particles belong 
now
to different species.


We consider in Section 4 some mathematical questions related to the systems
(1.23)-(1.25). We do not perform in detail the general
study of the steady states
since that would be mainly a repetition of what is done in Section 3 and
in \cite{EMV}. Here again, to give a sense to the integral collision $Q_{1,2}$
and $Q_{2,1}$ is the first question to be considered. Since the kernels
$Q_{1,1}(F,F)$ and
$Q_{2,2}(f,f)$ have already been treated in
the precedent section, we focus in the collision terms
$Q_{1,2}(F,f)$ and $Q_{2,1}(f,F)$. Let us notice that in the Fermi case,
$\tau  > 0$, the Fermi density $f$ satisfies an a priori bound in
$L^\infty$. Nevertheless, even with this extra estimate,the problem of existence 
of
solutions and their asymptotic behaviour for generic interactions, even with
strong unphysical truncation kernel and for radially symmetric distributions 
$f$, remains an open
question.


In order to get some insight on these problems, we
consider two simpler situations which are  important from a physical point
of view and still mathematically interesting since, in particular, they display 
Bose condensation in infinite time.
These are the equations describing boson-fermion interactions with fermions at 
equilibrium, and
photon-electron Compton scattering. Of course the deductions of
these two reduced models are well known in the physical literature but we 
believe
nevertheless that it may be interesting to sketch them here. In the first one,
still considered in Section 4, we suppose that the Fermi particles are at rest 
at
isothermal equilibrium. This is nothing but to fix the distribution of
Fermi particles $f$ in the system to be a Maxwellian or a Fermi state.
Without any loss of generality, this may be chosen to be centered at the origin, 
so
that it is radially symmetric. Moreover,
the boson-fermion interaction is  short range and  we
may  consider the ``slow particle interaction'' approximation of the
differential cross section $w=1$ (see Appendix \ref{A3}). The
system reduces then to a single equation which moreover is quadratic and not
cubic. Namely:
$$
{\partial F \over \partial t} =
\int_0^\infty  S (\varepsilon,\varepsilon') \bigl[ F' (1 + F) \,
e^{-\epsilon} - F (1 + F')
\, e^{-\epsilon'} \bigr] \, d\epsilon',\eqno{(1.26)}
$$
for some kernel S (see Section 4).

We prove in Section 4 the following result about existence, uniqueness and 
asymptotic
behaviour of global solutions for the Cauchy problem associated to (1.26) .

\begin{theorem} \label{thm4}
For any initial datum $F_{\rm in} \in L^1_1(\mathbb{R}_+)$, $F_{\rm in}\ge 0$,
there exists a solution $F \in C([0,\infty),L^1_{1/2}))$ to the equation
(1.26) such that
$$
\lim_{t\to 0}\|F(t)-F_{\rm in}\|_{L^1(\mathbb{R}^+)}=0.
$$
Moreover, if $\overline{f}=\mathcal{B}_N$ is the unique solution to the 
maximization
problem
\begin{gather*}
H(\overline{f})=\max\{H(g); \int \,g(\varepsilon)\,\varepsilon^2\,d\varepsilon= 
\int
\,F_{\rm in}(\varepsilon)\,\varepsilon^2\,d\varepsilon=: N\},\,\\
H(F)=\int_0^{\infty }h(f,
\varepsilon)\varepsilon^2d\varepsilon
\end{gather*}
with $h(x, \varepsilon)=(1+x)\ln (1+x)-x\ln x -\varepsilon x$,
$$
\begin{gathered}
 F(t,.) \mathop{\rightharpoonup}_{t \to \infty} \overline{f}
 \hbox{ weakly} \star \hbox{ in } \ \bigl( C_c({\mathbb{R}_+}) \bigr)' \\
\lim_{t\to \infty } \| F(t,.)-\overline{f} \|_{L^1((k_0,\infty))}=0 \quad
\forall k_0 > 0.
\end{gathered}\eqno(4.51)
$$
\end{theorem}

This result shows that the density of bosons $F$ underlies a Bose condensation
asymptotically in infinite time if its initial value is large enough.
The phenomena was already predicted by
Levich \& Yakhot in \cite{LY1, LY2}, and was described as condensation driven by 
the interaction
of bosons with a cold bath (of fermions) (see also Semikoz \& Tkachev
\cite{ST1,ST2}).

\subsubsection{Compton scattering}

In Section 5 we consider the equation describing the photon-electron interaction 
by
Compton scattering. This equation, that we call Boltzmann-Compton
equation, has been extensively studied in the physical and mathematical 
literature
(see in particular the works by Kompaneets \cite{K}, Dreicer \cite{D},
 Weymann \cite{W}, Chapline, Cooper and  Slutz \cite{CCS}). We show how it can
be derived starting from the system which describes the photon electron
interaction via Compton scattering. This interaction is described, in
the non relativistic limit, by the Thomson cross section, (see Appendix 
\ref{A3}).


It is important in this case to start with
the full relativistic quantum formulation since photons are relativistic
particles. Even if, later on, the electrons are considered at non
relativistic classical equilibrium. Finally, The equation has the same form as 
in (1.26) where
the only difference lies in the kernel
$S$.


The possibility of some kind of ``condensation'' for this Compton Boltzmann 
equation was
already considered in physical literature by  Chapline,  Cooper and  Slutz in 
\cite{CCS},
and  Caflisch \&  Levermore \cite{CL} for the Kompaneets equation
(see Section 5 and also \cite{EHV}).


We end with the so called Kompaneets equation, which is the limit of the 
Boltzmann-Compton equation
in the range $|p|, |p'|\ll mc^2$.

\section{The entropy maximization problem} % 2

In this Section we describe the solution to the maximization problem for
the entropy function under the moment
constraint. This problem may be stated as follows.
\begin{quote}
 Given any of the entropies $H$ and of the energies
$\mathcal{E}$ defined in the introduction,
given three quantities $N > 0$, $E > 0$ and  $P \in \mathbb{R}^3$, find $F
\ge 0$ such that
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ \mathcal{E}(p) \end{pmatrix} F(p) 
\, dp =
\begin{pmatrix} N \\ P \\ E \end{pmatrix}
\eqno(2.1)
$$
and
$$
H(F) = \max\{ H(g): g \mbox{ satisfies  (2.1)}\}. \eqno(2.2)
$$
\end{quote}
We  consider successively the case of a relativistic
non quantum gas, then the case of a Bose-Enstein gas, and last the
case of a Fermi-Dirac gas. For each of these two kind of gases,
we first consider in detail the
relativistic case, where the energy is given by
$$
\mathcal{E}(p) =  \sqrt{ 1\,  + {|p|^2\over c^2m^2}}.
\eqno(2.3)
$$
Then we describe the non relativistic case, $\mathcal{E}(p) = |p|^2/2m$, which 
is
simplest since by Galilean invariance it can be reduced to $P=0$.


In order to avoid lengthy technical details which are unnecessary for our 
purpose here, we
only present the results for each of the different cases. For the detailed
proofs the reader is referred to \cite{EMV}.


The relativistic non quantum case was completely solved by Glassey and
Strauss in \cite{GS1}, see also  Glassey in [Gl, even in
the non homogeneous case with periodic spatial dependence. However,
the proof that we give in \cite{EMV} is different, uses in a crucial way the 
Lorentz
invariance and may be adapted to the quantum relativistic case. Finally, notice 
that we do
not consider this entropy problem for a gas of photons ($\mathcal{E}(p) = |p|$ 
and
$H$  the Bose-Einstein entropy) since it would not have  physical meaning. In
Section 5 we discuss the entropy
problem for a gas constituted of electrons and photons.

\subsection{Relativistic non quantum gas} %2.1

In this subsection, we consider the Maxwell-Boltzmann entropy
$$
H(g) = - \int_{\mathbb{R}^3} g  \ln g \, dp.
\eqno(2.4)
$$
of a non quantum gas. From the heuristic argument presented in the
introduction, the solution to
(2.1)-(2.2) is expected to be a relativistic Maxwellian distribution:
$$
\mathcal{M}(p) = e^{ - \beta^0 p^0 + \beta \cdot p - \mu }.
\eqno(2.5)
$$
The result is the following.

\begin{theorem} \label{thm2.1} \begin{itemize}
\item[(i)] Given $E,N > 0$, $P \in \mathbb{R}^3$, there
exists a least one function $g \ge 0$
which solves the moments equation (2.1) if, and only if,
$$
m^2 c^2 \, N^2 + |P|^2 < E^2.
\eqno(2.6)
$$
When (2.6) holds we will say that $(N,P,E)$ is admissible.

\item[(ii)] For an admissible $(N,P,E)$ there exists at least one
relativistic Maxwellian distribution $\mathcal{M}$  satisfying (2.1).

\item[(iii)] Let $\mathcal{M}$ be a relativistic Maxwellian distribution. For 
any
function $g \ge 0$ satisfying
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, g(p) \, dp =
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \mathcal{M}(p) 
\, dp, \eqno(2.7)
$$
one has
$$
H (g) - H (\mathcal{M}) = H(g|\mathcal{M}) := \int_{\mathbb{R}^3} [g \ln {g 
\over \mathcal{M}} - g + \mathcal{M}
] \, dp. \eqno(2.8)
$$
Moreover, $H(g|\mathcal{M}) \le 0$ and vanishes if, and only if, $g = 
\mathcal{M}$.

\item[(iv)] As a conclusion, for an admissible $(N,P,E)$, the relativistic 
Maxwellian
constructed in (ii)  is the
unique solution to the entropy maximization problem (2.1)-(2.4).
\end{itemize}
\end{theorem}

\subsection{Bose gas} %2.2

We consider now a gas of Bose particles. As it has been said before, Bose 
\cite{B}
and Einstein \cite{E1,E2}, noticed that in this case, the set of steady 
distributions had to include solutions
containing a Dirac mass. It is then necessary to extend the entropy function $H$ 
defined in (1.11) with
$\tau = 1$ to such distributions. The way to do this may be well understood with 
the
following remark from \cite{CL}.

Let $a\in \mathbb{R}^3$ be any fixed vector and $(\varphi _n)_{n\in \mathbb{N}}$  
an
approximation of the identity:
$$
(\varphi _n)_{n\in \mathbb{N}};\quad \varphi _n\to \delta _a.
$$
For any $f\in L^1_2$ and every $n\in \mathbb{N}$ the quantity $H(f+\varphi _n)$
is well defined by (1.11). Moreover,
$$
H(f+\varphi _n)\to H(f)\quad \hbox{as}\quad n\to \infty.
$$
Suppose, for the sake of simplicity
that $\varphi _n\equiv 0$ if $|p-a|\ge 2/n$, then
$$
H(f+ \varphi _n)=\int_{|p-a|\ge 2/n}h(f(p, t), p)\,dp
+\int_{|p-a|\le 2/n}h((f(p, t)+ \varphi_n(p), p)\,dp.
$$
Let us call, $\sigma (z)=(1+z)\ln(1+z)-z\ln z$.
Using $|\sigma (z)|\le c\sqrt{z}$ we obtain
$$
\int_{|p-a|\le 2/n}|\sigma (f(p, t)+ \varphi_n(p))|dp
\le c{2\over \sqrt{ n^3}}\bigl(
\int_{|p-a|\le 2/n}(f(p, t)+ \varphi_n(p)dp\bigr)^{1/2}\to 0.
$$
Then
$$
\big|\int_{\mathbb{R}^3}h(f(p, t), p)dp-\int_{|p-a|\ge 2/n}\! h(f(p, t), 
p)dp\big|
\le \int_{|p-a|\le 2/n}|h(f(p, t), p)|dp\to 0
$$
which completes the proof.

This indicates that the expression of $H$ given in (1.11) may be extended to 
nonnegative measures and
that the singular part of the measure does not contributes to the entropy. More 
precisely,
for any non negative measure
$F$ of the form
$F=gdp+G$, where
$g\ge 0$ is an integrable function and
$G\ge 0$ is singular with respect to the Lebesgue measure $dp$,
we define the Bose-Einstein entropy of
$F$ by
$$
H(F) := H(g)= \int_{\mathbb{R}^3} \bigl[ (1+g) \ln (1+g) - g \ln g
\bigr] \, dp.\eqno{(2.9)}
$$
On the other hand, as we have seen in the Introduction, the
regular solutions to the entropy maximization problem should be the Bose
relativistic distributions
$$
b(p) = {1 \over  e^{ \nu(p) } - 1 }
\quad \hbox{ with } \quad
\nu(p) = \beta^0 p^0 - \beta \cdot p + \mu.
\eqno(2.10)
$$

The following result explains where the Dirac masses have now to be placed
(see \cite{EMV} for the proof).

\begin{lemma} \label{lm2.2}
The Bose relativistic distribution $b$ is non
negative and belongs to $L^1(\mathbb{R}^3)$
if, and only if, $\beta^0 > 0$, $|\beta| < \beta^0$ and $\mu \ge \mu_\mathbf{b}$ 
with
$\mu_\mathbf{b} := - m c \mathbf{b}$, $\mathbf{b} > 0$ and $\mathbf{b}^2= 
(\beta^0)^2 - |\beta|^2$.
In this case, all the moments of $b$ are well defined. Finally,
$$
\nu(p) > \nu(p_{m,c}) \ge 0 \quad \forall p \not= p_{m,c} :=
{m c \beta \over \sqrt{\beta^{0 \, 2} - |\beta|^2}}.
\eqno(2.11)
$$
\end{lemma}

We define now the generalized Bose-Einstein relativistic distribution 
$\mathcal{B}$ by
$$
\mathcal{B}(p) = b + \alpha \delta_{p_{m,c}}
= {1 \over  e^{ \nu(p) } - 1 } + \alpha \delta_{p_{m,c}}
\eqno(2.12)
$$
with
$$
\nu(p) = \beta^0 p_0 - \beta \cdot p + \mu > \nu(p_{m,c}) \ge 0
\quad \forall p \not= p_{m,c},
\eqno(2.13)
$$
and the condition $\alpha \nu(p_{m,c}) = 0$.


\begin{theorem} \label{thm2.3}
\begin{itemize}
\item[(i)] Given $E,N > 0$, $P \in \mathbb{R}^3$, there
exists at least one measure $F \ge 0$
which solves the moments equation (2.1) if, and only if,
$$
m^2 c^2 \, N^2 + |P|^2 \le E^2.
\eqno(2.14)
$$
When (2.14) holds we will say that $(N,P,E)$ is a admissible.

\item[(ii)] For any admissible $(N,P,E)$ there exists at least one
relativistic Bose-Einstein distribution $\mathcal{B}$ satisfying (2.1).

\item[(iii)]  Let $\mathcal{B}$ be a relativistic  Bose-Einstein distribution. 
For any
measure $F \ge 0$ satisfying
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, dF(p) =
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, 
d\mathcal{B}(p),
\eqno(2.15)
$$
one has
$$
H(F) - H(\mathcal{B}) = H_1(g|b) + H_2(G|b)
\eqno(2.16)
$$
where
$$
\begin{gathered}
H_1(g|b) := \int_{\mathbb{R}^3} \bigl( (1+g)  \ln {1+g \over 1+b} - g  \ln {g
\over b} \bigr) \, dp\,,\\
H_2(G|b) := - \int_{\mathbb{R}^3} \nu(p) \, dG(p). \end{gathered}
\eqno(2.17)
$$
Moreover, $H(F|\mathcal{B}) \le 0$ and vanishes if, and only if, $F = 
\mathcal{B}$. In
particular,
$H(F) < H(\mathcal{B})$ if $F \not= \mathcal{B}$.

\item[(iv)] As a conclusion, for an admissible $(N,P,E)$,
the relativistic Bose-Einstein distribution constructed in (ii) is the
unique solution to the entropy maximization problem
(2.1)-(2.3), (2.9).
\end{itemize}\end{theorem}


\subsubsection{Nonrelativistic Bose particles}
For non relativistic particles  the energy is
$\mathcal{E}(p)=|p|^2/2m$. By
Galilean invariance the problem (2.1) is then equivalent to the following 
simpler one: given three
quantities
$N>0$,
$E>0$, $P\in \mathbb{R}^3$ find $F(p)$ such
that
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ p-{P\over N} \\ |p-{P\over N}|^2/(2m)
\end{pmatrix}  F(p) \, dp =
\begin{pmatrix} N \\ 0 \\ E-(|P|^2/(2mN)) \end{pmatrix}.\eqno{(2.18)}
$$
It is rather simple, using elementary calculus, to prove that for any $E, N>0$,
$P\in \mathbb{R}^3$ there exists a distribution of the from
$$
F(p)={1\over e^{a|p-{P\over N}|^2+b^+}-1}-b^-\delta _{{P\over N}}\eqno{(2.19)}
$$
with
$a\in \mathbb{R}$, $b\in \mathbb{R}$, $\nu \in \mathbb{R}$, $b^+=\max (b, 0)$, 
$b^-=-\max (-b, 0)$
which satisfies (2.18).
Once such a solution (2.18) of (2.19) is obtained, the following
Bose-Einstein distribution
$$
\mathcal{B}(p)={1\over e^{\nu (p)}-1}+\alpha \delta _{p_{m,c}},\quad
\nu (p)=a|p|^2-{2a\over N}\cdot p+(b^++{a|P|^2\over N^2}) \eqno{(2.20)}
$$
solves (2.1).
This shows that for non relativistic  particles Theorem \ref{thm2.3} remains 
valid
under the unique following change: the statements (i) and (ii) have to be
replaced by
\begin{itemize}
\item[(i')] For every $E, N>0$, $P\in \mathbb{R}^3$,
there exists one relativistic Bose Einstein distribution defined by (2.19)
corresponding to these moments, i.e. satisfying (2.1).
\end{itemize}
Statements (iii) and (iv) of Theorem \ref{thm2.3} remain unchanged.

\subsection{Fermi-Dirac gas} %2.3

In this subsection we consider the entropy maximization problem for the
Fermi-Dirac entropy
$$
H_{FD}(f) := - \int_{\mathbb{R}^3} \bigl( (1-f) \ln (1-f) + f \ln f \bigr) \, 
dp.
\eqno(2.21)
$$
In particular, this implies the constraint $ 0 \le f \le 1$ on the density
$f$ of the gas.

From the heuristics argument presented in the introduction, we know that
the solution $\mathcal{F}$ of
(2.1)-(2.3), (2.21) is the Fermi-Dirac distribution
$$
\mathcal{F}(p) = {1 \over  e^{ \nu(p) } + 1 }
\quad \hbox{ with } \quad
\nu(p) = \beta^0 p^0 - \beta \cdot p + \mu.
\eqno(2.22)
$$
We also introduce the ``saturated" Fermi-Dirac (SFD) density
$$
\chi(p) = \chi_{\beta^0,\beta}(p) = {\bf 1}_{ \{ \beta^0 p^0 - \beta
\cdot p \le 1 \} } =
{\bf 1}_\mathcal{E} \hbox{ with } \mathcal{E} = \{ \beta^0 p^0 - \beta \cdot p 
\le 1 \} ,
\eqno(2.23)
$$
with $\beta \in \mathbb{R}^3$ and $\beta^0 > |\beta|$.

Our main result is the following.

\begin{theorem} \label{thm2.4}
\begin{itemize}
\item[(i)] For any $P$ and $E$ such that $|P| < E$
there exists an unique SFD state
$\chi = \chi_{P,E}$ such that $P(\chi) = P$ and $E(\chi) = E$. This one
realizes the maximum of particle number for
given energy $E$ and mean momentum $P$. More precisely, for any $f$ such
that $0 \le f \le 1$ one has
$$
P(f) = P, \quad E(f) = E \quad\hbox{ implies }\quad N(f) \le N(\chi_{P,E}).
\eqno(2.24)
$$
As a consequence, given $(N,P,E)$ there exists $F$ satisfying the
moments equation (2.1) if, and only
if, $E > |P|$ and $0 \le N \le N(\chi_{P,E})$. In this case, we say that
$(N,P,E)$ is admissible.

\item[(ii)] For any $(N,P,E)$ admissible there exists a Fermi-Dirac state
$\mathcal{F}$ (``saturated" or not) which
solves the moments equation (2.1).

\item[(iii)] Let $\mathcal{F}$ be a Fermi-Dirac state. For any $f$ such that $0 
\le f
\le 1$ and
$$
\int_{\mathbb{R}^3} f(p) \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} \, dp =
\int_{\mathbb{R}^3} \mathcal{F}(p) \begin{pmatrix} 1 \\ p \\ p^0 \end{pmatrix} 
\, dp,
$$
one has
$$
\begin{aligned}
H_{FD} (f) - H_{FD} (\mathcal{F}) =& H_{FD} (f | \mathcal{F}) \\
:=&
\int_{\mathbb{R}^3} \bigl( (1-f) \ln { 1-f \over 1-\mathcal{F}} - f \ln {f 
\over \mathcal{F}
} \bigr) \, dp.
\end{aligned} \eqno(2.25)
$$

\item[(iv)] As a conclusion, for any admissible $(N,P,E)$, the
relativistic Fermi-Dirac distribution
constructed in (ii) is the unique solution to the entropy
maximization problem
(2.1)-(2.3), (2.21).
\end{itemize}
\end{theorem}

The new difficulty with respect to the classic or the Bose case is to be
managed with the constraint $0\le f\le 1$.

\subsubsection{Nonrelativistic Fermi-Dirac particles}
Here again, since the energy is $\mathcal{E}(p)=|p^2|/2m$ the problem (2.1)
is equivalent to (2.18): given three quantities
$N>0$, $E>0$, $P\in \mathbb{R}^3$ find $f(p)$ such
that $0\le f\le 1$ and satisfying (2.18).
One may then check that for non relativistic  particles Theorem \ref{thm2.4}
remains valid under the unique following change: the
statements (i) and (ii) have
to be replaced by
\begin{itemize}
\item[(i')] For every $E, N>0$, $P\in \mathbb{R}^3$, satisfying
$5E\ge 3^{5/3}(4\pi )^{2/3}\, N^{5/3}$
there exists a non relativistic fermi Dirac state, saturated or not, defined by
$$
\mathcal{F}(p)=\begin{cases}
{1\over e^{\nu (p)}+1}, \mbox{where }\\
\nu (p)=a|p|^2-{2a\over N}P\cdot p+(b+{a|P|^2\over N^2})
& \mbox{if }E>{3^{5/3}(4\pi )^{2/3}\over 5}N^{5/3},\\
{\bf 1}_{\{|p-{P\over N}|\le c\}}
&\hbox{if }  E={3^{5/3}(4\pi )^{2/3}\over 5}N^{5/3}
\end{cases}
$$
corresponding to these moments, i.e., satisfying (2.1).
\end{itemize}
Statements (iii) and (iv) of Theorem \ref{thm2.4} remain unchanged.

\section{The Boltzmann equation for one single specie of quantum particles} %3

We consider now the homogeneous Boltzmann equation for quantum non relativistic 
particles, and treat
both Fermi-Dirac and Bose-Einstein particles.  We begin with the Fermi-Dirac 
Boltzmann equation for
which we may slightly improve the existence result of  Dolbeault [JD]áand Lions 
\cite{L}.
We also state a very simple (and weak) result concerning the long time behavior 
of solutions.
We finally consider the Bose-Einstein Boltzmann equation. We discuss the work of 
Lu \cite{Lu} and
slightly extend some of its results to the natural framework of measures

Consider the non relativistic quantum Boltzmann equation
$$
\begin{gathered}
{\partial f \over \partial t}  = Q(f)
= \iiint_{\mathbb{R}^9} w \delta_{\mathcal{C}} q(f) \, dv_* dv' dv'_*, \\
q(f)=\bigl( f' f'_* (1 + \tau \,f) (1 + \tau \,f_*)-
f f_* (1 + \tau \,f') (1 + \tau \,f'_*) \bigr)
\end{gathered}\eqno(3.1)
$$
where $\tau = \pm 1$ and $\mathcal{C}$ is the non relativistic collision 
manifold
of $\mathbb{R}^{12}$, $(\mathcal{C})$:
$$
\begin{gathered}
v_* + v'_* = v + v_* \\
{|v_*|^2 \over 2} + {|v'_*|^2 \over 2}= {|v|^2 \over 2} + {|v_*|^2 \over 2}.\\
\end{gathered} \eqno(3.2)
$$
We assume, without any loss of generality in this Section that the mass
$m$ of the particles is one.

\subsection{The Boltzmann equation for Fermi-Dirac particles} %3.1

We first want to give a mathematical sense to the
collision operator $Q$ in (3.1) under the physical natural bounds on the
distribution $f$.
Of course, if $f$ is smooth (say $C_c(\mathbb{R}^3)$) and $w$ is smooth (for 
instance $w=1$) the collision term
$Q(f)$ is defined in the distributional sense as it has been mentioned in the 
Introduction.
But, as we have already seen, the physical space for the densities of Fermi-
Dirac particles
is  $L^1_2 \cap L^\infty(\mathbb{R}^3)$.

To give a pointwise sense to the formula (3.1), we first recall the following
elementary argument from \cite{Gl}. After integration with respect to the $v'_*$
variable in (3.1) we have
$$
Q(f) = \int \!\! \int_{\mathbb{R}^6} w q(f)  \delta_{ \{ {|v'|^2 \over 2} + 
{|v+v_*-v'|^2 \over 2} - {|v|^2
\over 2} - {|v_*|^2 \over 2} = 0 \} }  \, dv_* dv'.
$$
By the change of variable $v' \to (r,\omega)$ with
$r \in \mathbb{R}$, $\omega \in S^2$, and $v' = v + r \omega$,
and using Lemma \ref{A1.1}, we obtain
$$
\begin{aligned}
Q(f) &= {1 \over 2} \int_{\mathbb{R}^3} \!\! \int_{S^2} \!\! \int_0^\infty w 
q(f)
\delta_{ \{ r (r - (v_*-v) \cdot \omega )  = 0 \} } \,  r^2 \,dr d\sigma
dv_* \\
&=  \int_{\mathbb{R}^3} \!\! \int_{S^2} w {|(v_*-v,\omega)| \over 2} q(f)  \,
d\omega dv_*,
\end{aligned} \eqno(3.3)
$$
where $v'$ and $v'_*$ are defined by
$$
v' = v + (v_*-v,\omega)  \omega, \quad
v'_* = v_* - (v_*-v,\omega) \omega. \eqno(3.4)
$$
Formula (3.3) gives a pointwise sense to $Q(f)$, say for $f \in 
C_c(\mathbb{R}^3)$.

To extend the definition of $Q(f)$ to measurable functions we
make the following assumptions on the cross-section:
\begin{gather*}
B = {1 \over 2} w |(v_*-v,\omega)|
\ \hbox{ is a function of }v_* - v \hbox{ and } \omega ,\tag{3.5}\\
B \in L^1(\mathbb{R}^3 \times S^2). \tag{3.6}
\end{gather*}
Though (3.5) is a natural assumption in view of Section 3.1, (3.6) is a strong 
restriction, in particular it does not hold when $w = 1$.
With these assumptions, first introduced in \cite{D}, we now explain how to give 
a sense to the collision
term $Q(f)$ when $f \in L^1 \cap L^\infty$.
In one hand, since $\Phi \ : \  (v,v_*,\omega) \  \mapsto \ (v',v'_*,\omega)$ 
(with $v'$, $v'_*$ given by
(3.4)) is a  $C^1$-diffeomorphism on  $\mathbb{R}^3 \times \mathbb{R}^3 \times 
S^2$ with jacobian $\hbox{Jac} \Phi = 1$,
we clearly have that $(v,v_*,\omega) \  \mapsto \ f' f'_*$ is a measurable 
function of
$\mathbb{R}^3 \times \mathbb{R}^3 \times S^2$.
On the other hand, performing a change of variable, we get
$$
\begin{aligned}
\iiint_{\mathbb{R}^3 \times \mathbb{R}^3 \times S^2}  B f' f'_* \, dv dv_* 
d\omega
&= \iint_{\mathbb{R}^3 \times \mathbb{R}^3} f f_* \Bigl( \int_{S^2}  B \, 
d\omega \Bigr) (v_* - v) \, dv dv_* \\
&\le \| f \|_{L^1}á\|áf \|_{L^\infty} \|áB \|_{L^1} < \infty,
\end{aligned}
$$
and by the Fubini-Tonelli Theorem,
$$
\int_{S^2}  B f' f' \, d\omega \ \in \ L^1(\mathbb{R}^3 \times \mathbb{R}^3).
$$
That gives a sense to the gain term
$$
Q^+(f) = \int_{\mathbb{R}^3} (1-f) (1-f_*)
\Bigl( \int_{S^2} B f' f' \, d\omega \Bigr) \, dv_*
$$
as an $L^1$ function. The same argument gives a sense to the loss term
$Q^-(f)$ as an $L^1$ function.

Note that, under the assumptions $B \in L^1_{\rm loc}(\mathbb{R}^3 \times S^2)$
and
$$
{1 \over 1+|z|^2} \iint_{B_R^v \times S^2}\!\!\! B(z+v,\omega) \, d\omega dv
\ \mathop{\longrightarrow}_{|z| \to \infty}  0 \quad \forall R > 0,
\eqno(3.7)
$$
one may give a sense to $Q^\pm(f)$ as a function of $L^1(B_R)$ ($\forall R > 0$) 
for any
$f \in L^1_2 \cap L^\infty$, see \cite{L}. In particular, the cross-section $B$ 
associated to $w = 1$
satisfies (3.7).


Finally, we can make a third assumption on the cross-section, namely
$$
0 \le B(z,\omega) \le (1+|z|^\gamma) \zeta(\theta), \hbox{ with } \gamma \in (-
5,0),
\quad \int_0^{\pi/2} \theta \zeta(\theta) \, d\theta < \infty.
\eqno(3.8)
$$
This assumption allows singular cross-sections, both in the $z$ variable and the
$\theta$ variable, near the origin. In that case, the collision term may be 
defined as a distribution
as follows:
$$
\int_{\mathbb{R}^3} Q(f) \varphi \, dv
= \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f f_* (1 - f' - 
f'_*) \, B \, K_\varphi \,
dv dv_*  d\omega
\eqno(3.9)
$$
with the notation
$$
K_\varphi = \varphi'+\varphi'_*-\varphi-\varphi_*. \eqno(3.10)
$$
To see that (3.9) is well defined we note that (see e.g. \cite{V}),
$$
| K_\varphi |\le C_\varphi |v-v_*|^2 \theta,
\eqno(3.11)
$$
from where,
$$
| f f_* (1 - f' - f'_*) \, B \, K_\varphi | \le
C_\varphi \theta \zeta(\theta) f f_* (|v-v_*|^2 + |v-v_*|^{\gamma+2})
\in L^1(\mathbb{R}^3 \times \mathbb{R}^3 \times S^2).
\eqno(3.12)
$$
To see this last claim, we just put $g(z) = |z|^a$; if $a \in (-3,0]$ one has $g 
\in L^1 + L^\infty$ and
therefore  $f (f \star g) \in L^1$ when $f \in L^1 \cap L^\infty$, and if $a \in 
[0,2]$ then
writing
$g(v-v_*) \le (1+|v|^2) (1 + |v_*|^2)$ we see that $f (f \star g) \in L^1$ when 
$f \in L^1_2$.


\begin{theorem} \label{thm3.1}
Assume that one of the conditions (3.6), (3.7) or (3.8) holds. Then, for any 
$f_{\rm in} \in
L^1_2(\mathbb{R}^3)$ such that $0 \le f_{\rm in} \le 1$ there exists a solution
$f \in C([0,+\infty);L^1(\mathbb{R}^3))$ to
equation (3.1). Furthermore,
$$
\begin{aligned}
&\int_{\mathbb{R}^3} f(t,v) \, dv = \int_{\mathbb{R}^3} f_{\rm in} \, dv =: N, 
\\
&\int_{\mathbb{R}^3} f(t,v) v \, dv = \int_{\mathbb{R}^3} f_{\rm in}(v) v \, dv 
=: P, \\
&\int_{\mathbb{R}^3} f(t,v) {|v|^2 \over 2} \, dv \le \int_{\mathbb{R}^3} f_{\rm 
in}(v) {|v|^2 \over 2} \, dv =: E,
\end{aligned}
\eqno(3.13)
$$
and
$$
\int_0^\infty \tilde D (f) \, dt \le C(f_{\rm in}),
\eqno(3.14)
$$
where
$$
\tilde D (f) := \iiint_{\mathbb{R}^3 \times \mathbb{R}^3 \times S^2} 
\!\!\!\!\!\!\!\! B_2 (f' f'_* (1 - f) (1 - f_*) - f f_* (1 - f') (1 - f'_*))^2 
\,
d\omega dv_* dv.
\eqno(3.15)
$$
\end{theorem}


\begin{remark} \label{rmk3.2} \rm
 In the non homogeneous context, the existence of solutions has been proved
by  Dolbeault \cite{D} under the assumption (3.6) and by  Lions \cite{L} under
the assumption (3.7). Existence in
a classical context under assumption (3.8) (without Grad's cut-off) has been
established by  Arkeryd \cite{Ar2}, Goudon \cite{G} and  Villani \cite{V}.
Finally, bounds on modified entropy dissipation term of the
kind of (3.15) have been introduced by Lu \cite{Lu} for the
Boltzmann-Bose equation.
\end{remark}

Concerning the behavior of the solutions we prove the following result.

\begin{theorem} \label{thm3.3}
For any sequence $(t_n)$ such that $t_n \to +\infty$ there
exists a subsequence $(t_{n'})$ and a stationary solution $\mathcal{S}$ such 
that
$$
f(t_{n'}+.,.) \ \mathop{\rightharpoonup}_{n' \to \infty} \ \mathcal{S}
\quad \hbox{ in } C([0,T];L^1 \cap L^\infty (\mathbb{R}^3) - weak) \quad \forall 
T > 0
\eqno(3.16)
$$
and
$$
\int_{\mathbb{R}^3} \mathcal{S} \, dv =  N, \quad
\int_{\mathbb{R}^3} \mathcal{S} v \, dv = P, \quad
\int_{\mathbb{R}^3} \mathcal{S} {|v|^2 \over 2} \, dv \le E.
\eqno(3.17)
$$
\end{theorem}

\begin{remark} \label{rmk3.4} \rm
 By stationary solution we mean a function $\mathcal{S}$ satisfying
$$
\mathcal{S}' \mathcal{S}'_* (1-\mathcal{S}) (1-\mathcal{S}_*) =
\mathcal{S} \mathcal{S}_* (1-\mathcal{S}') (1-\mathcal{S}'_*).
\eqno(3.18)
$$
Of course, such a function is, formally at least, a Fermi-Dirac distribution, we 
refer to \cite{D} for
the proof of this claim in a particular case.
Theorem \ref{thm3.3} also holds for the non homogeneous Boltzmann-Fermi 
equation, when, for example, the
position space is the torus, and under assumption (3.7) on $B$
(in order to have existence).
\end{remark}

\paragraph{Open questions:}
\begin{enumerate}
\item Is Theorem \ref{thm3.1} true under assumption (3.8) for all $\gamma \in (-
5,-2]$ ?

\item Is it possible to prove the entropy identity (1.14) instead of the 
modified
entropy dissipation bound (3.14)? Of course (1.14) implies the dissipation 
entropy bound (3.14) as it
will be clear in the proof of Theorem \ref{thm3.1}.

\item Is any function satisfying (3.18) a Fermi-Dirac distribution?

\item Is it true that
$$
\sup_{[0,\infty)} \int_{\mathbb{R}^3} f(t,v) |v|^{2 + \epsilon} \, dv \le 
C(f_{\rm in},\epsilon)
$$
for some $\epsilon > 0\,$? Notice that with such an estimate one could prove the 
conservation of the energy
(instead of (3.17)). If we could also give a positive answer to the question 3, 
we could then prove the
convergence to the Fermi-Dirac distribution
$\mathcal{F}_{N,P,E}$.

\item  Finally, is it possible to improve  the convergence (3.16), and prove for 
instance strong $L^1$
convergence?
\end{enumerate}

\paragraph{Proof of Theorem \ref{thm3.1}}
 Suppose that $B$ satisfies (3.6), (3.7) or (3.8) and define
$B_\epsilon = B {\bf 1}_{\theta > \epsilon}{\bf 1}_{\epsilon < |z| < 
1/\epsilon}$. Notice that $B_\epsilon$ satisfies
(3.6), $0 \le B_\epsilon \le B$ and $B_\epsilon \to B$ a.e.
From \cite{D} there exists a sequence of solutions $(f_\epsilon)$ to (3.1) 
corresponding to $(B_\epsilon)$.
Moreover, for any $\epsilon > 0$, the solution $f_\epsilon$ satisfies (3.13) and 
(1.14).
As it is shown by Lions in \cite{L}:
$$
\begin{aligned}
&a_\epsilon {f'_\epsilon} {f'_{\epsilon*}} (1 - f_\epsilon - f_{\epsilon *} )
\ \mathop{\rightharpoonup}_{n' \to \infty}
a {f' f'_*} (1 - f - f_* )
\quad L^1 \ \hbox{ weak}, \\
&a_\epsilon f_\epsilon f_{\epsilon*} {(1 - f'_\epsilon - f'_{\epsilon *} )}
\ \mathop{\rightharpoonup}_{n' \to \infty}
a {f f_* (1 - f' - f'_* )}
\quad L^1 \ \hbox{ weak},
\end{aligned}
\eqno(3.19)
$$
for any sequence $(f_\epsilon)$ such that $f_\epsilon \rightharpoonup f$ in $L^1 
\cap L^\infty$,
and any sequence $(a_\epsilon)$ satisfying (3.8) uniformly in $\epsilon$ and 
such that $a_\epsilon \to a$ a.e.
In particular, under the assumption (3.8) on $B$ and taking $a_\epsilon = 
B_\epsilon, a = B$, it is possible to
pass to the limit $\epsilon \to 0$ in the equation (3.1).
That gives existence of a solution $f$ to the Fermi-Boltzmann equation for $B$ 
satisfying (3.8).


Now, for $B$ satisfying (3.8), we just write
$$
\begin{aligned}
&\int_{\mathbb{R}^3} Q_\epsilon(f_\epsilon) \varphi \, dv\\
&= \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f_\epsilon 
f_{\epsilon *} {(1 - f_\epsilon' - f'_{\epsilon *})}
\, B_\epsilon  \, K_\varphi \,  {\bf 1}_{ \{ \theta \ge \delta, |v-v_*| \ge 
\delta \} } \, dv dv_*  d\omega
\\
&+ \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f_\epsilon 
f_{\epsilon *} {(1 - f_\epsilon' - f'_{\epsilon *})}
\, B_\epsilon \, K_\varphi \,  \bigl( {\bf 1}_{ \{ \theta \le \delta, |v-v_*| 
\ge \delta \}} +
{\bf 1}_{ \{ |v-v_*| \le \delta \}} \bigr) \, dv dv_*  d\omega \\
& \equiv \mathcal{Q}_{\delta,\epsilon} + r_{\delta,\epsilon}.
\end{aligned}
$$
For the first term $\mathcal{Q}_{\delta,\epsilon}$ we easily pass to the limit 
$\epsilon \to 0$ using (3.19).
For the second term $r_{\delta,\epsilon}$, using (3.11), we have
$$
\begin{aligned}
&\lim_{\epsilon \to 0}ár_{\delta,\epsilon}\\
&\le  \lim_{\epsilon \to 0} C_\varphi \int_{\mathbb{R}^3}\!\!\! 
\int_{\mathbb{R}^3}\!\!\!\! f_\epsilon f_{\epsilon *} (|v-v_*|^2 +
|v-v_*|^{\gamma+2})
\Bigl( \int_{S^2}  \theta \zeta(\theta) {\bf 1}_{ \{ \theta \le \delta \} } \, 
d\omega \Bigr) \, dv
dv_* \\
&+ \lim_{\epsilon \to 0} C_\varphi \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3} 
f_\epsilon  f_{\epsilon *} (|v-v_*|^2 +
|v-v_*|^{\gamma+2})  {\bf 1}_{ \{ |v_*-v| \le \delta \} }
\Bigl( \int_{S^2}  \theta \zeta(\theta) \, d\omega \Bigr) \, dv dv_*\\
&\le C_{\varphi,f} \int_{S^2}  \theta \zeta(\theta) {\bf 1}_{ \{ \theta \le 
\delta \} }
\, d\omega\\
&\quad+ C_{\varphi,\zeta} \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3} 
f_\epsilon f_{\epsilon *} (|v-v_*|^2 +
|v-v_*|^{\gamma+2})  {\bf 1}_{ \{ |v_*-v| \le \delta \} } \, dv dv_*.
\end{aligned}
$$
Therefore,
$$
\begin{aligned}
&\lim_{\epsilon \to 0} \int_{\mathbb{R}^3} Q_\epsilon(f_\epsilon) \varphi \, 
dv\\
&= \int_{\mathbb{R}^3}\!\!\int_{\mathbb{R}^3}\!\! \int_{S^2} f f_* {(1 - f' - 
f'_*)}
\, B  \, K_\varphi \,  {\bf 1}_{ \{ \theta \ge \delta, |v-v_*| \ge \delta \} } 
\, dv dv_*  d\omega
+ r_\delta,
\end{aligned}
$$
with $r_\delta \to 0$ when $\delta \to 0$. Since we also have
$$
\int_{\mathbb{R}^3} Q(f) \varphi \, dv
= \int_{\mathbb{R}^3}\!\!\! \int_{\mathbb{R}^3}\!\!\! \int_{S^2} f f_* (1 - f' - 
f'_*)
\, B  \, K_\varphi \,  {\bf 1}_{ \{ \theta \ge \delta, |v-v_*| \ge \delta \} } 
\, dv dv_*  d\omega
+ \tilde r_\delta,
$$
with $\tilde r_\delta \to 0$ when $\delta \to 0$, we conclude that
$$
\lim_{\epsilon \to 0} \int_{\mathbb{R}^3} Q_\epsilon(f_\epsilon) \varphi \, dv
= \int_{\mathbb{R}^3} Q(f) \varphi \, dv
$$
and $f$ is a solution to (3.1). We refer to \cite{L,V} for more details.
To establish (3.14), we take $a_\epsilon = a = \sqrt{B_\delta}$ in (3.19) and
we then have
$$
\begin{aligned}
\sqrt {B_{\delta }}[ f'_\epsilon f'_{\epsilon*}
(1 - f_\epsilon -& f_{\epsilon *} )
-f_\epsilon f_{\epsilon*} (1 - f'_\epsilon - f'_{\epsilon *} )] \\
& \mathop{\rightharpoonup}_{\epsilon \to 0}
\sqrt {B_{\delta }} [ f' f'_* (1 - f - f_* ) - f f_* (1 - f' - f'_* ) ].
\end{aligned}
\eqno(3.20)
$$
Using the elementary inequality
$$
(b-a)^2 \le j(a,b) \quad \forall \ a,b \in [0,1],
$$
we have for $\epsilon \in (0,\delta]$
$$
\int_0^\infty \tilde D_\delta(f_\epsilon) \, dt
\le \int_0^\infty \tilde D_\epsilon(f_\epsilon) \, dt
\le \int_0^\infty D_\epsilon(f_\epsilon) \, dt \le C(f_{\rm in}).
\eqno(3.21)
$$
Gathering (3.20) and (3.21) we obtain, using the convexity of $s \mapsto s^2$,
$$
\int_0^\infty \tilde D_\delta(f) \, dt
\le  \liminf_{\epsilon \to 0}á\int_0^\infty \tilde D_\delta(f_\epsilon) \, dt
\le C(f_{\rm in}),
$$
and we recover (3.15) letting $\delta \to 0$. \hfill$\square$\smallskip


\paragraph{Proof of Theorem \ref{thm3.3}}
 Let consider $f_n = f(t+t_n,.)$ as in the statement of the Theorem 
\ref{thm3.3}.
We know that there exists $n'$ and $\mathcal{S}$ such that
$f_{n'} \ \mathop{\rightharpoonup}_{n' \to \infty}  \mathcal{S}$ weakly in
$L^1 \cap L^\infty ((0,T) \times \mathbb{R}^3)$ for all $T > 0$, and we only 
have to
identify the limit $\mathcal{S}$. On one hand, $\mathcal{S}$
satisfies the moments equation (3.17). On the other hand by (3.21) and
lower semicontinuity we get
$$
\int_0^T \tilde D_\delta (\mathcal{S}) \, ds \le \liminf \int_0^T \tilde D 
(f_{n'}) \, ds
\le \liminf \int_{t_{n'}}^{T+t_{n'}} \tilde D (f) \, ds = 0
$$
for any $\delta > 0$, so that
$$
\mathcal{S}' \mathcal{S}'_* (1 - \mathcal{S}) (1 - \mathcal{S}_* ) - \mathcal{S} 
\mathcal{S}_* (1 - \mathcal{S}') (1 - \mathcal{S}'_* )
=
\mathcal{S}' \mathcal{S}'_* (1 - \mathcal{S} - \mathcal{S}_* ) - \mathcal{S} 
\mathcal{S}_* (1 - \mathcal{S}' - \mathcal{S}'_* ) =0
$$
for a.e. $(v,v_*,\omega) \in \mathbb{R}^3 \times \mathbb{R}^3 \times S^2$. In 
particular,
${\partial \mathcal{S} \over \partial t} = Q(\mathcal{S}) = 0$
and $\mathcal{S}$ is a constant function in time. We finally improve the 
convergence
of $f_{n'}$ to $\mathcal{S}$ and establish (3.16).
To this end we note that for any  $\psi \in C_c(\mathbb{R}^3)$,
$$
{d \over dt} \int_{\mathbb{R}^3} f_{n'} \psi \, dv = \int_{\mathbb{R}^3} 
Q(f_{n'}) \psi \, dv
$$
and the right hand side is bounded in $L^1(0,T)$. Therefore, 
$\int_{\mathbb{R}^3} f_n \psi \, dv$ is bounded
in $BV(0,T)$.  We can then extract a second subsequence
(not relabeled) such that
${ \bigl(\int_{\mathbb{R}^3} f_{n'} \psi \, dv \bigr) }$
converges strongly $L^1(0,T)$ and a.e. on
$(0,T)$ and the limit is obviously
${ \int_{\mathbb{R}^3} \mathcal{S} \psi \, dv }$.
Let  $\tau \in (0,T)$ be such that ${ \bigl(\int_{\mathbb{R}^3} f_{n'} \psi \, 
dv \bigr) }
\to { \int_{\mathbb{R}^3} \mathcal{S} \psi \, dv }$. We deduce that
$$
\int_{\mathbb{R}^3} f_{n'}(t,.) \psi \, dv = \int_{\mathbb{R}^3} f_{n'}(\tau) 
\psi \, dv
- \int_\tau^t \int_{\mathbb{R}^3} Q(f_{n'}) \psi \, dv ds
$$
and using P. L. Lions' result we get
$$
\int_0^T \int_{\mathbb{R}^3} Q(f_{n'}) \psi \, dv ds \to \int_0^\tau 
\int_{\mathbb{R}^3} Q(\mathcal{S}) \psi \, dv ds = 0,
$$
and therefore
$$
\sup_{[0,T]} \bigl| \int_{\mathbb{R}^3} f(t_{n'}) \psi \, dv - 
\int_{\mathbb{R}^3} \mathcal{S} \,
 \psi \, dv \bigr| \ \to \ 0.
$$
\hfill$\square$

\subsection{Bose-Einstein collision operator for isotropic density} %3.2

We consider now the Boltzmann equation for Bose-Einstein particles and take
$\tau = 1$ in (3.1).
We start with an elementary remark. Assume that $w = 1$ and consider a sequence 
$(f_\epsilon)$ defined by
$f_\epsilon(v) := \epsilon^{-3} f(v/\epsilon)$ for a given $0 \le f \in 
C_c(\mathbb{R}^3)$, $f \not\equiv 0$.
An elementary change of variables leads to
$$
\|Q^\pm(f_\epsilon) \|_{L^1} = {1 \over \epsilon^3} \|Q^\pm(f) \|_{L^1} \to 
+\infty.
\eqno
$$
Therefore, no a priori estimate of the form
$$
\| Q(f) \| \le \Phi (\|f \|_{L^1}), \hbox{ with }\Phi \in C(\mathbb{R}_+),
\eqno
$$
can be expected. In particular we will not be able to give a sense to the kernel
$Q(f)$ under the only physical bound $f \in M^1_2(\mathbb{R}^3)$ for such a $w$.


This first remark motivates the two following simplifications, originally 
performed by Lu \cite{Lu}
to give a sense to $Q(f)$: we assume that the density is isotropic and we make a 
strong (and unphysical)
truncation assumption on $w$.  We use them here, in a slightly different way 
that we believe to be simpler.


We then assume, until the end of this Section, that the density $f$ only depends 
on the quantity $|v|$, and
denote
$f(v) = f(|v|) = f(r)$ with
$r = |v|$. For a given function $q = q (r,r_*;r',r'_*)$ we define
$$
\mathcal{Q} [q] (v) : = \int_{\mathbb{R}^3} \!\! \int_{\mathbb{R}^3} \!\! 
\int_{\mathbb{R}^3} w q \delta_{\mathcal{C}} \, dv_* dv'
dv'_*.
\eqno(3.22)
$$
By introducing the new function
$$
\hat{w} (r,r_*,r',r'_*) := \iiint_{S^2 \times S^2 \times S^2}
w(v,v_*,v',v'_*) \delta_{\{ v+v_* = v' + v'_* \} } \, d\sigma_* d\sigma' 
d\sigma'_*
\eqno(3.23)
$$
and the notation $v_* = r_* \sigma_*$, $v_* = r' \sigma'$, $v'_* = r'_* 
\sigma'_*$,
$d\hat{r}_* = r_*^2 \, dr_*$, $d\hat{r}' = {r'}^2 \, dr'$, $d\hat{r}'_* = 
{r'_*}^2 \, dr'_*$,
$$
\widehat{\mathcal{C}}= \{(r,r_*,r',r'_*) \in \mathbb{R}_+^4, \  r^2 + r_*^2 = 
{r'}^2 + {r'}_*^2 \},
\eqno(3.24)
$$
we may write $\mathcal{Q}[q]$ in the simple way
$$
\mathcal{Q} [q] (r) = \iiint_{\mathbb{R}_+^3} \hat{w} 
\,\delta_{\widehat{\mathcal{C}}} \,q \, d\hat{r}_* d\hat{r}' d\hat{r}'_*.
\eqno(3.25)
$$
Let us emphasize that $\hat{w}$ is indeed a function of $r = |v|$ (and not on 
the whole variable $v$)
since, for any
$R \in SO(3)$, we have
$$
\begin{aligned}
&\iiint_{S^2 \times S^2  \times S^2}
w(R v,v_*,{v'},{v'_*}) \delta_{\{ R v+v_* = v' + v'_* \} } \, d\sigma_*
{d\sigma'} {d\sigma'_*} \\
&= \iiint_{S^2 \times S^2 \times S^2}
w(R v,R v_*,R {v'},R {v'_*}) \delta_{\{ R v+R v_* = R v' + R v'_* \} } \, d\sigma_*
{d\sigma' d\sigma'_*} \\
&= \iiint_{S^2 \times S^2 \times S^2}
w( v,v_*,{v'},{v'_*}) \delta_{\{ v+v_* = v' + v'_* \} } \, d\sigma_*
{d\sigma' d\sigma'_*}
\end{aligned}
$$
by the change of variables $(\sigma_*,\sigma',\sigma'_*) \to (R \sigma_*,R 
\sigma',R \sigma'_*)$ and
the rotation invariance of $w$. Moreover, $\hat{w}$ clearly satisfies the 
property
$$
\hat{w} (r,r_*,r',r'_*) =  \hat{w} (r_*,r,r',r'_*) =  \hat{w} (r',r'_*,r,r_*).
\eqno(3.26)
$$
     %%%%%%%%%   L^1   %%%%%%%

\begin{lemma} \label{lm3.5}
Assume that $w$ is such that $B$ defined by (3.5) satisfies
$$
\sup_{z \in \mathbb{R}^3} {1 \over 1 + |z|^s} \int_{S^2} B(z,\omega) \, d\omega
< \infty \eqno(3.27)
$$
with $s = 2$ and that $\hat{w}$ defined by (3.26) satisfies
$$
\hat{w} (r+r_*+r'+r'_*) \in L^\infty(\mathbb{R}^4_+).
\eqno(3.28)
$$
Then for any isotropic function $f \in L^1_2(\mathbb{R}^3)$ the kernel 
$Q^\pm(f)$
is well defined and
$$
\|Q^\pm (f) \|_{L^1} \le  C_B \|f \|^2_{L^1_2(\mathbb{R}^3)} + C_{\hat{w}} \|f 
\|^3_{L^1(\mathbb{R}^3)}.
\eqno(3.29)
$$
\end{lemma}
A refined version of the bound (3.29) has been used, by Lu \cite{Lu}, in order 
to prove a global existence
result when $s = 0$ in (3.27). For $s = 1$, X. Lu also proves an existence 
result under the
additional assumption that $B$ has the  particular form:
$B(z,\omega) = |z|^\gamma \zeta(\theta)$ with $\gamma \in [0,1]$, $\zeta \in 
L^1$.
Here condition (3.28) has to be understood as a truncation assumption
near the origin
$$
\exists B_0 \in (0,\infty), \quad B(z,\omega) \le B_0 (\cos \theta)^2 \sin 
\theta |z|^3,
\eqno(3.30)
$$
introduced in \cite{Lu}. In order to clarify the assumption (3.28) let us state
the following result.

\begin{lemma} \label{lm3.6}
\begin{enumerate}
\item For $w = 1$,
$$
\hat w = {4 \pi^2 \over r \, r_* \, r' \, r'_*} \min (r,r_*,r',r'_*).
\eqno(3.31)
$$
\item As a consequence, any cross-section $w$ such that
$$
\exists w_0 \in (0,\infty), \quad
w \le w_0 ((|v'-v| |v'_* - v|) \wedge 1)
\eqno(3.32)
$$
satisfies (3.27)-(3.28).
\end{enumerate}
\end{lemma}

Note that condition (3.32) is exactly the X. Lu's assumption (3.30) near the 
origin. This condition
kills the interaction between particles with low energy. We emphasize that this 
assumption is never
satisfied by the physically relevant cross-sections.

\paragraph{Proof of Lemma \ref{lm3.5}} We may write
$$
f' f'_* (1 + f) (1 + f_*) - f f_* (1 + f') (1 + f'_*)
= f' f'_* (1 + f + f_*) - f f_* (1 + f' + f'_*).
\eqno(3.33)
$$
Then, we have to define $\mathcal{Q} [q]$ for two kinds of terms $q$: for the 
quadratic terms $q = f f_*$
and
$q = f' f'_*$ and for cubic terms $q = f' f'_* f$, $q = f' f'_* f_*$,
$q = f f_* f'$ and $q = f f_* f'_*$. The quadratic terms may be defined in the 
same way
that for the Fermi-Dirac Boltzmann equation in Section 3.1 thanks to assumption 
(3.28) and they are
bounded by the first term in the right hand side of estimate (3.29).

We then focus on the cubic terms. We define them by performing one integration 
more in the
representation formula (3.25) (with respect to  one of the variables $r_*$, 
$r'$, $r'_*$) but we
still use the formula (3.25) in order to preserve the symmetries of 
$\mathcal{Q}$.

Let us first assume that moreover, $f \in C_c(\mathbb{R}^3)$. Performing in 
(3.25) the integration in
the $r_*$ variable we get, using Lemma \ref{lmA1.1},
$$
\mathcal{Q}[q] (r) = \iint_{\mathbb{R}_+^2} \hat{w} {r_* \over 2} {\bf 1}_{ \{ 
{r'}^2+ {r'_*}^2 \ge r^2 \} }
\,  q \, d\hat{r}' d\hat{r}'_*.
\eqno(3.34)
$$
where  $r_* = \sqrt{{r'}^2+{r'_*}^2-r^2}$.
It is then clear that (3.33) defines $\mathcal{Q}[q]$ as measurable function.

Moreover, we write
$$
\int_{\mathbb{R}_+} \mathcal{Q} [q] (r) d\hat{r} = \iiiint_{\mathbb{R}_+^4} 
\hat{w} \,\delta_{\widehat{\mathcal{C}}} \,q
\, d\hat{r} d\hat{r}_* d\hat{r}' d\hat{r}'_*
\eqno
$$
and first, perform an integration with respect to the variable which does not
appear in $q$ (namely, $r_*$ if $q = f' f'_* f$,  $r$
if $q = f' f'_* f_*$, and so on) and we get (using Lemma \ref{lmA1.1} again)
$$
\begin{aligned}
&\int_{\mathbb{R}_+} \mathcal{Q} [q] (r) d\hat{r}\\
&= \iiint_{\mathbb{R}_+^3} \hat{w}(r_1,r_2,r_3,r_4) {r_4 \over 2} {\bf 1}_{ \{ 
r_1^2+r_2^2 \ge r_3^2 \} }  \,  f(r_1) f(r_2) f(r_3) \,
 d\hat{r}_1 d\hat{r}_2 d\hat{r}_3 \\
& \le \| \hat{w} {(r+r_*+r'+r'_*)} \|_{ L^\infty(\mathbb{R}^4_+)} \|f 
\|^3_{L^1(\mathbb{R}^3)},
\end{aligned} \eqno(3.35)
$$
where in the first inequality we have used the notation
$r_4 = \sqrt{ r_1^2 + r_2^2 - r_3^2 }$.
We finally deduce (3.29) using a density-continuity argument.
\hfill$\square$ \smallskip


\paragraph{Proof of Lemma \ref{lm3.6}}
 We start by proving (3.31). Using that
$$
\delta (v+v_*-v'-v'_*) = \int_{\mathbb{R}^3} e^{i (z, v+v_*-v'-v'_*)} { dz \over 
(2\pi)^3}
=\int_0^\infty \!\! \int_{S^2} e^{i (z, v+v_*-v'-v'_*)} \, d\sigma {r^2 \, dr  
\over (2\pi )^3}
$$
where $z =|z| \sigma$ and $\epsilon = |z|$ in polar coordinates,
we obtain
$$
\begin{aligned}
&\int_{S^2}\!\int_{S^2}\!\int_{S^2} \delta {(v+v_*-v'-v'_*)} \, {d\sigma_* d\sigma'} 
{d\sigma'_*} \\
& = \int_0^\infty \int_{S^2}\!\int_{S^2}\!\int_{S^2}\!\int_{S^2}
e^{i (z, v+v_*-v'-v'_*)} \, {d\sigma d\sigma_* d\sigma' d\sigma'_*} {\epsilon^2 \, 
d\epsilon
\over(2\pi)^3} \\
& = \int_0^\infty {\epsilon^2 \, d\epsilon \over(2\pi)^3} \int_{S^2} e^{i (z, 
v)} \, d\sigma
\int_{S^2} e^{i (z,v_*)} \, d\sigma_* \int_{S^2} e^{i (z, v')} \, {d\sigma'}
\int_{S^2} e^{i (z, v'_*)} \, {d\sigma'_*} \\
&={16 \pi \over r \, r_* \, r' \, r'_*} \int_0^\infty {d\epsilon \over (2\pi)^3 
\epsilon^2} {\sin }(\epsilon \, r) {\sin }(\epsilon \, {r_*}) {\sin }(\epsilon \, {r'}) 
{\sin (\epsilon \, r'_*)}.
\end{aligned}
$$
Then (3.31) follows by an lengthy elementary trigonometric computation.
It is clear, thanks to (3.25), that (3.32) implies (3.27).
We then have to prove that (3.32) implies (3.28).
For any  $r,r_*,r',r'_*$ given, we set
$m_1 = \min (r,r_*,r',r'_*)$,
$m_2 = \min (\{ r,r_*,r',r'_*\}\backslash \{ m_1 \})$,
$m_3 = \min (\{ r,r_*,r',r'_*\}\backslash \{ m_1,m_2 \})$
and finally, $m_4 = \max ( r,r_*,r',r'_*)$.
Since $|v-v'| = |v_*-v'_*|$, $|v-v'_*| = |v_*-v'|$ and
$\{ r,r_* \}\not= \{ m_1,m_2 \}$, $\{ r,r_* \}\not= \{ m_3,m_4 \}$,
the assumption (3.32) leads to
$$
\begin{aligned}
w &\le w_0 \bigl\{ \bigl[ 4 \min \bigl( \max(r,r'), \max(r_*,r'_*) \bigr) 
\min\bigl(
\max(r,r'_*),
\max(r_*,r')\bigr) \bigr] \wedge 1 \bigr\} \\
&\le w_0  \bigl\{ \bigl[ 4 \, m_2 \, m_3 \,  \bigr] \wedge 1 \bigr\}.
\end{aligned}
$$
Gathering (3.31) and this last inequality  we deduce
$$
\hat{w} (r+r_*+r'+r'_*) \le
{4 \pi^2 \, m_1 \over r \, r_* \, r' \, r'_*} (4 \, w_0 \, m_2 \, m_3) 
(r+r_*+r'+r'_*)
\le 64 \pi^2 \, w_0,
$$
and that concludes the proof.
\hfill$\square$ \smallskip

We want now to extend the previous arguments and give a sense to $Q(F)$ for $F 
\in M^1(\mathbb{R}^3)$. For the
quadratic term that problem has been solved by Povzner in \cite{P}. For the 
cubic term we write for a
radial measure $dF = f \, r^2 \, dr$
$$
\langle \mathcal{Q}[q],\phi\rangle = \langle F \otimes F \otimes F , 
B_{\hat{w}}[\psi]
\rangle \eqno(3.36)
$$
with
$$
B_{\hat{w}}[\psi](r_1,r_2,r_3) := \hat{w}(r_1,r_2,r_3,r_4) {r_4 \over 2}
{\bf 1}_{ \{ r_1^2+r_2^2 \ge r_3^2 \}} \psi
$$
and $r_4 = \sqrt{ r_1^2 + r_2^2 - r_3^2 }$ and $\psi (r_1,r_2,r_3) = \phi(r_    
i)$ $i = 1,2,3$ or $4$
depending on $q$.
Under the condition
$$
B_{\hat{w}}[1] \in C (\mathbb{R}_+^3)
\eqno(3.37)
$$
we clearly have $B_{\hat{w}}[\psi] \in C_c (\mathbb{R}_+^3)$ for any $\phi \in 
C_c(\mathbb{R}_+)$ and then  the right
hand side term of (3.36) is well defined.
The assumption
$$
B \in C(\mathbb{R}^3 \times S^2), \eqno(3.38)
$$
which is satisfied by the hard potentials, guaranties that the
quadratic term is well defined and that $\hat{w} \in C(E  \backslash \{á0 \})$ 
where
$E = \{ (r_1,r_2,r_3,r_4) \in \mathbb{R}^4_+, \, r_1^2 + r_2^2 - r_3^2 - r_4^2 = 
0 \}$.
The condition (3.37) is then satisfied if moreover
$$
\lim_{ (r_1,r_2,r_3) \to 0}  \hat{w}(r_1,r_2,r_3,r_4) \, r_4 = 0\,.
$$
From the computations performed in the proof of Lemma \ref{lm3.6}. it is clear 
that the last condition holds
if we assume, for instance
$$
w  \le w_0 \,  |v'- v|^\gamma \wedge 1, \quad \gamma > 1.
\eqno(3.39)
$$
As a conclusion, under assumption (3.27), (3.38), (3.39) we may define
the collision kernel for general non negative bounded and isotropic measures.
From all the above one easily deduces the following existence result for (3.1)
with $\tau =1$ in the framework of non negative bounded measures.

\begin{theorem} \label{thm3.7}
 Assume $w$ satisfies (3.39). Let $F_{\rm in} \in M^1_{\rm rad}(\mathbb{R}^3)$,
$F_{\rm in}\ge 0$. Then, there exists a unique global solution
$F = g + G \in C([0,\infty),M^1_{\rm rad}(\mathbb{R}^3))$ to (3.1) with
$\tau=1$.
\end{theorem}

\begin{remark} \label{rmk3.8} \rm
It is straightforward to check that in the radially
symmetric case Theorem \ref{thm2.4} gives:
\end{remark}

\begin{theorem} \label{rmk2.5'}
Let $0 \le F \in M^1_{\rm rad}(\mathbb{R}^3)$ an isotropic measure on 
$\mathbb{R}^3$ such
that
$$
\int_{\mathbb{R}^3} \begin{pmatrix} 1 \\ |v|^2 / 2 \end{pmatrix} \, dF(v) =
\begin{pmatrix} N  \\ E \end{pmatrix}
\quad \hbox{and of course} \quad
\int_{\mathbb{R}^3} v \, dF(v) = 0,\eqno{(3.41)}
$$
with $N > 0$, $E > 0$.
Therefore the two following assertions are equivalent:
\begin{itemize}
\item[(i)] $F$ is the Bose-Einstein distribution $\mathcal{B}[N,0,E]$,

\item[(ii)] $F$ is the solution of the maximization problem:
$$\mathcal{H}(F) = \max \{ \mathcal{H}(F') \mbox{ with $F'$ satisfying (3.41)}.
$$
\end{itemize}
\end{theorem}


\paragraph{Open questions:}
In the $L^1$ setting X. Lu has proved in \cite{Lu}:
\begin{itemize}
\item[(i)] For any $(t_n)$ such that $t_n \to \infty$ there exists $m' \le m$, 
$E' \le E$ and a subsequence
$(t_{n'})$ such that
$$
g(t_{n'})  \rightharpoonup  b_{m',0,E'} \quad \hbox{biting } L^1_{\rm rad} \ 
\hbox{áweak.}
$$

\item[(ii)] For a given $E$ there exists $N_c = N_c(E)$ such that if
$N(f_{\rm in}) < N_c$ and $E(f_{\rm in}) = E$ then
$$
g(t)  \rightharpoonup  \mathcal{B}_{N,0,E} = b_{N,0,E} \quad L^1_{\rm rad} \
\hbox{weak.}
$$
\end{itemize}
Where the distributions $b$ and $\mathcal{B}$ are defined in (2.22) and (2.25).
The two following questions are then natural.

\noindent
1. Is it possible to  construct (global?) solutions to (3.1) for $\tau =1$ 
without
the strong truncation condition (3.39); for instance for $w =1$? What is the 
qualitative behavior of
such  solutions ?

\noindent
2. Is it possible to prove that, under the strong truncation condition (3.39),
$$
F(t) \rightharpoonup \mathcal{B}_{N,0,E} \quad
\hbox{weakly } \sigma (M^1,C_c)
\quad\hbox{ and }\quad
g \to b_{N,0,E} \quad\hbox{strongly}L^1(\mathbb{R}^3 \backslash \{ 0 \})
$$
as it may be expected from the stationary analysis?
If $N(g_{\rm in}) < N_c$, is it possible to prove strong convergence instead of 
result (ii) in Theorem
3.5?

\begin{remark} \label{rmk3.9}\rm
The Boltzmann-Compton equation, introduced in Section 5 below, is a
particular case of (3.1) with $\tau =1$. It has been proved in
\cite{EM1,EM2} that it also has
global solutions $F=g+G\in C([0,\infty),M^1_{\rm rad}(\mathbb{R}^3))$, where $g$ 
is the regular and $G$ the
singular part of the measure $F$ with respect to the Lebesgue measure. Moreover 
it was proved that the
Boltzmann-Compton may be splitted as a system of two coupled equations for the 
pair $(g, G)$. This
allows in particular for a detailed study of the asymptotic behavior of the 
solutions. Let us
briefly show that this is not true for the general isotropic solutions
$F=g+G$ of the equation (3.1) unless $G$ is one single Dirac mass.
\end{remark}

Let us write $F = g +G$ with
$g$ regular with respect to the Lebesgue measure  ($g \prec dv$) and $G$ 
singular with respect to
the Lebesgue measure ($G \perp dv$). We have then
\begin{align*}
&F' F'_* \,(1 + F) (1+F_*) - F F_* (1 + F') (1+F'_*) =\\
&= (1+g) (1+ g_* + G_*) (g' + G') (g'_* + G'_*) + G (1+F_*) F'
 F'_* \\
& \quad- g (g_* + G_*) (1+ g' + G') (1 + g'_* + G'_*) - G F_* (1 +F') (1+F'_*)\\
&= (1 + g) \bigl\{ (1+g_*) \, g' \, g'_* + (1+g_*) \, g' \, G'_* + (1+g_*) \, G' 
\, g'_* +
(1+g_*) \, G' \, G'_* \\
&\quad + G_* \, g' \, g'_* + G_* \, g' \, G'_* + G_* \, G' \, g'_* + G_*
\, G' \, G'_*\bigr\} \\
&\quad - g \bigl\{ g_* (1 + g') (1 + g'_*) + g_* (1 + g')
\, G'_* + g_* \, G' (1 + g'_*) + g_* \, G' \, G'_* \\
&\quad + G_* (1 + g') (1 + g'_*) + G_* (1 + g') \, G'_* + G_* \,
G' (1 + g'_*)+ G_* \, G' \, G'_* \bigr\} \\
&\quad + G (1+F_*) F' F'_* - G F_* (1 +F') (1+F'_*)\\
&= q(g) + q_1(g,G) + q_2(g,G) + q_3(G,g),
\end{align*}
with
\begin{align*}
q(g) &:= (1+g) (1+g_*) \, g' \, g'_* - g \, g_* (1+g') (1+g'_*),\\
q_1(g,G) &:=  G_* [ (1+g) \, g' \, g'_* - g (1+g') (1+g'_*) ]\\
&\quad+ G' [ (1+g) (1+g_*) \, g'_* - g \, g_* (1+g'_*)\\
&\quad+ G'_* [ (1+g) (1+g_*) \, g' - g \, g_* (1+g'),\\
q_2(g,G) &:= G_* \, G'[ (1+g) \, g'_* - g (1+g'_*) ]\\
&\quad+ G_* \, G'_* [(1+g) \, g' - g (1+g') ] \\
&\quad+ G' \, G'_* [ (1+g) (1+g_*) - g \, g_* ], \\
q_3(G) &:= (1+g) \,G_* \, G' \, G'_*  - g \, G_* \, G' \, G'_*, \\
q_4(G,g) &:= G (1+F_*) F' F'_* - G F_* (1 +F') (1+F'_*).
\end{align*}
Defining
$$
Q_i(g,G) =
\iiint_{\mathbb{R}^9} w \delta_\mathcal{C} q_i(g,G) \, dv_* dv' dv'_*,
$$
we may write
$$
Q(F) = Q(g) + Q_1(g,G) +  Q_2(g,G) + Q_3(G) + Q_4(g,G).
$$
The key result in all our analysis is the following result.

\begin{lemma} \label{3.10}
Assume (3.27), (3.38) and (3.39).
For any $F \in M^1_{\rm rad}(\mathbb{R}^3)$, $F_{\rm in}\ge 0$, we have
\begin{align*}
&Q(g), \ Q_1(g,G), \ Q_2(g,G) \in L^1_{\rm rad}(\mathbb{R}^3), \\
&Q_4(g,G) \in M^1_{\rm rad}(\mathbb{R}^3) \hbox{ and } Q_4(g,G) \perp dv.
\end{align*}
For any finite sum of Dirac masses $G$ the kernel $Q_3(G)$ is also finite sum of 
Dirac masses and, moreover, if $G$
is not single Dirac mass then $\hbox{supp} \, G \backslash \{ 0 \}$ is strictly 
contained in $\hbox{supp} \,
Q_3(G)$.  Finally, there exists $G$ singular such that $Q_3(G)$ is regular.
\end{lemma}

\paragraph{Proof of Lemma \ref{lm3.5}}
We already know from the above that
$Q(g) \in L^1_{\rm rad}(\mathbb{R}^3)$ and $Q_1(g,G)$, $Q_2(g,G)$, $Q_4(g,G)$, 
$Q_3(G) \in M^1_{\rm rad}(\mathbb{R}^3)$.
Let us denote by $q$ the first term in $q_1(g,G)$ and write, for any  $\phi$
\[
\langle \mathcal{Q}[q], \phi\rangle = \iiiint_{\mathbb{R}^4_+} \hat{w} 
\,\delta_{\widehat{\mathcal{C}}}
\, g' \, g'_* \phi \, dG_* d\hat{r}' d\hat{r}'_*
= \iint_{\mathbb{R}^2_+} \psi\, g' \,  d\hat{r}' \, dG(r_*),
\]
with
$$
\psi = \int_{\mathbb{R}_+} \hat{w} \, r {\bf 1}_{ \{{r'}^2 + {r'_*}^2 \ge r^2_* 
\} }
\,\phi(r) \, g'_* \,  d\hat{r}'_*
$$
where in the expression of $\psi$ we have used the notation $r = \sqrt{ {r'}^2 + 
{r'_*}^2 - r^2_*}$.
Observe that, by assumption, $(r_*,r',r'_*) \mapsto \hat{w} \, r {\bf 1}_{ 
\{{r'}^2 + {r'_*}^2 \ge r^2_*
\}}$ is continuous so that $(r_*,r') \mapsto \psi$ is continuous for any $\phi 
\in L^\infty_{\rm rad}(\mathbb{R}^3)$.
Moreover, taking $\phi = {\bf 1}_A$ for any set $A$ with Lebesgue measure equal 
zero we get $\psi = 0$,
so that the Radon-Nykodim Theorem implies that $\mathcal{Q}[q] \in L^1$.
By the same way we prove that all the terms in $Q_1(g,G)$ and $Q_2(g,G)$
belongs to $L^1_{\rm rad}(\mathbb{R}^3)$.

It is clear that taking $q =  F' F'_* - F_* - F_* F' - F_* F'_*$ we have
$\mathcal{Q}[q] \in C_b(\mathbb{R}^3)$ so that $Q_4(g,G) \perp dv$.
For given $R_*,R',R'_* \ge 0$ we set
$q = \delta_{R_*}(r_*) \delta_{R'}(r') \delta_{R'_*}(r'_*)$ and we
verify
$$
\langle \mathcal{Q}[q], \phi \rangle =  \phi(R) \, R \hat{w}(R,R_*,R',R'_*) {\bf 
1}_{ \{{R'}^2 + {R'_*}^2 \ge R^2_* \}}
$$
where $R$ is defined by $R = \sqrt{ {R'}^2 + {R'_*}^2 - R^2_*}$ if ${R'}^2 + 
{R'_*}^2 \ge R^2_*$.
In particular, if $G = \alpha \delta_0$ then $Q_3(G) = 0$ and if
$G = \alpha \delta_a$ with $a,\alpha \not = 0$ then
$Q_3(G) = \beta \delta_a$ with $\beta > 0$.
If
$$
G = \sum_{a \in E} \alpha_a \delta_a
$$
then
$$
Q_3(G) = \sum_{b \in E'} \beta_b \delta_b,
$$
with $E' = \{ b \ge 0, \ \exists a_*,a',a'_* \in E \ s.t. (b,a_*,a',a'_*) \in 
\widehat{\mathcal{C}} \}$.
That proves the claim of the Lemma since $E'$ strictly contain $E$ if $E$ is not 
a single point.

Finally, let $G$ be a measure supported by $\sqrt{C}$, where $C$ is a Cantor 
set, constructed for example as
follows.  For every $n\in \mathbb{N}$ consider the set
$$C_n := \{ x \in [0,1], \exists k \in \mathbb{N}, 2 \, k \le 3^n \, x \le 2 \, 
k + 1 \},$$
so  that $C_n \searrow C$ as $n\to \infty $. Then $g_n := |C_n|^{-1} {\bf 
1}_{C_n}\rightharpoonup H$, a singular non
negative  measure  whose support is  $C$. If we take now $G := H \circ s$ with 
$s:\mathbb{R} \to \mathbb{R}$, $s(r) =
\sqrt{r}$ we have, for any $\phi \in C_{rad,c}(\mathbb{R}^3)$:
$$
\langle Q_3(G) = \int \!\! \int \!\! \int_{\mathbb{R}_+^3} dG(r_*) \, dG(r') \, 
dG(r'_*) [ \phi(r) \, r \hat{w} {\bf 1}_{ \{{r'}^2 + {r'_*}^2 \ge r^2_* \} } ].
$$
Since
$$
\{ z \in \mathbb{R}_+; \ \exists \epsilon_*,\epsilon',\epsilon'_* \in C z = 
\epsilon'+\epsilon'_*-\epsilon_* \}
= \mathbb{R}_+
$$
we see that $\langle Q_3(G);\phi\rangle >  0$ for any non negative and not 
vanishing $\phi$, and thus $Q_3(G)$ has a
regular
part which is not equal to $0$.
\hfill$\square$

\begin{remark} \label{rmk3.11} \rm
If we assume that for every time $t>0$, $G(t)\equiv \alpha(t)\,\delta _{0}$,
then the equation (3.1) with $\tau =1$ may be split into a coupled system of
equations for the pair $(g, \alpha )$.
\end{remark}

\begin{remark} \label{rmk3.12}\rm
The equation (3.1) with $\tau =1$ has deserved some interest in the recent
physical literature in the context of Bose condensation. We particularly refer 
here to the works by
Semikoz \& Tkachev \cite{ST1,ST2}, and by Josserand \& Pomeau \cite{JP} due to 
their strong
mathematical point of view (but see also the references therein). In these two 
articles the authors present a
possible ``scenario'' to describe the occurrence of Bose condensation in finite 
time, based on the isotropic
version of the equation (3.1) with
$\tau =1$. Their arguments are based on formal asymptotics and, in
\cite{ST1,ST2}, also supported by numerical simulations.
\end{remark}

\section{Boltzmann equation for two species} %sec 4.

In this Section, we consider a system of two coupled homogeneous equations
describing a Bose gas interacting with a heat bath, chosen to be a
Fermi gas. This is a particular case of a gas composed of two species and has
already been considered in the physical literature (see references below). One 
of the
species are Bose particles and the other are either Fermi-Dirac particles or
non quantum particles. In this Section we only deal with non
relativistic particles. Relativistic particles, in particular
photons will be considered in the next Section. In order to avoid lengthy 
repetitions,
we do not specify the energy  (relativistic or not relativistic) unless
necessary.

From a mathematical point of view, the study of the Boltzmann equation for
a gas of Bose particles  is rather difficult, as we have already seen it in the 
preceding Section.
But it is  possible to derive, from the Boson-Fermion interaction system, a 
physically
relevant model which turns out to be a sort of  ``linearization''
of equation (3.1). This model is then simpler and gives some insight into the
behavior of the Boltzman equations for quantum particles. It describes the 
interaction
of Bose particles with isotropic distribution and non quantum Fermi particles at
isotropic equilibrium with non truncated cross-section. The equation is now 
quadratic
instead of cubic and its mathematical analysis is easier. We prove that  Bose- 
Einstein
condensation takes place in infinite time, in contrast with the finite time
condensation which is expected for the Bose-Bose interaction equation. Similar 
results
had previously been obtained in the physical literature, using formal and 
numerical
methods for similar situations (cf. \cite{LY1,LY2, ST1,ST2}).

We thus consider in what follows a gas composed of two species of particles.
The first are Bose particles.  The second are either Fermi particles or their
non quantum approximation but will always be designed as Fermi particles.
We suppose that when a Bose particle of momentum $p$ collides with a Fermi
particle of momentum $p_*$ they
undergo an elastic collision, so that the total momentum and the
total energy of the system
constituted by that pair of particles are conserved. More precisely,
denoting by
$\mathcal{E}_1(p)$ the energy of Bose particles with momentum $p$ and by
$\mathcal{E}_2(p_*)$ the energy of Fermi particles
with momentum $p_*$ we assume that after collision the particles have
momentum $p'$ (for Bose particles)
and $p'_*$ (for Fermi particles) which satisfy
$\mathcal{C}_{12}$:
$$\begin{gathered}
p' + p'_* = p + p_* \\
\mathcal{E}_1(p') + \mathcal{E}_2(p'_*) = \mathcal{E}_1(p) + \mathcal{E}_2(p_*). 
\\
\end{gathered}  \eqno(4.1)
$$

The gas is described by the density $F(t,p) \ge 0$ of Bose particles and
the density $f(t,p) \ge 0$ of
Fermi particles. We assume that the evolution of the gas is given by the
following Boltzmann equation (see for instance \cite{CC})
$$\begin{gathered}
{\partial F \over \partial t} = Q_{1,1}(F,F) + Q_{1,2}(F,f), \quad F(0,.)
= F_{\rm in}, \\
{\partial f \over \partial t} = Q_{2,1}(f,F) + Q_{2,2}(f,f), \quad f(0,.)
= f_{\rm in}.
\end{gathered} \eqno(4.2)
$$
The collision terms $Q_{1,1}(F,F)$ and $Q_{2,2}(f,f)$ stand for collisions
between particles of the same specie and therefore are given by (1.5).
The collision terms $Q_{1,2}(F,f)$ and $Q_{2,1}(f,F)$ stand for collisions
between particles of the
two different species, they are given by
$$
\begin{gathered}
Q_{1,2}(F,f) =  \int_{\mathbb{R}^3} \!\! \int_{\mathbb{R}^3} \!\! 
\int_{\mathbb{R}^3} w_{1,2} \delta_{\mathcal{C}_{1,2}} q_{1, 2}
 \, dp_* dp' dp'_*,\\
q_{1,2} = F' f'_* (1+F) (1 + \tau  \,f_*)
-  F f_* (1 + F') (1 + \tau  f'_*)
\end{gathered}\eqno(4.3)
$$
with $\tau = -1$ when the second specie is composed of true Fermi
particles and $\tau  = 0$ when the second specie is constituted of non quantum
particles. The collision kernel $Q_{2,1}(f,F)$ is given by an obvious similar 
expression.
The measure 
$w_{1,2}\delta_{\mathcal{C}_{1,2}}=w_{1,2}(p,p_*,p',p'_*)\delta_{\mathcal{C}_{1,
2}}$
satisfies the
micro-reversibility hypothesis
$$
w_{1,2}(p',p'_*,p,p_*)\delta_{\mathcal{C}_{1,2}} = 
w_{1,2}(p,p_*,p',p'_*)\delta_{\mathcal{C}_{1,2}},
\eqno(4.4)
$$
but not the indiscernibility $w_{1,2}(p_*,p,p',p'_*)\delta_{\mathcal{C}_{1,2}} =
w_{1,2}(p,p_*,p',p'_*)\delta_{\mathcal{C}_{1,2}}$ as in (1.7)
since the two species are now different. When both
energies are non relativistic
$w_{1,2}$ is invariant by Galilean transformations, and when both
energies are relativistic it is invariant by Lorentz transformations.
In the mixed case of one non relativistic specie and one
relativistic specie, the situation is a little more complicated and we
postpone the analysis to the next Section.


We start with some simple formal properties of the solutions of (4.2).
Thanks to symmetry (4.4),
performing a change of variables $(p',p'_*,p,p_*) \to (p,p_*,p',p'_*)$ we
get the fundamental formula:
for any $\psi = \psi(p)$,
$$\begin{aligned}
\int_{\mathbb{R}^3} Q_{1,2}(F,f) \psi \, dp = &{1 \over 2} 
\iiiint_{\mathbb{R}^{12}}
w_{1,2} \delta_{\mathcal{C}_{1,2}}  \bigl( F' f'_* (1+F) (1 +\tau
\,f_*) \\
&-  F f_* (1 + F') (1 +\tau  f'_*) \bigr) \bigl(\psi -
\psi' \bigr) \, dp\, dp_* dp'\, dp'_*\,.
\end{aligned}\eqno(4.5)
$$
A similar formula holds for $Q_{2,1}(f,F)$.


After integration of the two  equations of (4.2) separately, and using (1.7) and 
(4.5) we
formally get the particle number conservation
of each specie:
$$
\int_{\mathbb{R}^3} F(t,p) \, dp = \int_{\mathbb{R}^3} F_{\rm in}(p) \, dp\,, 
\quad
\int_{\mathbb{R}^3} f(t,p) \, dp = \int_{\mathbb{R}^3} f_{\rm in}(p) \, dp.
 \eqno(4.6)
$$
Multiplying both equations in (4.2) by $\psi(p) = p$, summing up and
using (1.7), (4.5) and $w_{1,2} =
w_{2,1}$ we obtain the global momentum conservation
$$
\int_{\mathbb{R}^3} \bigl( F(t,p) + f(t,p) \bigr) p  \, dp
= \int_{\mathbb{R}^3} \bigl( F_{\rm in}(p) + f_{\rm in}(p) \bigr) p  \, dp.
\eqno(4.7)
$$
Multiplying the first equation of (4.2) by $\mathcal{E}_1$, the second
equation of (4.2) by $\mathcal{E}_2$, using (1.7), (4.5) and the collision
invariance (4.1) we
get the global energy conservation
$$
\int_{\mathbb{R}^3} \bigl( F(t,p) \mathcal{E}_1(p) + f(t,p) \mathcal{E}_2(p) 
\bigr)\, dp
= \int_{\mathbb{R}^3} \bigl( F_{\rm in}(p) \mathcal{E}_1(p) + f_{\rm in}(p) 
\mathcal{E}_2(p) \bigr)
\, dp\,.
\eqno(4.8)
$$
Finally, we define the entropy of the system by
$$
H_{\mathcal{S}}(F,f) := H_1 (F) + H_{\tau } (f)
\eqno(4.9)
$$
with $H_\tau$ given by (1.1). Multiplying the first equation by $h_1'(F)$
and the second equation by
$h'_{\tau } (f)$ we obtain using (4.5)
$$
{d \over dt} H_{\mathcal{S}}(F,f) = D_\mathcal{S}(F,f) := \frac14  D_1(F) + {1 
\over 2}
D_{1,2} (F,f) +
\frac14  D_{\tau }(f) \ge 0,
 \eqno(4.10)
$$
where $D_1(F)$ and $D_{\tau }(f)$ are the usual dissipation entropy
production of one specie, and
$D_{1,2} (F,f)$ is the mixed dissipation entropy production given by
$$
\begin{aligned}
&D_{1,2} (F,f) :=\\
&\iiiint_{\mathbb{R}^{12}} w_{1,2} \delta_{\mathcal{C}_{1,2}}
j\bigl( {F' f'_* (1+F) (1 +\tau  f_*) , F'_* (1+F') (1 +\tau  f'_*) }
\bigr) \, dp dp_* {dp' dp'_*}.
\end{aligned}
\eqno(4.11)
$$

We list now several questions that one may naturally consider about the
system (4.2).

1. For the single quantum equation, one may consider the maximization
entropy problem under particle number, momentum and energy restriction i.e.: for 
any
given
$N_B, N_F, E > 0$,
$P
\in
\mathbb{R}^3$, to find a pair of functions
$(F,f)$ such that
$$
\begin{gathered}
\int_{\mathbb{R}^3} dF(p)= N_B, \quad \int_{\mathbb{R}^3} f(p) \, dp = N_F\\
 \int_{\mathbb{R}^3} p (F(p) +f(p)) \, dp = P, \quad
\int_{\mathbb{R}^3} (F(p) \mathcal{E}1(p) + f(p) \mathcal{E}_2(p)) \, dp = E.
\end{gathered}\eqno(4.12)
$$
and satisfying
$$
H_\mathcal{S}(F,f) = \max_{G,g s.t. (4.12)} H_\mathcal{S}(G,g),
\quad\hbox{with } H_\mathcal{S}(G,g) = H_{BE}(G) + H_\tau(g).
\eqno(4.13)
$$
We believe that a complete analysis of this problem can be done using
the same ideas exposed in the Section 2 and used in \cite{EMV}.

2. One  can also address the well posedness of the Cauchy problem for the
system (4.2). Of course the situation
here is the same as for the Boltzmann Bose equation. The
analysis performed in the Section 3.2 may be readily extended to prove the 
existence
of solutions under the assumption of isotropy of the distribution and with Lu's 
truncation
on the cross sections. A simpler question would be to consider the case when
$Q_{1,1}(F,F)$ and $Q_{2,2}(f,f)$
vanish and to address the well posedness of the Cauchy problem in this case.
Even  when $\tau  > 0$ (which gives an $L^\infty$ a priori bound on the Fermi
density $f$) we do not know if it is
possible to give a sense to the collision terms $Q_{1,2}(F,f)$ and
$Q_{2,1}(f,F)$ without Lu's truncation  on the cross sections.

We do not try to go further in any of these two directions and consider instead
the following question. In the study of gases formed by Bose and Fermi 
particles, it is particularly relevant
to consider the case where the Fermi particles are at equilibrium and where the
collisions between Bose particles can not distort significantly their 
distribution
function (cf. \cite{LY1,LY2}). This moreover constitutes a first important 
simplification
from a mathematical point of view. The system (4.2) reduces then to a unique 
equation on
the Bose distribution $F$. Moreover this equation is quadratic and not cubic
(c.f. sub Section 4.1)

A second simplification arises if we consider non relativistic, isotropic 
densities
and we assume  on physical grounds, that $w _{1,2}$ is constant, (c.f below). In 
that
case, we keep the same quadratic structure for the
equation on the Bose distribution $F$, but we obtain an explicit and quite 
simple
cross-section (c.f. sub Section 4.2).

In both situations, our main concern is to understand if it is possible to
obtain a global existence result
without the Lu's truncation on the cross sections and then to describe the long 
time
asymptotic behaviour of the solutions.

\subsection{Second specie at thermodynamical equilibrium} %4.1

Let us assume that $f = \mathcal{F}$ is at thermodynamical
equilibrium, which means that it is a Fermi or a
Maxwellian distribution defined by
$$
\mathcal{F}(p) = {1 \over e^{\nu(p)} -\tau }, \quad
\nu(p) = \beta^0 \mathcal{E}_2(p) - \beta \cdot p + \mu.\eqno{(4.14)}
$$
This greatly simplifies the situation since the system (4.2) reduces now to a 
single
equation for the Bose distribution
$F$ which reads
$$
{\partial F \over \partial t} = Q_{BQ}(F) := 
Q_{1,2}(F,\mathcal{F})\eqno{(4.15)}
$$
with
$$
Q_{BQ}(F) =  \int_{\mathbb{R}^3} S(p,p') \bigl( F' (1 + F) \, e^{- \beta^0 
\mathcal{E}_1(p)}
- F (1 + F') \, e^{- \beta^0 \mathcal{E}_1(p')} \bigr) \,  dp' ,
\eqno(4.16)
$$
$$
S(p,p') = \iint_{\mathbb{R}^3 \times \mathbb{R}^3} \!\!\! \!\!\! w_{1,2} 
\delta_{\mathcal{C}_{12}}
e^{\beta^0 \mathcal{E}_1(p)} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, dp_* 
dp'_*.
\eqno(4.17)
$$
To establish (4.16)-(4.17) we have used the elementary
identity
$$
e^{\beta^0 \mathcal{E}_1(p)} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*)
= e^{\beta^0 \mathcal{E}_1(p')} \mathcal{F}_* (1 +\tau  \mathcal{F}'_*)
\eqno(4.18)
$$
which holds on $\mathcal{C}_{12}$. Notice that, using the micro-reversibility
symmetry (4.4) and the identity
(4.18) we have
\begin{align*}
S(p',p) &= \iint_{\mathbb{R}^3 \times \mathbb{R}^3} \!\!\! \!\!\!
w_{1,2}(p',p_*,p,p'_*) \delta_{\mathcal{C}_{12}}
e^{\beta^0 \mathcal{E}_1(p')} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, dp_* 
dp'_* \\
&= \iint_{\mathbb{R}^3 \times \mathbb{R}^3} \!\!\! \!\!\! w_{1,2}(p,p_*,p',p'_*)
\delta_{\mathcal{C}_{12}}
e^{\beta^0 \mathcal{E}_1(p)} \mathcal{F}_* (1 +\tau  \mathcal{F}'_*)
\, dp_* dp'_* \\
&= S(p,p'),
\end{align*}
so that $S$ is symmetric.


Using the symmetry above one can observe that, at least formally, a
solution $F$ of equation (4.15) still
satisfies the qualitative properties
$$
\int_{\mathbb{R}^3} F \, dp = \int_{\mathbb{R}^3} F_{\rm in} \, dp
\eqno(4.19)
$$
and
$$
{d \over dt} H_{BQ}(F)  = D_{BQ} (F),\eqno{(4.20)}
$$
with
$$
H_{BQ}(F) = \int_{\mathbb{R}^3} \bigl((1+F)  \ln (1+F) - F \ln F - F \beta^0
\mathcal{E}_1(p) \bigr) \, dp\eqno{(4.21)}
$$
and
$$
D_{BQ} (F) = \iint_{\mathbb{R}^3 \times \mathbb{R}^3} S (p,p') \, j\bigl( F' (1 
+ F) \, e^{- \beta^0 \mathcal{E}_1(p)}  - F (1 + F') \, e^{- \beta^0 
\mathcal{E}_1(p')} \bigr) \,
dpdp'.\eqno{(4.22)}
$$
In other words, the particle number is preserved along the trajectories and 
$H_{BQ}$ is
a Lyapunov function (the relative entropy $H_{BQ}$ is a decreasing function 
along
the trajectories).


\subsubsection{Non relativistic particles, fermions at isotropic Fermi Dirac
equilibrium}

It is possible, under further simplifications of the model, to obtain more 
explicit
expressions of the cross section
$S$. As a first step in that direction we consider nonrelativistic
particles.  We also assume, without any loss of generality, that the two 
particles
have the same mass $m=1$ from where their energies are
$\mathcal{E}_i(p)=\mathcal{E}(p)
= |p|^2/2$, $i=1, 2$. We assume moreover that fermions are at isotropic 
equilibrium (see (2.39)):
$$\mathcal{F}(p)={1\over e^{\beta ^0{|p|^2\over 2\ }+b}-\tau }\quad
\hbox{for some}\,\theta 0, b\ge 0.$$
We introduce now the Carleman parametrization of the
collision manifold (4.1). Starting
by performing the $dp_*$ integration one finds
$$
S(p,p') =\! \int_{\mathbb{R}^3} w_{1,2} (p,p_*,p',p'_*) \delta_{|p'|^2 + 
|p'_*|^2
- |p|^2 - |p_*|^2 = 0}
\, e^{\beta ^0{|p|^2\over2 }} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, 
dp'_*,
\eqno(4.23)
$$
where now $p_*$ is defined by
$$
p_* := p' + p'_* - p_*.
\eqno(4.24)
$$
An elementary computation shows that
$$
|p'|^2 + |p'_*|^2 - |p|^2 - |p_*|^2 = - 2 (p'_* - p) \cdot (p'-p),
$$
so that in (4.23) $p'_*$ describes all the plane $E_{p,p'}$ orthogonal
to $p'-p$ and containing  $p$. Then, using the distributional Lemma 
\ref{lmA1.1}, we get
$$
S(p,p') = \int_{E_{p,p'}} {w_{1,2} (p,p_*,p',p'_*)  \over |p'-p|}
\, e^{\beta ^0{|p|^2\over2 }} \mathcal{F}'_* (1 +\tau  \mathcal{F}_*) \, 
dE(p'_*),
\eqno(4.25)
$$
where $dE(p'_*)$ stands for the Lebesgue measure on $E_{p,p'}$ and $p_*$ is
again defined thanks to
(4.24).

A further simplification may be performed, noticing that, on physical ground 
(cf.
Appendix \ref{A3}), the
boson-fermion interaction is a short range interaction and that the velocities 
of
the particles undergoing scattering are small . This allows to consider that
$w_{1,2}$ is constant, assuming without loss of generality that $w_{1,2}=1$. We 
then
obtain a more explicit expression of $S$ in (4.25). Namely:
$$
S(p,p') = \int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over 2 }}\over |p'-p|}
 {1\over e^{\beta ^0{|p'_*|^2\over 2 }+b} -\tau } {e^{\beta ^0{|p_*|^2\over 2 
}+b}\over e^{\beta ^0{|p_*|^2\over 2 }+b}-\tau }\, dE(p'_*).
$$
Note that
$${e^{b}\over 1+e^{b}}\le {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta 
^0{|p_*|^2\over 2
}+b}-\tau }<1$$
and
\begin{align*}
&\delta {e^{b}\over 1+e^{b}}\int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over
2}}\over |p'-p|} \, e^{-\beta ^0{|p'_*|^2\over 2 }-b} \, dE(p'_*)\\
&\le S(p,  p')<\int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over 2 }}\over
|p'-p|}  \, e^{-\beta ^0{|p'_*|^2\over 2}-b} \, dE(p'_*).
\end{align*}
Let us consider then the function
$$
S_0(p,p')\equiv\int_{E_{p,p'}} {e^{\beta ^0{|p|^2\over 2 }}\over
|p'-p|}
 \, e^{-\beta ^0{|p'_*|^2\over 2 }} \, dE(p'_*)$$
In the orthonormal basis
$(\vec {k},\vec{i},\vec{j})$ where
$\vec{k} := (p'-p)/|p'-p|$, $p'_\star$ may be written as follows
$p'_\star = p + s \vec{i} + t \vec{j}$ with
$s,t \in \mathbb{R}$.
\begin{align*}
|p'_\star|^2 &=  |p|^2 + (s + p \cdot i)^2 - (p \cdot i)^2 + (t + p \cdot
j)^2 - (p \cdot j)^2 \\
&= \Bigl(p ,{p' - p \over |p'-p|} \Bigr)^2 + (s + p \cdot i)^2 + (t + p
\cdot j)^2,
\end{align*}
and, integrating (4.25), we find
\begin{align*}
S_0(p,p') &= {e^{\beta ^0{|p|^2\over 2}}  \over |p'-p|} \int \!\! 
\int_{\mathbb{R}^2}
e^{-\beta ^0|p'_*|^2/2} \, dsdt \\
&= {2 \pi  \over \beta^0 |p'-p|} \exp \Bigl[ {\beta ^0 \over 2 } \Bigl( |p|^2 - 
\bigl(p ,{p' - p \over
|p'-p|} \bigr)^2 \Bigr) \Bigr].
\end{align*}
This can also be written in the symmetric form
$$
S_0(p,p') = {2 \pi  \over \beta ^0 |p'-p|} \exp \Bigl[ {\beta ^0 \over 4 } 
\Bigl( |p|^2 + |p'|^2 - \bigl(p ,{p' - p
\over |p'-p|} \bigr)^2
- \bigl(p' ,{p' - p \over |p'-p|} \bigr)^2 \Bigr) \Bigr].\eqno{(4.26)}
$$

\begin{remark} \label{rmk4.1}\rm
If one considers the system
(4.2) when collisions between Boson particles are very weak (so
that they may be neglected) and collisions between Fermi particles are very
strong (in such a way that the
distribution of Fermi particles goes through thermodynamical
equilibrium very rapidly) we may consider
as relevant the following scaling
$$
\begin{aligned}
{\partial F_\epsilon \over \partial t} &= \epsilon \, 
Q_{1,1}(F_\epsilon,F_\epsilon) +
Q_{1,2}(F_\epsilon,f_\epsilon) \\
{\partial f_\epsilon \over \partial t} &= Q_{2,1}(f_\epsilon,F_\epsilon) + 
{1\over\epsilon}
\, Q_{2,2}(f_\epsilon,f_\epsilon),
\end{aligned} \eqno(4.27)
$$
in the limit $\epsilon \to 0$. Notice that the particle number conservation 
(4.6)
and energy conservation (4.8) provide the a priori bounds
$$
\sup_{\epsilon > 0} \sup_{t \in [0,\infty)} \int_{\mathbb{R}^3} F_\epsilon(t,p) 
(1 +
\mathcal{E}_1(p)) \, dp, \quad
\int_{\mathbb{R}^3} f_\epsilon(t,p) (1 + \mathcal{E}_2(p)) \, dp \le 
C.\eqno{(4.28)}
$$
Moreover, since
$$
{d \over dt} H_{\mathcal{S}}(F_\epsilon,f_\epsilon) = {\epsilon \over 4} 
D_1(F_\epsilon) + {1 \over
2} D_{1,2} (F_\epsilon,f_\epsilon) +
{1 \over 4 \epsilon} \, D_{\tau }(f_\epsilon),
$$
and all the terms at the right hand side are positive, we obtain
$$
\int_0^\infty D_{\tau }(f_\epsilon) \le \epsilon \, C(F_{\rm in},f_{\rm 
in}).\eqno{(4.29)}
$$
Formally, these bounds imply that, up to the extraction of a subsequence,
we have
$$
f_\epsilon \to \mathcal{F}, \quad F_\epsilon \to F,\eqno{(4.30)}
$$
where $\mathcal{F}$ has same momentum that $f_{\rm in}$ and $D_{\tau 
}(\mathcal{F}) = 0$ so
that $\mathcal{F}$ is the Fermi or Maxwellian
distribution associated to $f_{\rm in}$ given by (4.4) and $F$ solves the
Quadratic Bose equation (4.15)-(4.17).
\end{remark}

\begin{remark} \label{rmk4.2} \rm
It is important to notice that there is
no conservation of the energy for equation (4.15)-(4.16). The conserved
quantity in the system (4.2) is the total energy of the bosons-fermions gas
but not of any of the two species as it is shown by (4.8).
\end{remark}

\paragraph{Open questions:}
\begin{enumerate}
\item Establish rigorously (4.30).

\item Solve the Cauchy problem (4.15)-(4.17) for physically relevant $S$.
\end{enumerate}

\subsection{Isotropic distribution and second specie at the
thermodynamical equilibrium} %4.2

We now consider the Fermi particles
at non relativistic isotropic equilibrium and assume that the Bose distributions
are also isotropic. In other words we suppose that
$$
f(p, t)\equiv f(|p|)= {1\over e^{\beta ^0{|p|^2\over 2 }+b}-\tau }\quad
\hbox{and}\quad F(p, t)=F(|p|, t).\eqno{(4.31)}
$$
The interaction of an isotropic gas of Bose particles and Fermi particles at
equilibrium has been considered by Levich and  Yakhot in \cite{LY1} and
\cite{LY2}, by means of formal arguments. They were interested in particular
in the occurrence of Bose Einstein
condensation. Numerical simulations showing Dirac mass formation in infinite 
time for a
related equation have been obtained by  Semikoz and  Tkachev in \cite{ST2,ST2}.
The Fermi distributions considered in that case are  quantum, isotropic,
saturated Fermi Dirac distributions (SFD in Section 2). This corresponds to the 
choice
$$f(p)\equiv f(|p|)={\bf 1}_{\{0\le |p|\le \mu \}}$$
for some $\mu >0$, and gives rise to a slightly different equation than ours.

The Cauchy problem for the resulting quadratic Bose equation is a
``linearized model'' of the Bose-Bose interaction equation considered in
Section 3, where moreover, the function $S(p, p')$ may be calculated  
explicitly.

Similar equations have been considered in \cite{EM1,EM2}. Nevertheless, the 
global
existence results obtained in these references do not apply to our case, because
the  collision kernel does not fulfill the required hypothesis. Therefore, we 
end
this Section proving  global existence of solutions, with integrable
initial data, to our problem. Finally, the long time behaviour of these global 
solutions
may be addressed exactly as in \cite{EM2} and then, we only state the result for 
the sake of
completeness.

We start with the following:

\begin{proposition} \label{prop4.3}
If the function $F$ is radially symmetric then so is $Q_{BQ}(F)$.
In that case we write $Q_{BQ}(F)(p)=Q_{BQ}(F)({{|p|^2\over 2\theta}})$
by abuse of notation. Moreover,
$$
Q_{BQ}(F) (\varepsilon) =
\int_0^\infty S \bigl[ F' (1 + F) \, e^{-\epsilon} - F (1 + F') \,
e^{-\epsilon'} \bigr] \,  d\epsilon',
\eqno(4.32)
$$
with $S(\epsilon,\epsilon') = \Sigma(\epsilon,\epsilon') / \sqrt{\epsilon}$ and
$$
\Sigma (\epsilon,\epsilon') =
\int_{\max(0,\epsilon-\epsilon')}^\infty \, e^{\epsilon-\epsilon'_*} 
{e^{\varepsilon' +
\varepsilon'_*- \varepsilon+b}\over  e^{\varepsilon' +\varepsilon'_*-
\varepsilon+b}-\tau }\min(\sqrt{\epsilon},\sqrt{\epsilon'+\epsilon'_*-
\epsilon},\sqrt{\epsilon'},\sqrt{\epsilon'_*}) \,
d\epsilon'_*.\eqno{(4.33)}
$$
\end{proposition}

\paragraph{Proof}
It follows from (4.17) and (4.31) that
\begin{align*}
S(p,p')&= e^{\beta ^0{|p|^2\over2 }} \int\int_{\mathbb{R}^3 \times \mathbb{R}^3}
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0}\,\delta _{p'+p'_*-p-p_*=0}\\
&\hskip 3cm \times {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta
^0{|p'_*|^2\over 2 }+b}-\tau } {dp_* dp'_*\over e^{\beta ^0{|p_*|^2\over 2 }+b}-
\tau }
\\
&=\int_0^{\infty }\int_0^{\infty }
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0}{e^{\beta ^0{|p|^2\over2 }}\over e^{\beta ^0{|p'_*|^2\over 
2
}+b}-\tau } {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta ^0{|p_*|^2\over 2
}+b}-\tau }\\
&\hskip 3cm \times
\int_{S^2}\int_{S^2}\delta _{p'+p'_*-p-p_*=0}\,d\omega _*d\omega
'_*|p'_*|^2d|p'_*\|p_*|^2d|p_*|.
\end{align*}
Therefore,
\begin{align*}Q_{BQ}(F)=&\int_0^{\infty }\int_0^{\infty }\int_0^{\infty }
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0} {e^{\beta ^0{|p|^2\over2 }}\over e^{\beta 
^0{|p'_*|^2\over 2
}+b}-\tau } \,  {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta ^0{|p_*|^2\over 2
}+b}-\tau }\\
& \times  [F'(1+F)e^{-\beta ^0{|p|^2\over 2 }}-
 F(1+F')e^{-\beta ^0{|p'|^2\over 2 }}]\cdot \\
& \times \int_{S^2}\int_{S^2}\int_{S^2}\delta _{p'+p'_*-p-p_*=0}
\,d\omega _*
d\omega '_* d\omega '|p'_*|^2d|p'_*\|p_*|^2d|p_*\|p'|^2d|p'|.
\end{align*}
As we have already seen in Section 3,
$$
\int_{S^2}\int_{S^2}\int_{S^2}\delta _{p'+p'_*-p-p_*=0}\,d\omega _* d\omega '_*
d\omega '={4 \pi ^2\over |p\|p'\|p_*\|p_*'|}\min \bigl[|p|,|p'|, |p_*|, |p_*'|
\bigr]
$$
from where,
\begin{align*}
Q_{BQ}(F)=&{4 \pi ^2\over |p|}\int_0^{\infty }
[F'(1+F)e^{-\beta ^0{|p|^2\over 2 }}-F(1+F')e^{-{|p'|^2\over
2\theta }}]e^{\beta ^0{|p|^2\over2}}\\
&\Big\{\int_0^{\infty }\int_0^{\infty}
\delta_{|p'|^2 + |p'_*|^2
- |p|^2 - |p_*|^2 = 0} {e^{\beta ^0{|p|^2\over2 }}\over e^{\beta 
^0{|p'_*|^2\over 2
}+b}-\tau } \,  {e^{\beta ^0{|p_*|^2\over 2 }+b}\over e^{\beta ^0{|p_*|^2\over 2
}+b}-\tau }\\
&\min \bigl[|p|,|p'|, |p_*|,
|p_*'|\bigr]\,|p'_*|d|p'_*\|p_*|d|p_*|\Big\}|p'|d|p'|\\
=&\int_0^{\infty }
\big[F'(1+F)e^{-\beta ^0{|p|^2\over 2 }}-F(1+F')e^{-\beta ^0{|p'|^2\over
2 }}\big]\,\overline{S}(|p|,|p'|)|p'|d|p'|,
\end{align*}
\begin{align*}
\overline{S}(|p|, |p'|)&={4 \pi ^2\over |p|}\int_0^{\infty }\int_0^{\infty}
\delta_{|p'|^2 + |p'_*|^2- |p|^2 - |p_*|^2 = 0} {e^{\beta ^0{|p|^2\over2 }}\over
e^{\beta ^0{|p'_*|^2\over 2}+b}-\tau } \,  {e^{\beta ^0{|p_*|^2\over 2 }+b}\over
e^{\beta ^0{|p_*|^2\over 2 }+b}-\tau }\\
&\hskip 3cm \times \min[|p|,|p'|, |p_*|,
|p_*'|]|p'_*|d|p'_*\|p_*|d|p_*|\\
&={4 \pi ^2\over |p|}\int_0^{\infty }{e^{\beta ^0{|p|^2\over2 }}\over e^{\beta 
^0{|p'_*|^2\over
2 }+b}} {e^{\beta ^0{{|p'|^2 + |p'_*|^2- |p|^2}\over 2 }+b}\over e^{\beta 
^0{{|p'|^2 +
|p'_*|^2- |p|^2}\over 2 }+b}-\tau }{\bf 1}_{\{|p'|^2+|p'_*|^2\ge |p^2|\}} \\
&\hskip 3cm \times\min \bigl[|p|,|p'|,
\sqrt{|p'|^2 + |p'_*|^2- |p|^2}, |p_*'|\bigr]\,|p'_*|d|p'_*|
\end{align*}
By defining
$$
\varepsilon=\beta ^0{|p|^2\over 2 },
\quad \varepsilon'=\beta ^0{|p'|^2\over 2},
\varepsilon_*=\beta ^0{|p_*|^2\over 2 },
\quad \varepsilon'_*=\beta ^0{|p'_*|^2\over 2 },
$$
we have
\begin{align*}
\overline{S}(\sqrt {2\varepsilon/\beta ^0 }, \sqrt {2\varepsilon'/\beta ^0})
&=4\pi ^2\sqrt{2}(\beta ^0)^{-3/2}e^{-b} \int_0^{\infty
}e^{\varepsilon-\varepsilon'_*}{e^{\varepsilon' +
\varepsilon'_*- \varepsilon+b}\over  e^{\varepsilon' +\varepsilon'_*-
\varepsilon+b}-\tau}{\bf 1}_{ \{ \epsilon' + \epsilon'_* \ge
\epsilon \} }\\
&\quad\times {\min(\sqrt{\epsilon},\sqrt{\varepsilon' +
\varepsilon'_*- \varepsilon},\sqrt{\epsilon'},\sqrt{\epsilon'_*})\over \sqrt 
{\varepsilon}}
d\varepsilon'_*
\equiv S(\varepsilon, \varepsilon').
\end{align*}
Finally
$$
Q_{BQ}(F)= \int_0^\infty \bigl[ F' (1 + F) \, e^{-\epsilon} - F (1 + F') \,
e^{-\epsilon'} \bigr] \, S \,  d\epsilon'
$$
with $S(\epsilon,\epsilon') = \Sigma(\epsilon,\epsilon') / \sqrt{\epsilon}$ and
$$
\Sigma (\epsilon,\epsilon') =
\int_{\max(0,\epsilon-\epsilon')}^\infty \, e^{\epsilon-\epsilon'_*} 
{e^{\varepsilon' +
\varepsilon'_*- \varepsilon+b}\over  e^{\varepsilon' +\varepsilon'_*-
\varepsilon+b}-\tau }\min(\sqrt{\epsilon},\sqrt{\epsilon'+\epsilon'_*-
\epsilon},\sqrt{\epsilon'},\sqrt{\epsilon'_*}) \,
d\epsilon'_*.
$$
By a change of variables in the integral definition of $\Sigma$  see
that it is a symmetric
function.
We may then write equation (4.15) as
$$
\sqrt{\epsilon} {\partial F \over \partial t} =
\int_0^\infty  \Sigma \bigl[ F' (1 + F) \, e^{-\epsilon} - F (1 + F')
\, e^{-\epsilon'} \bigr] \, d\epsilon'.
$$
and perform the usual change of variable
$$
\sqrt{\epsilon} F \to F, \quad b(\epsilon,\epsilon') = {\Sigma 
(\epsilon,\epsilon') \over
\sqrt{\epsilon} \sqrt{\epsilon'}}
$$
to obtain the more suitable form
$$
{\partial F \over \partial t} =
\int_0^\infty  b(\epsilon,\epsilon') \bigl[ F' (\sqrt{\epsilon} + F) \, e^{-
\epsilon}
- F (\sqrt{\epsilon'} + F') \, e^{-\epsilon'} \bigr] \, d\epsilon',
\eqno(4.34)
$$
Notice that $b$ is singular near the origin and as a consequence we can not
apply the results obtained in \cite{EM1, EM2}.

 We first want to give a precise mathematical sense
to the collision term in (4.34). Notice that it can be written t in the 
following way
\begin{align*}
Q(F) = &\int_0^\infty b(\epsilon,\epsilon') \sqrt{\epsilon} \, e^{-\epsilon} F' 
\, d\epsilon'
- F \int_0^\infty b(\epsilon,\epsilon') \sqrt{\epsilon'} \, e^{-\epsilon'} \, 
d\epsilon' \\
&+ \int_0^\infty b(\epsilon,\epsilon') (e^{-\epsilon} - e^{-\epsilon'} ) \,  F 
F' \,
d\epsilon'.
\end{align*}

\begin{lemma} \label{lm4.4}
The function
$$
\ell(\epsilon) := \int_0^\infty b(\epsilon,\epsilon') \sqrt{\epsilon'} \, e^{-
\epsilon'} \,
d\epsilon'.
\eqno(4.37)
$$
satisfies $\ell \in C([0,\infty))$, $\ell\le C_1(1+\sqrt{\varepsilon})$ for
some positive constant $C_2$, $\ell
\to C_2(b, \tau )$ when $\epsilon
\to 0$ and
$\ell \sim \gamma \sqrt{\epsilon}$
when $\epsilon \to \infty$, with $\gamma > 0$. Moreover, we have
$$
\chi(\epsilon,\epsilon') := ( e^{-\epsilon} - e^{-\epsilon'}) 
b(\varepsilon,\varepsilon')
\in C_b([0,\infty) \times [0,\infty)).
\eqno(4.38)
$$
As a conclusion, for any $F$ such that $(1 +\sqrt{\epsilon}) F \in L^1$,
$Q(F)$ belongs to $L^1$ and the map $F\mapsto Q(F)$ is continuous from
$L_{1/2}^1$ to $L^1$.
\end{lemma}


\paragraph{Proof of Lemma \ref{lm4.4}}
In order to prove the properties of $\ell$ we write
$$
\ell(\epsilon)
= {1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta_b (\epsilon') \, e^{-
\epsilon'} \, d\epsilon'
+  {1 \over \sqrt{\epsilon}} \int_0^\epsilon \xi _b (\epsilon, \varepsilon') \, 
e^{-\epsilon'} \, d\epsilon'
$$
where,
\begin{gather*}
\zeta _b(z)=\int_0^{\infty }e^{z-k}{e^{k+b}\over
e^{k+b}-\tau }\min(\sqrt{z},\sqrt{k})\,dk \\
\xi _{b}(z, y)=\int_0^{\infty }e^{z-k}{e^{k-z+y+b}\over
e^{k-z+y+b}-\tau }\min(\sqrt{z},\sqrt{k})\,dk.
\end{gather*}
Note that
$$
\zeta _b(z)\le \int_0^{\infty }e^{z-k}\min(\sqrt{z},\sqrt{k})dk= \gamma
(z)e^{z}+\sqrt {z}\equiv\zeta (z)
$$
with $\gamma (x)=\int_0^{x}e^{-y}\sqrt{y}dy$,
and $\xi _b(z, y)\le \zeta (z)$.
Assume first that $\varepsilon\to 0$. Then,
$$
{1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta_b (\epsilon') \, e^{-\epsilon'} 
\, d\epsilon' \le
{1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta (\varepsilon') e^{-\epsilon'} \, 
d\epsilon'\to 0\quad
\hbox{as}\,\varepsilon\to 0.
$$
On the other hand,
\begin{align*}
&{1 \over \sqrt{\epsilon}} \int_\epsilon ^{\infty }\xi _b (\epsilon, 
\varepsilon') \,
e^{-\epsilon'} \,d\epsilon'\\
&={1\over \sqrt{\epsilon}}\int_{\varepsilon}^{\infty
}e^{-\varepsilon'}\int_0^{\varepsilon}e^{\varepsilon-\varepsilon'_*}
{e^{\varepsilon'+\varepsilon'_*-\varepsilon+b}\over
e^{\varepsilon'+\varepsilon'_*-\varepsilon+b}-\tau }\sqrt
{\varepsilon'_*} d\varepsilon'_*d\varepsilon'+\\
&\quad+\int_{\varepsilon}^{\infty }e^{-\varepsilon'}\int_{\varepsilon}^{\infty }
e^{\varepsilon-\varepsilon'_*}{e^{\varepsilon'+\varepsilon'_*-
\varepsilon+b}\over
e^{\varepsilon'+\varepsilon'_*-\varepsilon+b}-\tau }\sqrt
{\varepsilon'_*} d\varepsilon'_*d\varepsilon'\\
&\quad
\to \int_0^{\infty }e^{-\varepsilon'}\int_0^{\infty }
e^{-\varepsilon'_*}{e^{\varepsilon'+\varepsilon'_*+b}\over
e^{\varepsilon'+\varepsilon'_*+b}-\tau }\sqrt
{\varepsilon'_*} d\varepsilon'_*d\varepsilon'=C_2(b, \tau )\quad
\hbox{as}\,\varepsilon\to 0.
\end{align*}
Suppose now that $\varepsilon\to \infty $. We first note that,
$$
{1 \over \sqrt{\epsilon}} \int_0^\epsilon \xi _b (\epsilon, \varepsilon') \, 
e^{-\epsilon'}
\, d\epsilon'\le {1 \over \sqrt{\epsilon}} \int_0^\epsilon \zeta (\varepsilon')
\, e^{-\epsilon'}
\, d\epsilon'\to 0\quad \hbox{as}\,\ \varepsilon\to \infty.
$$
Finally,
\begin{align*}
{1\over \sqrt{\varepsilon}}\int_0^{\varepsilon}e^{-\varepsilon'}\zeta
_b(\varepsilon')d\varepsilon'
=&{1\over\sqrt{\varepsilon}}\int_0^{\varepsilon}
\int_0^{\varepsilon'}e^{-k}{e^{k+b}\over e^{k+b}-\tau }
\sqrt{k}dk\,d\varepsilon'\\
&+{1\over \sqrt{\varepsilon}}\int_0^{\varepsilon}
\sqrt{\varepsilon'}\int_{\varepsilon'}^{\infty}
e^{-k}{e^{k+b}\over e^{k+b}-\tau }dkd\varepsilon'\equiv I_1+I_2
\end{align*}
and,
\begin{gather*}
I_2\le {1\over
\sqrt{\varepsilon}}
\int_0^{\varepsilon}\sqrt{\varepsilon'}e^{-\varepsilon'}d\varepsilon'=O({1\over
\sqrt{\varepsilon}}),\\
\lim_{\varepsilon\to \infty }{1\over \varepsilon}\int_0^{\varepsilon}
\int_0^{\varepsilon'}e^{-k}\sqrt{k}dkd\varepsilon'=\int_0^{\infty }e^{-k}
\sqrt{k}dk\equiv \gamma
\end{gather*}
To prove (4.38) we just have to consider the two cases:
If $\min(\epsilon,\epsilon') \to 0$ then
$$
\chi \ \sim \ { e^{-\max} - e^{-\min} \over \sqrt{\max} }  \
\mathop{\longrightarrow}_{\max \to 0,\infty} \ 0
\quad\hbox{ uniformly in } \min.
$$
If $\min(\epsilon,\epsilon') \to \infty$ then $\epsilon,\epsilon' \to \infty$ 
and
$$
|\chi| \ \le \ { e^{-\min} - e^{-\max} \over \sqrt{\max} \sqrt{\min} }
\, e^{\min} \le 1.
$$
\hfill$\square$ \smallskip

The particle number of the solutions to (4.34) is constant in time (at least
formally) and this gives a first a
priori bound in $L^1$. But we have still need to control higher moments of $F$.
We show in the next Lemma how this may be formally done for polynomial momments.
Exponential moments may be controlled in a  similar way.

\begin{lemma} \label{lm4.5}
Let $F$ be a solution to equation (4.34). Then, there exists
a positive constant $C$ such that, formally:
$$
{d \over dt} Y_1(F) + {\gamma \over 4} \int_0^\infty F (1 + \sqrt{\epsilon})
\epsilon \, d\epsilon \le C_{\rm in},
\eqno(4.39)
$$
with
$$
Y_\theta (F) := \int_0^\infty \epsilon^\theta \, d|F|(\epsilon).
\eqno(4.40)
$$
\end{lemma}

\paragraph{Proof of Lemma \ref{lm4.5}}
Let $F$ be a solution to (4.34). A formal straightforward calculation gives
\begin{align*}
{d \over dt} Y_1(F)
&= \int_0^\infty \!\!\! \int_0^\infty  b F F' [e^{-\epsilon} -
e^{-\epsilon'}] \epsilon \,
d\epsilon'd\epsilon \\
&\quad
+ \int_0^\infty \!\!\! \int_0^\infty  b [ \sqrt{\epsilon} F' \epsilon \,
e^{-\epsilon}
- \sqrt{\epsilon'} F \, e^{-\epsilon'} \epsilon ] \, d\epsilon'd\epsilon \\
&= \int_0^\infty \!\!\! \int_0^\infty { b \over 2} F \,  F' [e^{-\epsilon} - 
e^{-\epsilon'}] ( \epsilon - \epsilon') \,d\epsilon'd\epsilon \\
&\quad
+ \int_0^\infty \!\!\! \int_0^\infty  b \sqrt{\epsilon'} F \, e^{-\epsilon'} 
[\epsilon' - \epsilon] \, d\epsilon'd\epsilon.
\end{align*}
Since the first term is non positive, we just have to manage with the
second term.
In order to estimate it we compute
\begin{align*}
I(\epsilon)
&:= \int_0^\infty b \sqrt{\epsilon'} \, e^{-\epsilon'} \,  [ \epsilon' - 
\epsilon] \,
d\epsilon'  \\
&= \int_0^\epsilon {\zeta(\epsilon') \over \sqrt{\epsilon}} \, e^{-\epsilon'} \,  
[ \epsilon' -
\epsilon] \, d\epsilon'
+  {\zeta(\epsilon) \over \sqrt{\epsilon}} \int_\epsilon^\infty \,  e^{-
\epsilon'} \,  [
\epsilon' - \epsilon] \, d\epsilon' = I_1 + I_2.
\end{align*}
On the one hand,
$$
I_2 = {\zeta(\epsilon) \over \sqrt{\epsilon}} \Bigl( \int_\epsilon^\infty 
\epsilon' \,
e^{-\epsilon'} \, d\epsilon'
- \epsilon \int_\epsilon^\infty \, e^{-\epsilon'} \, d\epsilon' \Bigr)\\
= {\zeta(\epsilon) \over \sqrt{\epsilon}} \, e^{-\epsilon} = e^{-\epsilon} + 
{\gamma
(\epsilon) \over \sqrt{\epsilon}} \le C.
$$
On the other hand,
$$
I_1 \le {1 \over \sqrt{\epsilon}} \int_0^\epsilon \gamma(\epsilon') (\epsilon'-
\epsilon) \,
d\epsilon'
\le {\gamma(m) \over \sqrt{\epsilon}} \int_m^\epsilon (\epsilon'-\epsilon) \, 
d\epsilon' =
{\gamma(m) \over \sqrt{\epsilon}} \bigr(-{\epsilon^2 \over 2} + m \epsilon - 
{m^2
\over 2} \bigl).
$$
Since $\gamma(m) \to \gamma$ when $m \to \infty$ and the term of greater
order have a minus sign,
we obtain
$$
I \le C - {\gamma \over 4} (1+\sqrt{\epsilon}) \epsilon,
$$
and (4.39) follows. \hfill$\square$

We state now the global existence result for the equation (1.26).

\begin{theorem} \label{thm4.6}
For any initial datum $F_{\rm in} \in L^1_1(\mathbb{R}_+)$, $F\ge 0$, there 
exists a
solution $F \in C([0,\infty),L^1_{1/2}))$ to the equation
$$
{\partial F\over \partial t}=Q_{BQ}(F),\quad
\hbox{for}\quad \varepsilon>0,\,\,t>0
$$
with $Q_{BQ}$ defined in (4.32), and such that
$$
\lim_{t\to 0}\|F(t)-F_{\rm in}\|_{L^1(\mathbb{R}^+)}=0.$$
\end{theorem}

\paragraph{Proof of Theorem \ref{thm4.6}}
We first introduce the regularized problem:
$$
{\partial F_n \over \partial t} =
\int_0^\infty  b_n(\epsilon,\epsilon') \bigl[ F'_n (\sqrt{\epsilon_n} + F_n) \,
e^{-\epsilon}
- F_n (\sqrt{\epsilon'_n} + F'_n) \, e^{-\epsilon'} \bigr] \, d\epsilon',
\eqno(4.41)
$$
with
$$
b_n(\epsilon,\epsilon') = {\zeta(\epsilon \wedge \epsilon' \wedge n)  \over 
\sqrt{\epsilon
\wedge \epsilon' \wedge n} \sqrt{\epsilon \vee \epsilon' \vee 1/n} }, \quad
\epsilon_n = \epsilon \vee {1 \over n}.
$$
The existence of a solution $F_n$ to this equation follows from the existence
result in [EM] since $b_n \in L^\infty$. We obtain a priori estimates on the
sequence $(F_n)$ in the two following Lemmas and then pass to the limit
using Lemma \ref{lm4.4}

\begin{lemma} \label{lm4.7}
The sequence of solutions $(F_n)$ satisfies
$$
{d \over dt} Y_1(F_n)
+ {\gamma \over 3} \int_0^\infty F_n \sqrt{\epsilon} (\epsilon \wedge n) \, 
d\epsilon
\le C_{\rm in} + Y_1(F_n).
\eqno(4.42)
$$
for $n$ large enough. This implies that $Y_1(F_n)$ is bounded in
$L^\infty(0,T)$ and \hfill \break
$\int_0^\infty F_n \sqrt{\epsilon} (\epsilon \wedge n) \, d\epsilon$ is bounded
in $L^1(0,T)$ for any $T > 0$.
\end{lemma}


\paragraph{Proof of Lemma \ref{lm4.7}}
 By the same computation as in Lemma \ref{lm4.5}, which
makes now perfectly sense by the properties of $F_n$, we deduce
$$
{d \over dt} Y_1(F_n) = \int_0^\infty \!\!\! \int_0^\infty { b_n \over 2}
F_n \,  F'_n [e^{-\epsilon} - e^{-\epsilon'}] ( \epsilon - \epsilon') 
\,d\epsilon'd\epsilon
+ \int_0^\infty F_n \, I_n \, d\epsilon.
$$
The first term in the right hand side is non positive and the second satisfies:
\begin{align*}
I_n(\epsilon)
&:= \int_0^\infty b_n \sqrt{\epsilon'_n} \, e^{-\epsilon'} \,  [ \epsilon' - 
\epsilon ]
\, d\epsilon'  \\
&\le \int_0^\epsilon { \gamma (\epsilon' \wedge n) \sqrt{\epsilon'_n} \over
\sqrt{\epsilon_n} \sqrt{\epsilon' \wedge n} }
\, e^{\epsilon' \wedge n - \epsilon'} [\epsilon'-\epsilon] d\epsilon'
+ { \zeta(\epsilon \wedge n) \over \sqrt{\epsilon \wedge n} } \, e^{-\epsilon}.
\end{align*}
Note that the last term is bounded.
Let fix $\ell \ge 1$ such that $\gamma(\ell) > 2 \gamma/3$. For $n$ and
$\epsilon$ such that $\epsilon \wedge n \ge
\ell$ we write
\begin{align*}
&\int_0^\epsilon { \gamma (\epsilon' \wedge n) \sqrt{\epsilon'_n}
 \over \sqrt{\epsilon_n}\sqrt{\epsilon' \wedge n} }
 e^{\epsilon' \wedge n-\epsilon'} [\epsilon'-\epsilon] d\epsilon'\\
&\le {2 \gamma \over 3 \sqrt{\epsilon}} \int_0^{\epsilon} {\sqrt{\epsilon'} 
\over
\sqrt{\epsilon' \wedge n}}
\, e^{\epsilon'\wedge n - \epsilon'} (\epsilon'-\epsilon) \, d\epsilon' + \gamma 
\int_0^\ell  {\epsilon \over \sqrt{\epsilon'
\wedge n}} \, d\epsilon' \\
&\le {2 \gamma \over 3 \sqrt{\epsilon}} \int_0^{\epsilon \wedge n} (\epsilon'-
\epsilon) \, d\epsilon'
+ \gamma \, 2 \sqrt{\ell} \epsilon
\le 2 \sqrt{\ell} \gamma \epsilon - {\gamma \over 3} \sqrt{\epsilon} (\epsilon 
\wedge n).
\end{align*}
We then conclude as in the proof of Lemma \ref{lm4.5}. \hfill$\square$

\begin{lemma} \label{lm4.8}
 The set $(F_n)$ is a Cauchy sequence
in $C([0,T];L^1_{1/2}(\mathbb{R}_+))$ for any $T > 0$.
\end{lemma}

\paragraph{Proof of Lemma \ref{lm4.8}} We just compute
$$
{\partial \over \partial t} (F_m-F_n) = Q_n(F_m) - Q_n(F_n) + Q_m(F_m) -
Q_n(F_m)
$$
and
$$
\begin{aligned}
&{d \over dt} \| F_m-F_n \|_{L^1_{1/2}}\\
&\le \int_0^\infty |F_m - F_n| \int_0^\infty b_n {\sqrt{\epsilon_n'}} \, e^{-
{\epsilon'}} [{\sqrt{\epsilon} - \sqrt{\epsilon'} }] \, {d\epsilon'} \, d\epsilon
\\
&+ \int_0^\infty\!\!\!\int_0^\infty |{F_m F'_m - F_n F'_n}| |e^{-\epsilon} - e^{-
\epsilon'} \, b_n (1 +{\sqrt{\epsilon}}) \, {d\epsilon'}d\epsilon \\
&+ \int_0^\infty \!\!\! \int_0^\infty  (b_m - b_n) F_m \,  {F'_m} |e^{-\epsilon} - 
e^{-\epsilon'}| ( 1 + {\sqrt{\epsilon}} ) \,{d\epsilon'}d\epsilon\\
&+ \int_0^\infty \!\!\! \int_0^\infty  |b_m {\sqrt{\epsilon'_m}} - b_n 
{\sqrt{\epsilon'_n}}| F_m \, e^{-\epsilon'}
\,   [ 2 + {\sqrt{\epsilon'} + \sqrt{\epsilon}}] \, {d\epsilon'}d\epsilon,\\
&\equiv J_1+J_2+J_3+J_4
\end{aligned} \eqno(4.43)
$$
with $b_{m,n} = b_m - b_n$.
We now estimate each of the terms $J_i$, $i=1,2,3,4$ separately.


\noindent{\sl 1. Estimate of $J_1$.} The first term is nothing but
$$
J_1(n) = \int_0^\infty |F_m - F_n| \, I_n \, d\epsilon
$$
with
\begin{align*}
I_n(\epsilon):=& \int_0^\infty b_n \sqrt{\epsilon'_n} \, e^{-\epsilon'} [ 
\sqrt{\epsilon'} - \sqrt{\epsilon} ] \, d\epsilon' \\
\le& {\zeta(\epsilon\wedge n) \over \sqrt{\epsilon\wedge n}} 
\int_\epsilon^\infty
e^{-\epsilon'} (\sqrt{\epsilon'} - \sqrt{\epsilon}) \, d\epsilon' =: I'_n.
\end{align*}
For $\epsilon \ge 1$, integration by parts gives
$$
I'_n = {\zeta(\epsilon\wedge n) \over \sqrt{\epsilon\wedge n}} 
\int_\epsilon^\infty
{e^{-\epsilon'} \over
\sqrt{\epsilon'}}
\, d\epsilon' \le {\zeta(\epsilon\wedge n) \over \sqrt{\epsilon\wedge n}} \, 
e^{-\epsilon}
\in L^\infty (\mathbb{R}_+).
$$
Since $I'_n$ is bounded for $\epsilon \le 1$, there exists a constant $K_1 > 0$
such that
$$
J_1 \le K_1 \| F_m - F_n \|_{L^1}.
\eqno(4.44)
$$
{\sl 2. Estimate of $J_2$.} We notice that
\begin{align*}
\chi_n &:= b_n (e^{-\epsilon} - e^{\epsilon'}) \\
&\le {\zeta(\epsilon \wedge \epsilon') \over \sqrt{\epsilon \wedge \epsilon'}}
{e^{-\epsilon} - \epsilon^{-\epsilon'} \over \sqrt{\epsilon \vee \epsilon'}} 
{\bf 1}_{
\epsilon \wedge \epsilon' \le 1}
+ {\zeta(\epsilon \wedge \epsilon' \wedge n) \over \sqrt{\epsilon \wedge 
\epsilon'
\wedge n}}\max(e^{-\epsilon},\epsilon^{-\epsilon'} ) {\bf 1}_{ \epsilon \vee 
\epsilon'
\ge 1}
\end{align*}
is uniformly bounded in $\mathbb{R}_+^2$ as we have already shown in the proof
of Lemma \ref{lm4.4}. Then,  we deduce
$$
\begin{aligned}
J_2 \!\le& \| \chi_n\|_{L^\infty_{\epsilon,\epsilon'}} [ \| (F_m - F_n) 
(1+\sqrt{\epsilon}) \|_{L^1} \| F_m
\|_{L^1} + \| F_m - F_n \|_{L^1} \| F_m (1+\sqrt{\epsilon}) \|_{L^1} \\
\le& K_2 \| F_m \|_{L^1_{1/2}} \,  \| F_m - F_n \|_{L^1_{1/2}},
\end{aligned}\eqno{(4.45)}
$$
for some constant $K_2 > 0$.


\noindent{\sl 3. Estimate of $J_3$.} Since $x \mapsto \zeta(x)/\sqrt{x}$ is
increasing (at least for the
large values of $x$) we have $0 \le b_n \le b_m$. We then notice that
\begin{align*}
0 \le b_m - b_n = &{\zeta(\epsilon \wedge \epsilon') \over \sqrt{\epsilon \wedge 
\epsilon'}} \bigl( {1 \over \sqrt{\epsilon \vee \epsilon' \vee 1/m}} - \sqrt{n} 
\bigr) {\bf
1}_{\epsilon \vee \epsilon' < 1/n} \\
&+ {1 \over \sqrt{\epsilon \vee \epsilon'}} \bigl( {\zeta(\epsilon \wedge 
\epsilon'
\wedge m) \over \sqrt{\epsilon \wedge \epsilon'
\wedge m}} - {\zeta(n) \over \sqrt{n}} \bigr) {\bf 1}_{\epsilon \wedge \epsilon'
n}.
\end{align*}
On the one hand, we have
\begin{align*}
&(b_m - b_n) |e^{-\epsilon} - e^{-\epsilon'}| {\bf 1}_{\epsilon \vee
\epsilon' < 1/n}\\
&\le
{\zeta(\epsilon \wedge \epsilon') \over \sqrt{\epsilon \wedge \epsilon'}} \,  {1 
\over
\sqrt{\epsilon \vee \epsilon' \vee 1/m}}
| e^{-\epsilon} - e^{-\epsilon'} | {\bf 1}_{\epsilon \vee \epsilon' < 1/n} \\
&\le \| {\zeta(x) \over \sqrt{x}} \|_{L^\infty (0,1)}
 \sqrt{\epsilon \vee \epsilon'}  {\bf 1}_{\epsilon \vee \epsilon' < 1/n}
\le {K_3 \over\sqrt{n}}  {\bf 1}_{\epsilon \vee \epsilon' < 1/n}\,.
\end{align*}
On the other hand
\begin{align*}
(b_m - b_n) |e^{-\epsilon} - e^{-\epsilon'}| {\bf 1}_{\epsilon \wedge \epsilon' 
\ge n}
&\le
{\zeta(\epsilon \wedge \epsilon' \wedge m) \over \sqrt{\epsilon \wedge \epsilon' 
\wedge m}} {1 \over \sqrt{\epsilon \vee \epsilon'}}
| e^{-\epsilon} - e^{-\epsilon'} | {\bf 1}_{\epsilon \wedge \epsilon' \ge n} \\
&\le {2 \over \sqrt{\epsilon \vee \epsilon'}} \| {\zeta(x) \, e^{-x} \over
\sqrt{x}} \|_{L^\infty}
{\bf 1}_{\epsilon \wedge \epsilon' \ge n} \le {K_3 \over\sqrt{n}} {\bf
1}_{\epsilon \wedge \epsilon' > n}
\end{align*}
for some constant $K_3 > 0$. We deduce that
$$
J_3 \le {K_3 \over \sqrt{n}} \, Y_0(F_m) \| F_m \|_{L^1_{1/2}}.\eqno{(4.46)}
$$
{\sl 4. Estimate of $J_4$.} Since $0 \le \sqrt{\epsilon'_n} \, b_n \le
\sqrt{\epsilon'_m} \, b_m$ for $n$ large
enough, we may write
\begin{align*}
&\sqrt{\epsilon'_m} \, b_m - \sqrt{\epsilon'_n} \, b_n\\
&\le {\sqrt{\epsilon' \vee 1/m}  \over \sqrt{\epsilon \vee \epsilon' \vee 1/m}} 
{\zeta(\epsilon \wedge \epsilon') \over
\sqrt{\epsilon \wedge \epsilon'}} {\bf 1}_{\epsilon \vee \epsilon' < 1/n}
+ {\sqrt{\epsilon'} \over \sqrt{\epsilon \vee \epsilon'}} {\zeta(\epsilon \wedge 
\epsilon'
\wedge m) \over
\sqrt{\epsilon \wedge \epsilon' \wedge m}} {\bf 1}_{\epsilon \wedge \epsilon' > 
n}.
\end{align*}
On the one hand we compute
$$
\begin{aligned}
\int_0^\infty &(b_m \sqrt{\epsilon_m'} - b_n \sqrt{\epsilon_n'} ) \,
e^{-\epsilon'} [2 + \sqrt{\epsilon} +
\sqrt{\epsilon'}] {\bf 1}_{\epsilon \vee \epsilon' < 1/n}  \, d\epsilon' \le \\
&\le 4 \int_0^{1/n} (b_m \sqrt{\epsilon_m'} - b_n \sqrt{\epsilon_n'} )  \,
d\epsilon'
\le 4 \| {\zeta(x)  \over \sqrt{x}} \|_{L^\infty(0,1)}  {1 \over n}.
\end{aligned} \eqno(4.47)
$$
On the other hand,
\begin{align*}
&\int_0^\infty (b_m \sqrt{\epsilon_m'} - b_n \sqrt{\epsilon_n'} ) \,
e^{-\epsilon'} [2 + \sqrt{\epsilon} + \sqrt{\epsilon'}] {\bf 1}_{\epsilon \wedge 
\epsilon' \ge n} \,
d\epsilon' \\
&\le \int_0^\epsilon {\sqrt{\epsilon'} \over \sqrt{\epsilon}} {\zeta(\epsilon' 
\wedge m) \over \sqrt{\epsilon'\wedge m}} \,  e^{-\epsilon'} [2 + 
\sqrt{\epsilon} + \sqrt{\epsilon'}] \, d\epsilon' {\bf 1}_{\epsilon \ge n} \\
&\quad + {\zeta(\epsilon \wedge m) \over \sqrt{\epsilon\wedge m}} 
\int_\epsilon^\infty  \,
e^{-\epsilon'} [2 + \sqrt{\epsilon} + \sqrt{\epsilon'}] \, d\epsilon' {\bf 
1}_{\epsilon \ge n} \\
&\le \int_0^\epsilon {\zeta(\epsilon' \wedge m) \over \sqrt{\epsilon'\wedge m}} 
\,
e^{-\epsilon'} \,  \sqrt{\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon' {\bf 1}_{\epsilon 
\ge n}
+ \| {\zeta(x) \over \sqrt{x}} \,  e^{-x} \|_{L^\infty} (3 + 2 \sqrt{\epsilon}) 
{\bf 1}_{\epsilon \ge n}.
\end{align*}
To estimate the first term in the last right hand side, we
successively consider the cases
$\epsilon \le m$ and $\epsilon \ge m$ and make use of the following elementary
estimate
$$
\int_m^\epsilon (\epsilon')^\gamma \, e^{-\epsilon'} \, d\epsilon' \le 
(1+m^\gamma) \, e^{-m},
$$
with $\gamma = 1/2$ and $\gamma = 1$. When $\epsilon \le m$ we have
$$
\int_0^\epsilon {\zeta(\epsilon' \wedge m) \over \sqrt{\epsilon'\wedge m}} \,
e^{-\epsilon'} \,  \sqrt{\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon'
=\! \int_0^\epsilon \zeta(\epsilon' ) \,  e^{-\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilon}}] \, d\epsilon'
\le 4 \| \zeta(x) \,  e^{-x} \|_{L^\infty} \epsilon.
\eqno(4.48)
$$
Now, when $\epsilon \ge m$, we get
\begin{align*}
&\int_0^\epsilon {\zeta(\epsilon' \wedge m) \over \sqrt{\epsilon'\wedge m}} \,
e^{-\epsilon'} \,  \sqrt{\epsilon'}
[3 + {\sqrt{\epsilon'} \over \sqrt{\epsilo