information...this article is a more complete, improved version of the
Preprint Nr.604/11/93 Bielefeld Univesity (BiBoS).
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\documentstyle{article}
\author{C.P. Gr\"unfeld\thanks{Permantent adress: IGSS, Institute of Atomic
Physics, Bucharest-Magurele, P. O. Box MG-6, RO-76900, Romania, E-mail
grunfeld@ifa.ro}\\
Equipe de physique math\'ematique et g\'eom\'etrie, Institut de \\
Math\'ematique de Paris Jussieu, CNRS, Universit\'e Paris VII, \\
case 7012, couloir 45-55, 5-\`eme \'etage, 2 pl. Jussieu, \\
Paris, 75251, France
\and
E. Georgescu\\
IGSS, Institute of Atomic Physics, Bucharest-Magurele,\\
P. O. Box MG-6, RO-76900, Romania}
\title{ON A CLASS OF KINETIC EQUATIONS FOR REACTING GAS MIXTURES
}
\date{
}
\begin{document}
\maketitle
\begin{abstract}
We consider a general class of kinetic equations for real gases with
(possibly) multiple inelastic collisions and chemical reactions. We prove
the existence, uniqueness and positivity of solutions for the Cauchy problem
and obtain the conservation relations for mass, momentum and energy, the
H-Theorem as well as the law of the mass action.
\end{abstract}
\section{Introduction}
We investigate the mathematical properties of a class of Boltzmann-type
kinetic equations for a model of reacting gas composed of several species of
mass points with well-defined, unique internal energy state and
multi-particle (in)elastic collisions (reactions). The number of species and
the multiplicity of the collisions may be arbitrary. The gas particles move
freely between collisions. The gas collisions occur with energy and momentum
conservation according to the laws of the classical mechanics. For the
one-component gas, with elastic binary collisions, the model kinetic
equations can be reduced to the classical Boltzmann equation.
Our interest in this model is due to the following thing. Certain kinetic
equations for the real gas, which are important for applications, but less
understood mathematically, appear to belong to our class of Boltzmann-type
kinetic equations, as soon as they are written in convenient form. The main
example refers to the Wang Chang and Uhlenbeck \cite{wa} as well as the
Ludwig and Heil \cite{lu} equations, describing the real gas with inelastic
collisions and chemical reactions, respectively. The fact that the equations
introduced in Ref.\cite{wa}, \cite{lu} belong to the class examined in this
paper is the consequence of the point of view, implicitly adopted in Ref.%
\cite{wa}, \cite{lu} (see also \cite{ce}, \cite{ku}): in certain situations
a real gas particle (molecule, atom, etc.) with internal structure can be
considered as a mechanical system that differs from a mass point by a
succession of internal states; each internal state has a well-defined value
of the energy. It becomes convenient to treat different internal states of
the gas particle with internal structure, as distinct, structureless
point-objects, belonging to different species, of given mass and unique
internal energy state, and described by different distribution functions.
Consequently we can think of the gas of particles with internal structure as
a gas mixture of different mass-points, with unique internal energy, and
re-write the original kinetic equations in a suitable form by re-labeling
the original distribution functions (each original distribution function,
describing a succession of internal states of a particle with internal
structure, is replaced by a sequence of distribution functions associated to
each internal state).
The aim of the present paper is to solve the Cauchy problem for the
aforementioned class of reactive Boltzmann-type equations and to prove the
basic global conservation relations, the H-Theorem, as well as the law of
the mass action. The analysis reveals new mathematical difficulties, in
comparison with the classical Boltzmann equation and other rigorous models $%
\left[ 5-9\right] $. The difficulties are essentially due to the presence of
the internal energy. They are introduced by the reaction thresholds, and are
already visible in the case of the gas model with three-body collisions
(reactions). The situations with more than three-body collisions (reactions
) do not introduce additional conceptual problems. However, the mathematical
difficulties are better understood by investigating the general model than
particular cases that might contain irrelevant details.
The plan of the paper is as follows. In the next section we introduce the
class of reactive Boltzmann-type kinetic equations. The main result, Theorem
1, obtained in Section 3, proves the existence, uniqueness and positivity of
solutions (with small initial data) for the Cauchy problem associated to
this class of equations. The solutions are global (in time) in the case in
which the endo-energetic reactions are not present at the gas processes. In
the case of the simple gas with elastic binary collisions, Theorem 1 reduces
to known existence results on the classical Boltzmann equation \cite{be}.
The argument of Theorem 1 follows by fixed point techniques, due to
estimations based on the (local) conservation relations for mass, momentum
and energy. The key estimation is given in Lemma 1. In Section 4 we prove
the bulk conservation relations for mass, momentum and energy as well as the
H-Theorem.
Finally, the following fact should be remarked. The probability of
multi-particle Collisions is zero, in some sense (\cite{ai}), in the
dynamics of the classical hard sphere gas with elastic collisions (which
plays an essential role in the validation of the classical Boltzmann
equation). The situation seems being different in the case of the reacting
gas: reaction processes {\it producing more that two} {\it particles} could
be important to the gas evolution.
{\it \ }Some of the results presented here, have been announced in \cite{gr}.
\section{The frame}
Consider a model of reacting gas without external fields, composed of $N$ $%
\geq 1$, distinct species of mass-points, with one-state internal energy.
Each species of gas constituents will be labeled by some simple index $%
k=1,...,N$. The gas particles have a free classical motion, in the whole
space, between (in)elastic, instant collisions. By hypothesis, at most, $%
M\geq 2$ identical partners may participate in some in (out) collision
(reaction) channel. During the gas processes, the particles may change their
chemical nature (in particular, mass and internal energy) and velocity. It
is supposed that the collisions occur with the conservation of mass,
momentum and energy, respectively, according to the laws of classical
mechanics. The particles internal energies enter in the energy balance.
Let ${\cal M}:=\left\{ \gamma =(\gamma _n)_{1\le n\le N}\mid \gamma _n\in
\left\{ 0,1,\ldots ,M\right\} \right\} $ be a multi-index set. A certain gas
collision (reaction) process can be specified by a couple $(\alpha ,\beta
)\in {\cal M\times M}$. Here $\alpha =(\alpha _1,\ldots ,\alpha _N)$ is the
''in'' channel. It designates the pre-collision configuration, with $\alpha
_n\in \left\{ 0,1,\ldots ,M\right\} $ participants of the species $n$, $1\le
n\le N$. Further, $\beta =(\beta _1,\ldots ,\beta _N)$ denotes the ''out''
channel. It refers to the post-collision configuration, with $\beta _n\in
\left\{ 0,1,\ldots ,M\right\} $ participants of the species $n$, $1\le n\le
N $. For some $\gamma \in {\cal M}$, the total numbers of the particles in
channel $\gamma $ is $\mid \gamma \mid :=\sum_{n=1}^N\gamma _n$. The family
of those species present in the channel $\gamma \in {\cal M}$ can be
identified by ${\cal N}(\gamma ):=\left\{ n\mid 1\le n\le N,\;\gamma _n\ge
1\right\} $. Consequently, if $\gamma \in {\cal M}$, with $\mid \gamma \mid
\geq 1$, for each $n\in {\cal N}(\gamma )$, there are exactly $\gamma _n$
identical particles of the species $n$, participating in $\gamma $. Their
velocities will be denoted by ${\bf w}_{n,1},...,{\bf w}_{n,\gamma _n}\in
{\bf R}^3$. Also set ${\bf w}=(({\bf w}_{n,i})_{1\leq i\leq {\gamma _n}%
})_{n\in {\cal N}(\gamma )}$, understanding that ${\bf w\in R}^{3\mid \gamma
\mid }$. By $m_n>0$ and $E_n\in {\bf R}$, denote the mass and the internal
energy, respectively of a mass-point of the species $n=1,...,N$.
Let $V_\gamma ({\bf w})$ and $W_\gamma ({\bf w})$ be the classical mass
center velocity and the total energy, respectively, for the particles in
channel $\gamma $, i.e.,%
$$
V_\gamma ({\bf w}):=(\sum_{n=1}^N\gamma _nm_n)^{-1}\sum_{n\in {\cal N}%
(\gamma )}\sum_{i=1}^{\gamma _n}m_n{\bf w}_{n,i},
$$
$$
W_\gamma ({\bf w}):=\sum_{n\in {\cal N}(\gamma )}\sum_{i=1}^{\gamma
_n}(2^{-1}m_n{\bf w}_{n,i}^2+E_n).
$$
According to the previous conservation assumptions we are interested in
those gas processes $(\alpha ,\beta )\in {\cal M\times M}$, where
\begin{equation}
\label{15}\sum_{n=1}^Nm_n(\alpha _n-\beta _n)=0,\qquad V_\alpha ({\bf w}%
)=V_\beta ({\bf u}),\qquad W_\alpha ({\bf w})=W_\beta ({\bf u}),
\end{equation}
with ${\bf w}=(({\bf w}_{n,i})_{1\leq i\leq {\alpha _n}})_{n\in {\cal N}%
(\alpha )}$ and ${\bf u}=(({\bf u}_{n,i})_{1\leq i\leq {\beta _n}})_{n\in
{\cal N}(\beta )}$ defining the velocities of the particles in the channels $%
\alpha $ and $\beta $, respectively.
Suppose that one knows the transition law (\cite{wa}, \cite{lu}) $K_{\alpha
,\beta }$ of each reaction process $(\alpha ,\beta )$. Following the
standard Boltzmann procedure, we can formally write equations similar to
those introduced in \cite{wa}, \cite{lu}
\begin{equation}
\label{1}\partial _tf_k+{\bf v}\cdot \nabla f_k=P_k(f)-S_k(f),\qquad 1\le
k\le N.
\end{equation}
The unknowns are the functions $f_k:{\bf R}_{+}\times {\bf R}^3\times {\bf R}%
^3\rightarrow {\bf R}_{+}$,$\;1\le k\le N$, where ${\bf R}_{+}:=[0,\infty )$%
. Here $f_k=f_k(t,{\bf v},{\bf x})$ ($t$-time, ${\bf v}$ -velocity, ${\bf x}$%
-position) is the distribution function for species $k$ of mass-points and $%
f:=(f_1,\ldots ,f_N)$. The collision processes are described by the
nonlinear terms $P_k(f)$ and $S_k(f)$%
$$
P_k(f)(t,{\bf v},{\bf x}):=\sum_{\alpha ,\beta \in {\cal M}}\ \alpha _k\
\int_{{\bf R}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid
}}\,f_\beta (t,{\bf u},{\bf x})\,K_{\beta ,\alpha }({\bf u},{\bf w})\
$$
\begin{equation}
\label{2}\times \ \delta ({\bf w}_{k,\alpha _k}-{\bf v})\ \delta (V_\beta (%
{\bf u})-V_\alpha ({\bf w}))\ \delta (W_\beta ({\bf u})-W_\alpha ({\bf w}))\
d{\bf u\otimes }d{\bf w},
\end{equation}
\smallskip\
$$
S_k(f)(t,{\bf v},{\bf x}):=\sum_{\alpha ,\beta \in {\cal M}}\alpha _k\int_{%
{\bf R}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid }}\ \ f_\alpha
(t,{\bf w},{\bf x})K_{\alpha ,\beta }({\bf w},{\bf u})
$$
\begin{equation}
\label{2'}\times \ \delta ({\bf w}_{k,\alpha _k}-{\bf v})\ \delta (V_\beta (%
{\bf u})-V_\alpha ({\bf w}))\ \delta (W_\beta ({\bf u})-W_\alpha ({\bf w}))\
d{\bf u\otimes }d{\bf w},
\end{equation}
for all $t\ge 0$, ${\bf v}$, ${\bf x}\in {\bf R}^3$; $1\le k\le N$. Here, $%
(K_{\alpha ,\beta })_{(\alpha ,\beta )\in {\cal M}\times {\cal M}}$ is the
family of transition functions $K_{\alpha ,\beta }:{\bf R}^{3\mid \alpha
\mid }\times {\bf R}^{3\mid \beta \mid }\rightarrow {\bf R}_{+}$, $\alpha $,
$\beta \in {\cal M}$ and \
$$
f_\gamma (t,{\bf w},{\bf x})=\Pi _{n\in {\cal N}(\gamma )}\,\Pi
_{i=1}^{\gamma _n}\,f_n(t,{\bf w}_{n,i},{\bf x}),
$$
We introduce the following general assumptions:
a) $K_{\alpha ,\beta }\equiv 0\,\,$if $\mid \alpha \mid \le 1\,$or $\mid
\beta \mid \leq 1$.
b) If for some $\alpha $, $\beta \in {\cal M}$,$\;\sum_{n=1}^N\alpha _nm_n%
\not =\sum_{n=1}^N\beta _nm_n$, then $K_{\alpha ,\beta }\equiv 0$.
c) For each ${\bf w}$, ${\bf u}$ and $n\in {\cal N}(\alpha )$ fixed, $%
K_{\alpha ,\beta }({\bf w},{\bf u})$ is invariant at the interchange of
components ${\bf w}_{n,1},...,{\bf w}_{n,\alpha _n}$ of ${\bf w}$; a similar
statement is true with respect to the interchange of the components of ${\bf %
u}$.
d) For each $a\in {\bf R}^3$, define the map ${\bf w\rightarrow }T(a){\bf w}$
by setting $(T(a){\bf w})_{n,i}:={\bf w}_{n,i}+a$, for all $n,i$ ; then $%
K_{\alpha ,\beta }({\bf w},{\bf u})\equiv K_{\alpha ,\beta }(T(a){\bf w},%
{\bf u})\equiv K_{\alpha ,\beta }({\bf w},T(a){\bf u})$, for all $a\in {\bf R%
}^3$, $({\bf w,u)}\in {\bf R}^{3\left| \alpha \right| }\times {\bf R}^{3\mid
\beta \mid }$ and $\alpha $, $\beta \in {\cal M}.$
Assumption a) excludes the ''spontaneous decay'' $(\mid \alpha \mid \le 1)$
and the ''total fusion'' $(\mid \beta \mid \le 1)$. Condition b) states the
mass conservation during the gas processes. Moreover, c) expresses the
''indistinguishability'' of identical collision partners. Finally, d) claims
absence of external fields.
The presence of the Dirac $\delta $-''functions'' in (\ref{2}) and (\ref{2'}%
) expresses the conservation of the total energy and momentum, respectively,
during collisions.
It can be easily seen that the kinetic equations introduced in \cite{wa},
\cite{lu} can be written in the form (\ref{1}), by redefining the
distribution functions according to the remarks in the previous section.
For some channel $\gamma \in {\cal M}$, let%
$$
W_{r,\gamma }({\bf w}):=W_\gamma ({\bf w})-2^{-1}(\sum_{n=1}^N\gamma
_nm_n)V_\gamma ({\bf w})^2-\sum_{n=1}^N\gamma _nE_n,\qquad {\bf w\in {R}}%
^{3\mid \gamma \mid },
$$
be the corresponding mass center kinetic energy. Obviously, $W_{r,\gamma }(%
{\bf w})\ge 0$.
We suppose that, $\forall \alpha $, $\beta \in {\cal M}$, the transition law
$K_{\alpha ,\beta }$ is continuous on the set $\{({\bf w},{\bf u})\in {\bf R}%
^{3\mid \alpha \mid }\times {\bf R}^{3\mid \beta \mid }\mid W_{r,\alpha }(%
{\bf w})>0,W_{r,\beta }({\bf u})>0\}$.
We introduce the following hypothesis, extending a class of cut-off
conditions for elastic binary collisions \cite{be}.
{\bf ASSUMPTION.}{\it - There are some constants} $C>0$,$\;0\leq q\leq 1$%
{\it , such that} {\it for all}$\;\alpha $, $\beta $, ${\bf w\in {R}}^{3\mid
\alpha \mid }$,$\;{\bf u\in {R}}^{3\mid \beta \mid }$, {\it we have}
\begin{equation}
\label{3}K_{\alpha ,\beta }({\bf w},{\bf u})\le C\ \frac{1+W_{r,\alpha }(%
{\bf w})^{q/2}+W_{r,\beta }({\bf u})^{q/2}}{W_{r,\alpha }({\bf w})^{(3\mid
\alpha \mid -5)/2}+W_{r,\beta }({\bf u})^{(3\mid \beta \mid -5)/2}}.
\end{equation}
\
In the rest of this section we give a meaning to (\ref{2}), (\ref{2'}). Let $%
C_c({\bf R}{^3}\times {\bf R}^3)$ denote the space of continuous functions
with compact support on ${\bf R}{^3}\times {\bf R}^3$. For each $\tau \geq 0$%
, $n=1,...,N$, fixed, let $\dot C_{n,\tau }$ be the closure of $C_c({\bf R}{%
^3}\times {\bf R}^3)$-real in the norm%
$$
\left| h\right| _{n\tau }=\sup \left\{ \exp \left[ \tau m_n({\bf x}^2+{\bf v}%
^2)\right] \left| h({\bf v},{\bf x})\right| :{\bf v},{\bf x}\in {\bf R}%
^3\right\} ,\;h\in C_c({\bf R}{^3}\times {\bf R}^3).
$$
Set $\dot C_\tau =\Pi $$_{1\le n\le N}\,\dot C_{n,\tau }$, with norm $\mid
h\mid _\tau :=\max \limits_{1\le n\le N}\left| h_n\right| _{n,\tau }$, for $%
h=(h_1,\ldots ,h_N)\in \dot C_\tau $. Let $\delta _\epsilon :{\bf R}%
\rightarrow {\bf R}_{+}$, $\epsilon >0$, be an even mollifier with supp $%
\delta _\epsilon =[-\epsilon ,\epsilon ]$ (i.e. $\delta _\epsilon
(t)=:\epsilon ^{-1}J(t/\epsilon )$, for some even function $J\in C_c({\bf R};%
{\bf R}_{+})$, with $supp$ $J=[-1,1]$ and $\left\| J\right\| _{L^1}=1$). Set
$\delta _\epsilon ^3(y):=\delta _\epsilon (y_1)\cdot \,\delta _\epsilon
(y_2)\cdot \,\delta _\epsilon (y_3)$, with $y=(y_1,y_2,y_3)\in {\bf R}^3$.
For some $\tau >0$ and $f=(f_1,\ldots ,f_N)\in \dot C_\tau $, define%
$$
P_{k\epsilon \eta }(f)({\bf v},{\bf x}):=\sum_{\alpha ,\beta \in {\cal M}%
}\alpha _k\int_{{\bf R}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid
-3}}\ d{\bf u\otimes }d{\bf \tilde w}_{(k)}\ \left[ f_\beta ({\bf u},{\bf x}%
)\right.
$$
\begin{equation}
\label{4}\,\left. \times \,K_{\beta ,\alpha }({\bf u},{\bf w})\,\delta
_\epsilon ^3(V_\beta ({\bf u})-V_\alpha ({\bf w}))\,\delta _\eta (W_\beta (%
{\bf u}))-W_\alpha ({\bf w}))\right] _{{\bf w}_{k,\alpha _k}={\bf v}},
\end{equation}
\smallskip\ \
$$
R_{k\epsilon \eta }(f)({\bf v},{\bf x}):=\sum_{\alpha ,\beta \in {\cal M}%
}\alpha _k\int_{{\bf R}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid
-3}}\ d{\bf u\otimes }d{\bf \tilde w}_{(k)}\,\ \left[ f_{\alpha ,k}({\bf w},%
{\bf x})\right.
$$
\begin{equation}
\label{4'}\,\left. \times \,K_{\alpha ,\beta }({\bf w},{\bf u)\,}\delta
_\epsilon ^3(V_\beta ({\bf u})-V_\alpha ({\bf w}))\,\delta _\eta (W_\beta (%
{\bf u}))-W_\alpha ({\bf w}))\right] _{{\bf w}_{k,\alpha _k}={\bf v}}\;,
\end{equation}
for all ${\bf v}$, ${\bf x}\in {\bf R}^3$, $1\le k\le N$. Here by
definition, the terms with $\alpha _k=0$, vanish identically, $d{\bf \tilde w%
}_{(k)}$ is the Euclidean element of area induced by $d{\bf w}$ on the
manifold $\left\{ {\bf w\in R^{3\mid \alpha \mid }:w}_{k,\alpha _k}={\bf v}%
\right\} ${\it ,} while%
$$
f_\beta ({\bf u},{\bf x}):=\Pi _{n\in {\cal N}(\beta )}\Pi _{i=1}^{\beta
_n}f_n({\bf u}_{n,i},{\bf x}),
$$
$$
f_{\alpha ,k}({\bf w},{\bf x})=\Pi _{n\in {\cal N}(\alpha )\setminus \left\{
k\right\} }\Pi _{i=1}^{\alpha _n}f_n({\bf w}_{n,i},{\bf x})\cdot \Pi
_{p=1}^{\alpha _k-1}f_k({\bf w}_{k,p},{\bf x}).
$$
{\bf PROPOSITION 1}. Let $\tau >0$ and $f\in \dot C_\tau $.
{\it a) For each} $k=1,...,N$, {\it there exist the limits}%
$$
P_k(f)({\bf v},{\bf x})=\lim \limits_{\eta \rightarrow 0}\lim
\limits_{\epsilon \rightarrow 0}P_{k\epsilon \eta }(f)({\bf v},{\bf x}%
)\;and\;R_k(f)({\bf v},{\bf x})=\lim \limits_{\eta \rightarrow 0}\lim
\limits_{\epsilon \rightarrow 0}R_{k\epsilon \eta }(f)({\bf v},{\bf x}),
$$
$\forall $$({\bf v},{\bf x})\in {\bf R}^3\times {\bf R}^3$. {\it Also,} $%
P_k(f)\in C_{k,\mu }({\bf R}^3\times {\bf R}^3)$ {\it for all $\mu \in
\left[ 0,\tau \right) $}, {\it while} $\sup \limits_{{\bf v},{\bf x}}\left\{
(1+{\bf v}^2)^{-q/2}\left| R_k(f)({\bf v},{\bf x})\right| \right\} <\infty $.
{\it b) Let }$h\in C({\bf R}^3)$ {\it with} $\sup \limits_{{\bf v}}\left\{
(1+{\bf v}^2)^{-1}\left| h({\bf v})\right| \right\} <\infty ${\it . Then, }$%
\forall ${\it \ }${\bf x}\in {\bf R}^3$,
$$
\int_{{\bf R}^3}h({\bf v})\ P_k(f)({\bf v},{\bf x})\ d{\bf v}=\lim
\limits_{\eta \rightarrow 0}\lim \limits_{\epsilon \rightarrow 0}\int_{{\bf R%
}{^3}}h({\bf v})\ P_{k\epsilon \eta }(f)({\bf v},{\bf x})\ d{\bf v},
$$
$$
\int_{{\bf R}^3}h({\bf v})\ f_k({\bf v},{\bf x})R_k(f)({\bf v},{\bf x})\ d%
{\bf v}=\lim \limits_{\eta \rightarrow 0}\lim \limits_{\epsilon \rightarrow
0}\int_{{\bf R}{^3}}h({\bf v})\,f_k({\bf v},{\bf x})R_{k\epsilon \eta }(f)(%
{\bf v},{\bf x})\ d{\bf v},
$$
{\it for each} $k=1,...,N$ \
{\bf Proof.} Set $T_\beta ({\bf u})=W_\beta ({\bf u})-\sum_{n=1}^N\beta
_nE_n $. We associate Jacobi coordinates $(\underline{{\sl V}},\xi )\in {\bf %
R}^3\times {\bf R}^{3\mid \beta \mid -3}$ to the form $T_\beta ({\bf u})$ on
${\bf R}^{3\mid \beta \mid }$, with $\xi :=(\xi _1,\ldots ,\xi _{\mid \beta
\mid -1})$, $\,\xi _i\in {\bf R}{^3}$,${\ \;}i=1,\ldots ,\mid \beta \mid -1$
$\,$(see (A.2) Appendix A). Consider a representation of $\xi $ in spherical
coordinates on ${\bf R}^{3\mid \beta \mid -3}$,$\;\xi =r{\bf n}$ , with $(r,%
{\bf n})\in \left[ 0,\infty \right) \times \Omega _{3\mid \beta \mid -4}$,
where $\Omega _{3\mid \beta \mid -4}$ is the unit sphere in ${\bf R}^{3\mid
\beta \mid -3}$. In (\ref{4}) and (\ref{4'}) we choose $(\underline{V},r,%
{\bf n})$ as new integration variables such that ${\bf u}={\bf u}(\underline{%
V},r,{\bf n})$. Then the limits of Prop.1 follow by repeated application of
Lebesgue's dominated convergence theorem, using the properties of $K_{\alpha
,\beta }$, $\delta _\epsilon ^3$ and $\delta _\eta $. The continuity of $%
P_k(f)$ and $R_k(f)$ is a consequence of the continuity of $K_{\alpha ,\beta
}$.$\Box $\
The proof of Prop.1 provides the limits (\ref{4}), (\ref{4'}) in explicit
form. Define%
$$
t_{\beta ,\alpha }({\bf w})=\left\{
\begin{array}{l}
\left[ W_{r,\alpha }(
{\bf w})+\sum\limits_{n=1}^N(\alpha _n-\beta _n)E_n\right]
^{1/2}\,if\;W_{r,\alpha }({\bf w})+\sum\limits_{n=1}^N(\alpha _n-\beta
_n)E_n\geq 0,\medskip \\ 0,\;\;otherwise.
\end{array}
\right.
$$
If
\begin{equation}
\label{5}W_{r,\alpha }({\bf w})+\sum_{n=1}^N(\alpha _n-\beta _n)E_n\ge 0,
\end{equation}
then set
\begin{equation}
\label{6}{\bf u}_{\beta \alpha }({\bf w},{\bf n}):={\bf u}(\underline{V},r,%
{\bf n})_{\mid \underline{V}=V_\alpha ({\bf w}),\;r=t_{\beta ,\alpha }({\bf w%
})}.
\end{equation}
For the sake of simplicity, ${\bf u}_{\beta \alpha }$ will replace the
notation ${\bf u}_{\beta \alpha }({\bf w},{\bf n})$. Define
\begin{equation}
\label{7}p_{\beta \alpha }({\bf w},{\bf n}):=2^{-1}\Delta _\beta \cdot
t_{\beta ,\alpha }({\bf w})^{3\mid \beta \mid -5}K_{\beta ,\alpha }({\bf u}%
_{\beta \alpha },{\bf w}),
\end{equation}
\begin{equation}
\label{7'}r_{\beta \alpha }({\bf w},{\bf n}):=2^{-1}\Delta _\beta \cdot
t_{\beta ,\alpha }({\bf w})^{3\mid \beta \mid -5}K_{\alpha ,\beta }({\bf w},%
{\bf u}_{\beta \alpha }),
\end{equation}
where the constant $\Delta _\beta $ is introduced by the Jacobian of ${\bf %
w\rightarrow }(\underline{V},r,{\bf n})$. With the definitions (\ref{4}), (%
\ref{4'}), we can write%
$$
P_k(f)({\bf v},{\bf x})
$$
\begin{equation}
\label{8}=\,\sum_{\alpha ,\beta \in {\cal M}}\alpha _k\int_{{\bf R}^{3\mid
\alpha \mid -3}\times \Omega _{3\mid \beta \mid -4}}d{\bf \tilde w}%
_{(k)}\otimes d{\bf n\,}\left[ p_{\beta \alpha }({\bf w},{\bf n})f_\beta (%
{\bf u}_{\beta \alpha },{\bf x})\right] _{{\bf w}_{k,\alpha _k}={\bf v}},
\end{equation}
\smallskip\
$$
R_k(f)({\bf v},{\bf x})
$$
\begin{equation}
\label{8'}\,=\,\sum_{\alpha ,\beta \in {\cal M}}\alpha _k\int_{{\bf R}%
^{3\mid \alpha \mid -3}\times \Omega _{3\mid \beta \mid -4}}d{\bf \tilde w}%
_{(k)}\,\otimes d{\bf n\,}\left[ r_{\beta \alpha }({\bf w},{\bf n})f_{\alpha
,k}({\bf w},{\bf x})\right] _{{\bf w}_{k,\alpha _k}={\bf v}}.
\end{equation}
For $f$ as in Prop. 1, we define $S_k(f)({\bf v},{\bf x})=f_k({\bf v},{\bf x}%
)R_k(f)({\bf v},{\bf x})$ with $R_k(f)$ given by (\ref{8'}).
We point out the following simple relations resulting from the definition of
${\bf u}_{\beta \alpha }$, provided that, condition (\ref{5}), is fulfilled:
\begin{equation}
\label{9}
\begin{array}{c}
V_\beta (
{\bf u}_{\beta \alpha })=V_\alpha ({\bf w}),\quad W_\beta ({\bf u}_{\beta
\alpha })=W_\alpha ({\bf w}),\medskip \\ \;W_{r,\beta }({\bf u}_{\beta
\alpha })=W_{r,\alpha }({\bf w})+\sum_{n=1}^N(\alpha _n-\beta _n)E_n.
\end{array}
\end{equation}
By (\ref{3}) and (\ref{9}), there exists some constant $K>0$ such that if
condition (\ref{5}) is fulfilled, then (for $q\in \left[ 0,1\right] $
introduced in (\ref{3})),
\begin{equation}
\label{10}p_{\beta \alpha }({\bf w},{\bf n})\le K\left[ 1+W_{r,\alpha }({\bf %
w})^{q/2}\right] ,\quad r_{\beta \alpha }({\bf w},{\bf n})\le K\left[
1+W_{r,\alpha }({\bf w})^{q/2}\right] .
\end{equation}
{\bf REMARK:} In the definitions of $p_{\beta \alpha }$ and $r_{\beta \alpha
}$ , the presence of $t_\beta $ exhibits the contributions of the reaction
thresholds. \ \
\section{Existence theory}
In this paper we are interested to solve Eq.(\ref{1}) in $\dot C_0$ (the
space of continuous distribution functions, vanishing at infinity in the
velocity and position variables). With the notations of Prop.1, for some $%
\tau >0$ fixed, set $P(f)=(P_1(f),...,P_N(f))$ and $S(f)=(S_1(f),\ldots
,S_N(f))$, $\forall \,\,f\in \dot C_\tau \subset \dot C_0$. Then $%
f\rightarrow P(f)$ and $f\rightarrow S(f)$, considered as maps in $\dot C_0$%
, have extensions (also denoted $P$ and $S)$ to their natural domains in $%
\dot C_0$.
The Cauchy problem for Eq.(\ref{1}) formulated in $\dot C_0$ is
\begin{equation}
\label{11}d_tf=Af+P(f)-S(f),\quad f(t=0)=f_0,
\end{equation}
with $A$ the infinitesimal generator of the positivity preserving,
continuous group $\{U^t\}_{t\in {\bf R}}$ of isometries of $\dot C_0$, given
by its $\dot C_{n,0}$ components, $1\le n\le N,$%
\begin{equation}
\label{u}(U^tf)_n({\bf v},{\bf x}):=U_n^tf_n({\bf v},{\bf x})=f_n({\bf v},%
{\bf x}-t{\bf v}),\quad (t,{\bf v},{\bf x})\in {\bf R}\times {\bf R}^3\times
{\bf R}^3.
\end{equation}
We call $f\in C(0,T;\dot C_0)$ a mild solution, on $\left[ 0,T\right] $, of
Eq.(\ref{11}) (in $\dot C_0$ ) if $P(f)$, $S(f)\in C(0,T;\dot C_0)$, and $%
\,f $ satisfies
\begin{equation}
\label{12}f(t)=U^tf_0+\int_0^tU^{t-s}P(f(s))ds-\int_0^tU^{t-s}S(f(s))ds,
\end{equation}
(the integral being in $\dot C_0$ in the sense of Riemann ).
Our main result states the existence, uniqueness and positivity of mild
solutions, for initial data close to the vacuum state. These solutions are
(time) global in the case of the gas with purely exo-energetic reactions
and/or elastic (multiple) collisions.
For $T>0$ fixed, consider $C(0,T;\dot C_\tau )$ with the usual sup norm,
denoted $\left\| \circ \right\| _\tau $. If $g=(g_1,...,g_N)\in C(0,T;\;\dot
C_\tau )$ and $t\in \left[ 0,T\right] $, by $g_n(t,{\bf v},{\bf x})$, denote
the value of $g_n(t)\in \dot C_{n,\tau }$ , $1\leq n\leq N$, at $({\bf v},%
{\bf x})\in {\bf R}^3\times {\bf R}^3$. Let $\dot C_\tau ^{+}:=\left\{
g=(g_1,...,g_N)\in \dot C_\tau :\;g_n({\bf v},{\bf x})\geq 0,\forall \,\;(%
{\bf v},{\bf x})\in {\bf R}^3\times {\bf R}^3\,;n=1,...,N\right\} $.
Finally, for some $R>0$, put ${\cal H}_\tau (R)=\left\{ h\mid h\in
C(0,T;\dot C_\tau ^{+}),\left\| \;h\right\| _\tau \le R\right\} $.
{\bf THEOREM 1 }{\it Let} $\tau >0$ {\it and} $f_0\in \dot C_\tau ^{+}$.
{\bf \ }{\it a) For each} $T>0,\;\exists \,R_T$, $R_T^{*}>0$ {\it such that
if } $\mid f_0\mid _\tau \le R_T$, {\it then Eq.(\ref{11}) in $\dot C_0$},
{\it \ has a unique mild solution} $f$ {\it on $\left[ 0,T\right] $}, {\it %
satisfying} $U^{-t}f\in {\cal H}_\tau (R_T^{*})$.{\it \ The map }$%
f_0\rightarrow f$ {\it is continuous from }$\left\{ h\in \dot C_\tau
^{+}:\left| h\right| _\tau \,\leq R_T\right\} $ {\it to } $C(0,T;\;\dot C_0)$%
.
{\it b)} {\it Assume} {\it that} $K_{\alpha ,\beta }\equiv 0$ {\it for each
couple} $(\alpha ,\beta )$ {\it that yields} $\sum_{n=1}^N(\alpha _n-\beta
_n)E_n<0$ ({\it exo-energetic reactions). In this case}, $\exists
\,R,R^{*}>0 $ {\it such that if } $\mid f_0\mid _\tau \le R${\it , then for
each} $T>0,$ {\it Eq. (\ref{11}) in $\dot C_0$},{\it \ has a unique mild
solution} $f$, {\it on }$\left[ 0,T\right] $, {\it satisfying} $U^{-t}f\in
{\cal H}_\tau (R^{*})$. {\it The map} $f_0\rightarrow f$ {\it is continuous
from} $\left\{ h\in \dot C_\tau ^{+}:\left| h\right| _\tau \,\leq
R_T\right\} $ {\it to} $C(0,T;\;\dot C_0)$.
{\it c) In each situation, $U^{-t}f\in C^1(0,T;\dot C_0)$}. \
{\bf REMARK} - In the case of the simple gas, with binary elastic
collisions, the statements of Theorem 1.b) reduce to known results on the
classical Boltzmann equation.
The proof of Theorem 1 will be given in several steps. We would like to
apply the Banach fixed point theorem to Eq. (\ref{12}) in $C(0,T;\dot C_0)$.
This is not, directly, possible since, $P$ and $S$ may be unbounded.
However, writing Eq. (\ref{12}) more conveniently, the smothering properties
of the time integrals appear to play a compensating role. \ The argument
uses the following key estimation, extending certain energetic inequalities,
obtained for the classical Boltzmann equation in \cite{be}. For some $\gamma
\in {\cal M}$, define
\begin{equation}
\label{13}\Phi _\gamma (t,{\bf w},{\bf x},{\bf v}):=\sum_{n\in {\cal N}%
(\gamma )}\sum_{i=1}^{\gamma _n}m_n\{[{\bf x}-t({\bf w}_{n,i}-{\bf v})]^2+%
{\bf w}_{n,i}^2\},
\end{equation}
for all ${\bf w\in {R}}^{3\mid \gamma \mid },\;{\bf x},{\bf v}\in {\bf R}^3$%
, $t>0$. Also set%
$$
\Gamma _{k\gamma }(t,{\bf v},{\bf x}):=\gamma _k\exp [\tau m_k({\bf v}^2+%
{\bf x}^2){\bf ]}
$$
\begin{equation}
\label{14}\times \int_{{\bf R}^{3\mid \gamma \mid -3}}d{\bf \tilde w}%
_{(k)}\int_0^tds\left[ (1+W_{r,\gamma }({\bf w})^{q/2})\exp (-\tau \Phi
_\gamma (s,{\bf w},{\bf x},{\bf v}))\right] _{{\bf w}_{k,\gamma _k}={\bf v}%
},
\end{equation}
for all ${\bf v}$, ${\bf x}\in {\bf R}^3$,$\ t>0$; $q\in \left[ 0,1\right] $.
{\bf LEMMA 1}.{\it a) Under conditions (\ref{15}),}
\begin{equation}
\label{16}\Phi _\beta (t,{\bf u},{\bf x},{\bf v})=\Phi _\alpha (t,{\bf w},%
{\bf x},{\bf v})+2(1+t^2)\sum_{n=1}^N(\alpha _n-\beta _n)E_n.
\end{equation}
{\it b) }$\max \limits_{\gamma \in {\cal M,}1\leq k\leq N}$ $\sup \{\Gamma
_{k\gamma }(t,{\bf v},{\bf x})\mid (t,{\bf v},{\bf x})\in {\bf R}_{+}\times
{\bf R}^3\times {\bf R}^3\}=const<\infty $.\
The proof is given in Appendix B.
Let $T,\tau >0$ and $f_0\in \dot C_\tau ^{+}$. With the substitution $%
g(t):=U^{-t}f(t)$, Eq.(\ref{12}) becomes,
\begin{equation}
\label{17}g(t)=f_0+\int_0^tP^{\#}(g)(s)ds-\int_0^tS^{\#}(g)(s)ds,\;0\le t\le
T,
\end{equation}
Here $P^{\#}$ and $S^{\#}$ are considered as operators in $C(0,T;\dot C_0)$,
defined on their natural domains {\em D}$(P^{\#})$ and {\em D}$(S^{\#})$,
respectively, by%
$$
P^{\#}(g)(t):=U^{-t}P(U^tg(t)),\quad \;S^{\#}(g)(t):=U^{-t}S(U^tg(t)).
$$
It follows that, we can prove Theorem 1, by looking for those $g\in
C(0,T;\dot C_0^{+})$ solving Eq.(\ref{17}) in $\dot C_0$. Since $U^t$ leaves
$C(0,T;\dot C_0^{+})$ invariant, we may equivalently look for those $%
g=(g_1,...,g_N)\in {\em D}(P^{\#})\cap {\em D}(S^{\#})\cap C(0,T;\;\dot
C_0^{+})$, solving the system
\begin{equation}
\label{18}g_k(t,{\bf v},{\bf x})=I_k(g)(t,{\bf v},{\bf x}),\;k=1,...,N,
\end{equation}
$(t,{\bf v},{\bf x})\in \left[ 0,T\right] \times {\bf R}^3\times {\bf R}^3$.
Here $I_k(g)\in C(\left[ 0,T\right] \times {\bf R}^3\times {\bf R}^3)$ is
given by%
$$
I_k(g)(t,{\bf v},{\bf x})=f_{k,0}(t,{\bf v},{\bf x})\exp
[-\int_0^tR_k^{\#}(g)(\lambda ,{\bf v},{\bf x})d\lambda ]\
$$
\begin{equation}
\label{19}+\int_0^t\exp [-\int_s^tR_k^{\#}(g)(\lambda ,{\bf v},{\bf x}%
)d\lambda ]P_k^{\#}(g)(s,{\bf v},{\bf x})ds,\,
\end{equation}
with $R_k^{\#}(g)(t,{\bf v},{\bf x}):=U_k^{-t}R_k(U^tg(t))({\bf v},{\bf x})$%
, $k=1,...,N$ (the integrals being in the classical sense). Obviously, the
system (\ref{18}) represents a weak form of Eq.(\ref{17}). Due to the
assumptions on $f_0$, it will appear that Eq.(\ref{18}) has solutions given
by elements of $C(0,T;\dot C_\tau ^{+})$. Let $I(g):=(I_1(g),...,I_N(g))$.
We show that $g\rightarrow I(g)$ fulfills the conditions for applying the
Banach fixed point theorem in ${\cal H}_\tau (R)$, for $R$ small enough.\
{\bf PROPOSITION 2.}a){\it \ If }$g\in C(0,T;\dot C_\tau ^{+})$, {\it then
also }$I(g)\in C(0,T;\;\dot C_\tau ^{+})$.
{\it b)} {\it For each} $T>0$, {\it there exist }$R_T,R_T^{*}>0$, {\it with }%
$R_T^{*}\rightarrow 0$, {\it as }$R_T\rightarrow 0$, {\it such that if }$%
\mid f_0\mid _\tau \le R_T$, {\it then} $g\rightarrow I(g)$ {\it leaves }$%
{\cal H}_\tau (R_T^{*})$ {\it invariant. Moreover, if} $K_{\beta ,\alpha
}\equiv 0,$ {\it whenever} $\sum_{n=1}^N(\alpha _n-\beta _n)E_n<0,$ {\it %
then there exist }$R,R^{*}>0$,{\it \ independent of} $T$, {\it with }$%
R^{*}\rightarrow 0$, {\it as $R$} {\it $\rightarrow 0$}, {\it \ such that if
}$\mid f_0\mid _\tau \le R$, {\it then } $g\rightarrow I(g)$ {\it leaves }$\;%
{\cal H}_\tau (R^{*})$ {\it invariant.}\
{\bf Proof. }a) First, remark that $C(0,T;\dot C_{k,\tau })$, $k=1,...,N$,
can be identified with the set of those $h\in C([0,T]\times {\bf R}^3\times
{\bf R}^3)$ (real) with the property
\begin{equation}
\label{20}\sup \limits_{\left| {\bf x}\right| +\left| {\bf v}\right| \geq
r}\{\exp [\tau m_k({\bf x}^2+{\bf v}^2)]\left| h(t,{\bf v},{\bf x})\right|
\}\rightarrow 0\quad as\;r\rightarrow \infty ,
\end{equation}
uniformly in $t\in [0,T]$. We verify (\ref{20}). If $g\in C(0,T;\dot C_\tau
^{+}),$ $\gamma \in {\cal M}$, denote%
$$
G_\gamma ^{\#}(t,{\bf w,\,x},{\bf v})=\Pi _{n\in {\cal N}(\gamma )}\Pi
_{i=1}^{\gamma _n}g_n(t,\,{\bf w}_{n,i},\;{\bf x}-t({\bf w}_{n,i}-{\bf v}%
))\exp \left[ \tau \Phi _\gamma (t,{\bf w},{\bf x},{\bf v})\right] .
$$
Using the definitions of $P^{\#}$ and $P$, as well as Rel.(\ref{10}) and
Lemma 1.a), we estimate (\ref{19}): since $R_k^{\#}(g)(t,{\bf v},{\bf x})\ge
0$, $1\le k\le N$, the exponents are negative in (\ref{19}) ; moreover, $%
P_k^{\#}(g)(t,{\bf v},{\bf x})\geq 0$; then, with the notations of Rel. (\ref
{8}), (\ref{8'}), for some constant $K>0$,
$$
0\le I_k(g)(t,{\bf v},{\bf x})
$$
$$
\leq f_{k,0}(t,{\bf v},{\bf x})+K\sum_{\alpha ,\beta \in {\cal M}}\alpha
_k\int_{{R}^{3\mid \alpha \mid -3}\times \Omega _{3\mid \beta \mid -4}}d{\bf %
\tilde w}_{(k)}\otimes d{\bf n\;}\left[ (1+W_{r,\alpha }({\bf w}%
)^{q/2})\right. \
$$
\begin{equation}
\label{21}\left. \times \int_0^tds\;\Lambda _{\beta \alpha }(s)\,G_\beta
^{\#}(s,{\bf u}_{\beta \alpha },{\bf x},{\bf v})\ \exp [-\tau \Phi _\alpha
(s,{\bf w},{\bf x},{\bf v})]\right] _{_{{\bf w}_{k,\alpha _k}={\bf v}}},
\end{equation}
for all $\;(t,{\bf v},{\bf x})\in \left[ 0,T\right] \times {\bf R}^3\times
{\bf R}^3$. Here,
\begin{equation}
\label{22}\Lambda _{\beta \alpha }(t)=\left\{
\begin{array}{l}
\exp \left[ -2\tau (1+t^2)\sum_{n=1}^N(\alpha _n-\beta _n)E_n\right]
,\;if\;K_{\beta ,\alpha }
\not\equiv 0,\medskip\ \\ 0,\;if\;K_{\beta ,\alpha
}\equiv 0.
\end{array}
\right.
\end{equation}
Since $g\in C(0,T;\,\dot C_\tau ^{+})$, by (\ref{20}), (\ref{9}), there
exists $r\ge 0$ such that $G_\beta ^{\#}(t,{\bf u,x},{\bf v})$ $\le \epsilon
\left\| g\right\| _\tau ^{\mid \beta \mid -1}$, provided that $\Phi _\beta
(t,{\bf u},{\bf v},{\bf x})\ge r$. Observe that $m_k({\bf x}^2+{\bf v}^2)$ $%
\leq \Phi $$_\alpha (t,{\bf w},{\bf v},{\bf x})\mid _{_{{\bf w}_{k,\alpha
_k}={\bf v}}}$. Consequently, by Lemma 1a), for each$\;\epsilon >0$, there
exists $r_0>0$ (possibly depending on $T$) such that if $({\bf x}^2+{\bf v}%
^2)\ge r_0$, then%
$$
0\le G_\beta ^{\#}(t,\,{\bf u}_{\beta \alpha },\,{\bf x},\,{\bf v})_{\mid _{%
{\bf w}_{k,\alpha _k}={\bf v}}}\le \epsilon \left\| g\right\| _\tau ^{\mid
\beta \mid -1},
$$
uniformly in the rest of variables. We introduce the last inequality in (\ref
{21}). There exist two constants $K_1>0$ and $r_1>0$, such that, if ${\bf x}%
^2+{\bf v}^2\ge r_1$, then%
$$
0\le I_k(g)(t,{\bf v},{\bf x})\le f_{k,0}({\bf v},{\bf x})\ +
$$
\begin{equation}
\label{24}+\epsilon \,K_2\,\Lambda (T)\exp [-\tau m_k({\bf x}^2+{\bf v}%
^2)]\sum_{\mid \alpha \mid \ge 2,\,,\mid \beta \mid \ge 2}\Gamma _{k\alpha
}(t,{\bf v},{\bf x})\left\| g\right\| _\tau ^{\mid \beta \mid -1},
\end{equation}
for all $\;(t,{\bf v},{\bf x})\in \left[ 0,T\right] \times {\bf R}^3\times
{\bf R}^3$. Here,
\begin{equation}
\label{23}\Lambda (T):=\sup \nolimits_{\alpha ,\beta \in {\cal M}}\;[\sup
\nolimits_{0\le t\le T}\Lambda _{\beta \alpha }(t)].
\end{equation}
Since $f_{k,0}$ is in $\dot C_{k,\tau }$, it is now sufficient to apply
Lemma 1.b) to obtain that (\ref{20}) is satisfied. This concludes the proof
of a).
b). Since $0\le G_\beta ^{\#}\le \left\| g\right\| _\tau ^{\mid \beta \mid }$%
, the same procedure as before implies%
$$
\mid I_k(g)(t)\mid _{k,\tau }\le \mid f_0\mid _\tau +K_2\Lambda
(T)\sum_{\mid \beta \mid \ge 2}\left\| g\right\| _\tau ^{\mid \beta \mid },
$$
for some constant $K_2>0$. Now the argument can be easily concluded. $\Box $%
\
Let $I$ denote the map $g\rightarrow I(g)$, according to Prop 2.b). Clearly,
Eq.(\ref{18}) can be formulated in $C(0,T;\dot C_\tau ^{+})$ as
\begin{equation}
\label{nou}g=I(g).
\end{equation}
\
{\bf PROPOSITION 3.} {\it For each} $T>0$ {\it there exist} $R_T,R_T^{*}>0$,
{\it with} $R_T^{*}\rightarrow 0$, {\it as }$R_T\rightarrow 0$, {\it such
that if } $\mid f_0\mid _\tau \le R_T$, {\it then} $I$ {\it is a strict
contraction on} ${\cal H}_\tau (R_T^{*})${\it . Assume that} $K_{\beta
,\alpha }\equiv 0,$ {\it whenever} $\sum_{n=1}^N(\alpha _n-\beta _n)E_n\le 0$
{\it . In this case, there exist} $R$, $R^{*}>0${\it , with $%
R^{*}\rightarrow 0$},{\it \ as $R\rightarrow 0$, independent of }$T$, {\it \
such that if} $\mid f_0\mid _\tau \le R$, {\it then} $I$ {\it is a strict
contraction on} ${\cal H}_\tau (R^{*})${\it .}\
{\bf Proof.} {\it -} By (\ref{19}) , for $g,h\in C(0,T;\dot C_\tau ^{+})$,
we can write%
$$
\mid I_k(g)(t,{\bf v},{\bf x})-I_k(h)(t,{\bf v},{\bf x})\mid
$$
\begin{equation}
\label{25}\le Q_k^A(g,h)(t,{\bf v},{\bf x})+Q_k^B(g,h)(t,{\bf v},{\bf x}%
)+Q_k^C(g,h)(t,{\bf v},{\bf x}),
\end{equation}
with%
$$
Q_k^A(t,{\bf v},{\bf x}):=f_{k,0}({\bf v},{\bf x})\mid \exp
[-\int_0^tR_k^{\#}(g)(\lambda ,{\bf v},{\bf x})d\lambda ]-\exp
[-\int_0^tR_k^{\#}(h)(\lambda ,{\bf v},{\bf x})d\lambda ]\mid ,
$$
$$
Q_k^B(t,{\bf v},{\bf x})
$$
$$
:=\int_0^tds\,P_k^{\#}(g)(s,{\bf v},{\bf x})\left| \exp
[-\int_s^tR_k^{\#}(g)(\lambda ,{\bf v},{\bf x})d\lambda ]\ -\exp
[-\int_s^tR_k^{\#}(h)(\lambda ,{\bf v},{\bf x})d\lambda ]\right| ,
$$
$$
Q_k^C(t,{\bf v},{\bf x}):=\int_0^t\exp [-\int_s^tR_k^{\#}(h)(\lambda ,{\bf v}%
,{\bf x})d\lambda ]\mid P_k^{\#}(g)(s,{\bf v},{\bf x})-P_k^{\#}(h)(s,{\bf v},%
{\bf x})\mid ds.
$$
for all $(t,{\bf v},{\bf x})\in \left[ 0,T\right] \times {\bf R}^3\times
{\bf R}^3$; $k=1,\ldots ,N$.
First we estimate $Q_k^A(t,{\bf v},{\bf x})$ . Since $g_k(t,{\bf v},{\bf x})$%
, $h_k(t,{\bf v},{\bf x})\ge 0$, then $R_k^{\#}(g)(t,{\bf v},{\bf x})\ge 0$
and $R_k^{\#}(h)(t,{\bf v},{\bf x})\ge 0$, hence we can write%
$$
Q_k^A(g,h)(t,{\bf v},{\bf x})\le f_{k,0}({\bf v},{\bf x})\int_0^t\mid
R_k^{\#}(g)(\lambda ,{\bf v},{\bf x})-R_k^{\#}(h)(\lambda ,{\bf v},{\bf x}%
)\mid d\lambda ,
$$
for all$\;(t,{\bf v},{\bf x})\in \left[ 0,T\right] \times {\bf R}^3\times
{\bf R}^3$.
Using the definitions of $R_k^{\#},\;$by arguments similar to those in the
proof of Prop 2, there exists a constant $C_1>0$ such that%
$$
Q_k^A(g,h)(t,{\bf v},{\bf x})
$$
\begin{equation}
\label{26}\leq C_1f_{k,0}({\bf v},{\bf x})\left\| g-h\right\| _\tau \left(
\sum_{\mid \alpha \mid \ge 2}\sum_{n=0}^{\mid \alpha \mid -2}\left\|
g\right\| _\tau ^{\mid \alpha \mid -n-2}\left\| h\right\| _\tau ^n\Gamma
_{k\alpha }(t,{\bf v},{\bf x})\right) .
\end{equation}
for all$\;(t,{\bf v},{\bf x})\in \left[ 0,T\right] \times {\bf R}^3\times
{\bf R}^3$. Since%
$$
Q_k^B(g,h)(t,{\bf v},{\bf x)}
$$
$$
\leq \int_0^tP_k^{\#}(g)(s,{\bf v},{\bf x})\,ds\left( \int_0^t\mid
(R_k^{\#}(g)(\lambda ,{\bf v},{\bf x})-R_k^{\#}(h)(\lambda ,{\bf v},{\bf x}%
)\mid d\lambda \right) ,
$$
similar estimations give (for some constant $C_2>0$)%
$$
Q_k^B(g,h)(t,{\bf v},{\bf x)}
$$
$$
\leq C_2\Lambda (T)\exp [-\tau m_k({\bf x}^2+{\bf v}^2)]\left\| g-h\right\|
_\tau \left( \sum_{\mid \beta \mid \ge 2}\left\| g\right\| _\tau ^{\mid
\beta \mid }\right)
$$
\begin{equation}
\label{27}\times \left( \sum_{\mid \alpha \mid \ge 2}\Gamma _{k\alpha }(t,%
{\bf v},{\bf x})\right) \left( \sum_{\mid \alpha \mid \ge 2}\sum_{n=0}^{\mid
\alpha \mid -2}\left\| g\right\| _\tau ^{\mid \alpha \mid -n-2}\left\|
h\right\| _\tau ^n\Gamma _{k\alpha }(t,{\bf v},{\bf x})\right) ,
\end{equation}
with $\Lambda (T)$ defined by (\ref{23}); $(t,{\bf v},{\bf x})\in \left[
0,T\right] \times {\bf R}^3\times {\bf R}^3$.
In the same way, for some constant $C_3>0,$we obtain%
$$
Q_k^C(g,h)(t,{\bf v},{\bf x})\ \le C_3\Lambda (T)\exp [-\tau m_k({\bf x}^2+%
{\bf v}^2)]
$$
\begin{equation}
\label{28}\times \left\| g-h\right\| _\tau \left( \sum_{\mid \alpha \mid \ge
2}\Gamma _{k\alpha }(t,{\bf v},{\bf x})\right) \left( \sum_{\mid \beta \mid
\ge 2}\sum_{n=1}^{\mid \beta \mid -1}\left\| g\right\| _\tau ^{\mid \beta
\mid -n-1}\left\| h\right\| _\tau ^n\right) ,
\end{equation}
$(t,{\bf v},{\bf x})\in \left[ 0,T\right] \times {\bf R}^3\times {\bf R}^3$.
By inequalities (\ref{26})-(\ref{28}), applying Lemma 1.b), one can find a
constant $C_0>0$ and a polynomial $p(\cdot )$ with positive coefficients,
such that $\forall \,r>0$,
\begin{equation}
\label{29}\left\| I(g)-I(h)\right\| _\tau \le C_0[\mid f_0\mid _\tau
+r\Lambda (T)]\,p(r)\left\| g-h\right\| _\tau ,
\end{equation}
provided that $g,h\in {\cal H}_\tau (r)$.
Now, by Prop 2 and Rel. (\ref{29}), we can choose $R_T,R_T^{*}>0$ such that
if $\mid f_0\mid _\tau \le R_T$ then $I$ is a strict contraction on ${\cal H}%
_\tau (R_T^{*})$. By (\ref{23}), if $K_{\beta ,\alpha }\equiv 0$, whenever $%
\sum_{n=1}^N(\alpha _n-\beta _n)E_n\le 0,$ then $\Lambda (T)=1$, $\forall
T>0.$ Consequently there exist $R,\;R^{*}>0$, independent of $T$, such that
if $\mid f_0\mid _\tau \le R$, then $I$ is a strict contraction on ${\cal H}%
_\tau (R^{*})$. This concludes the proof of Prop. 3.$\Box $
The existence and uniqueness part in Theorem 1a) follows by Prop.2, Prop.3
and the Banach fixed point theorem: for $R_T,R_T^{*}>0$, small enough, Eq.(%
\ref{nou}), with $\left| f_0\right| \leq R_T$, can be uniquely solved in $%
{\cal H}_\tau (R_T^{*})$. To conclude the argument it is sufficient to
remark that ${\cal H}_\tau (R_T^{*})\subset {\em D}(P^{\#})\cap {\em D}%
(S^{\#})\subset $ $C(0,T,\dot C_0)$. To prove the rest of Theorem 1a),
namely the continuity of the solution in the initial datum, first remark by
Prop.2, that, for each $g\in {\cal H}_\tau (R_T^{*})$, fixed, the map $%
f_0\rightarrow I(g)$ is continuous from $\left\{ h\in \dot C_\tau
^{+}:\,\left| h\right| _\tau \leq R_T\right\} $ to $C(0,T,\dot C_\tau )$.
Then the proof follows by means of the inequality (\ref{29}).
Part b) of Theorem 1 can be similarly proved. \
Part c) of Theorem 1 is immediate: $g\in C^1(0,T;\,\dot C_\tau )$, by Eq.(%
\ref{17}), while the solution $f$ of Eq.(\ref{12}) is related to $g$ by $%
f=U^tg$. $\;\Box $
\section{Conservation relations, H-Theorem and the law of the mass action}
In this section we prove the global conservations relations for mass,
momentum and energy and a H-Theorem analogous to the results obtained in the
case of the classical Boltzmann equation with elastic binary collisions.
Set $\Psi _k^0({\bf v})=m_k,\Psi _k^4({\bf v})=2^{-1}m_k{\bf v}^2+E_k$ and $%
\Psi _k^i({\bf v})=m_k{\bf v}_i$ for all ${\bf v}=({\bf v}_1,{\bf v}_2,{\bf v%
}_3)\in {\bf R}^3$, with ${\bf v}_i\in {\bf R},\;i=1,2,3,\;1\le k\le N$. The
following result states the bulk momentum and energy conservation
relations.\ \
{\bf THEOREM 2}.{\it a) Let }$\tau >0$ {\it and $f\in \dot C_\tau $. Then for%
} $i=0,1,\ldots ,4$,
$$
\sum_{k=1}^N\int_{{\bf R}^3}\Psi _k^i({\bf v})\left( P_k(f)({\bf v},{\bf x}%
)-S_k(f)({\bf v},{\bf x})\right) d{\bf v}\equiv 0.
$$
$\forall {\bf x\in R}^3$.
b) {\it If} $f_0$ {\it and} $f$ {\it are as in Theorem 1, then, for each }$%
t\in \left[ O,T\right] $,
$$
\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}\Psi _k^i({\bf v})f_k(t,{\bf v},%
{\bf x})\,d{\bf v\otimes }d{\bf x}\equiv \sum_{k=1}^N\int_{{\bf R}^3\times
{\bf R}^3}\Psi _k^i({\bf v})f_{k,0}({\bf v},{\bf x})\,d{\bf v\otimes }d{\bf %
x.}
$$
\
{\bf Proof.} - a) We give the argument for $\Psi _k^4$, the other cases
being similar. By Prop.1.b),%
$$
\sum_{k=1}^N\int_{{\bf R}^3}\Psi _k^4({\bf v})P_k(f)({\bf v},{\bf x})\,d{\bf %
v}=\lim \limits_{\eta \rightarrow 0}\lim \limits_{\epsilon \rightarrow
0}\sum_{\alpha ,\beta \in {\cal M}}\sum_{k=1}^N\sum_{i=1}^{\alpha _k}\int_{%
{\bf R}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid }}\ (2^{-1}m_k%
{\bf w}_{k,i}^2+E_k)
$$
$$
\times \ f_\beta ({\bf u},{\bf x})K_{\beta ,\alpha }({\bf u},{\bf w}%
)\,\delta _\epsilon ^3(V_\beta ({\bf u})-V_\alpha ({\bf w}))\ \,\delta _\eta
(W_\beta ({\bf u})-W_\alpha ({\bf w}))\ d{\bf u\otimes }d{\bf w\,}
$$
$$
=\lim \limits_{\eta \rightarrow 0}\lim \limits_{\epsilon \rightarrow
0}\sum_{\alpha ,\beta \in {\cal M}}\sum_{k=1}^N\sum_{i=1}^{\beta _k}\int_{%
{\bf R}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid }}(2^{-1}m_k%
{\bf u}_{k,i}^2+E_k)
$$
$$
\times \ \,f_\alpha ({\bf w},{\bf x})K_{\alpha ,\beta }({\bf w},{\bf u})\
\delta _\epsilon ^3(V_\beta ({\bf u})-V_\alpha ({\bf w}))\,\delta _\eta
(W_\beta ({\bf u})-W_\alpha ({\bf w}))\ d{\bf u\otimes }d{\bf w\,,}
$$
where the last equality results by interchanging $\alpha $ and ${\bf w}$
with $\beta $ and ${\bf u}$ respectively, and using the symmetry of $\delta
_\epsilon ^3$ and $\delta _\eta $, respectively as well as the invariance of
$K_{\beta ,\alpha }$ at permutations. Then it is sufficient to remark that%
$$
\sum_{k=1}^N\int_{{\bf R}^3}\Psi _k^4({\bf v})\left( P_k(f)({\bf v},{\bf x}%
)-S_k(f)({\bf v},{\bf x})\right) \,d{\bf v}
$$
$$
=\lim \limits_{\eta \rightarrow 0}\lim \limits_{\epsilon \rightarrow
0}\sum_{\alpha ,\beta \in {\cal M}}\int_{{\bf R}^{3\mid \beta \mid }\times
{\bf R}^{3\mid \alpha \mid }}\left( W_\beta ({\bf u})-W_\alpha ({\bf w}%
)\right)
$$
$$
\times \,f_\alpha ({\bf w},{\bf x})K_{\alpha ,\beta }({\bf w},{\bf u}%
)\,\delta _\epsilon ^3(V_\beta ({\bf u})-V_\alpha ({\bf w}))\ \delta _\eta
(W_\beta ({\bf u})-W_\alpha ({\bf w}_\alpha ))\ d{\bf u\otimes }d{\bf w\,}%
\equiv 0.
$$
b) Let $f$ be as in Theorem 1. Note that for each $t$ fixed, $U_k^t$,
introduced in Rel.(\ref{u}) is a positivity preserving, linear isometry on $%
L^1({\bf R}^3\times {\bf R}^3,\,d{\bf v\otimes }d{\bf x})$; $k=1,...,N.$
Further, by Theorem 1c), $g=U^{-t}f$ is of class $C^1$ and verifies Eq.(\ref
{17}). Then for each $i=1,...,4$,%
$$
{\frac d{dt}}\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}\Psi _k^i\,({\bf v}%
)f_k(t,{\bf v},{\bf x})\,d{\bf v\otimes }d{\bf x=}{\frac d{dt}}%
\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}\Psi _k^i({\bf v})\,g_k(t,{\bf v}%
,{\bf x})\,d{\bf v\otimes }d{\bf x}\
$$
$$
=\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}\Psi _k^i({\bf v})\left(
P_k^{\#}(g)(t,{\bf v},{\bf x})-S_k^{\#}(g)(t,{\bf v},{\bf x})\right) \,d{\bf %
v\otimes }d{\bf x}
$$
$$
=\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}\Psi _k^i({\bf v})\left(
P_k(f)(t,{\bf v},{\bf x})-S_k(f)(t,{\bf v},{\bf x})\right) \,d{\bf v\otimes }%
d{\bf x}\equiv 0,
$$
using again the $L^1$- properties of $U_k^t$ and Part a). This concludes the
proof.$\Box $\ \
Let $C_1,\ldots ,C_N>0$ be constants and $C^\alpha :=C^{\alpha _1}\times
\ldots \times C^{\alpha _N},\;\alpha \in {\cal M}$. In the rest of this
section, we suppose the following detailed balance condition
\begin{equation}
\label{33b}C^\beta K_{\alpha ,\beta }({\bf w,u})\equiv C^\alpha K_{\beta
,\alpha }({\bf u,w)},\quad \forall \;{\bf w},\;{\bf u,\;}\alpha ,\;\beta ,
\end{equation}
First remark that if $f=(f_1,..,f_N)\in \dot C_\tau ^{+}$, $\tau >0$, with $%
f_n>0$, then $f_n\log f_n\in L^1({\bf R}^3\times {\bf R}^3,d{\bf v\otimes }d%
{\bf x})$, for all $n=1,...,N$. The argument is standard (\cite{be}): let $%
\log {}^{+}$ ($\log {}^{-}$) denote the positive (negative) part of the
function $\log $;\ clearly $f_n\log {}^{+}f_n\in L^1$; it is sufficient to
prove the same for $f_n\log {}^{-}f_n$; to this end, in the inequality $\xi
\log {}^{-}\xi \leq \eta -\xi \log \eta $, valid for $\xi >0$ and $0<\eta
\leq 1$ (\cite{be}), we take $\xi =f_n({\bf v},{\bf x})$ and $\eta =\exp (-%
{\bf v}^2-{\bf x}^2)$, obtaining.
\begin{equation}
\label{tos}(f_n\log {}^{-}f_n)({\bf v},{\bf x})\leq \exp (-{\bf v}^2-{\bf x}%
^2)+({\bf v}^2+{\bf x}^2)f_n({\bf v},{\bf x}).
\end{equation}
Therefore we can define the H-function
\begin{equation}
\label{31}H(f)=\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}\,f_k({\bf v},%
{\bf x})\,[\log \left( C_kf_k({\bf v},{\bf x})\right) -1]\,d{\bf v\otimes }d%
{\bf x.}
\end{equation}
{\bf PROPOSITION} 4.{\it a)}\ {\it Let }$\tau >0$, $f=(f_1,..,f_N)\in \dot
C_\tau ^{+}$, {\it such that} $f_n>0$ {\it and} $\sup (1+{\bf x}^2+{\bf v}%
^2)^{-1}\left| \log f_n({\bf v},{\bf x})\right| <\infty $ {\it for all} $%
n=1,...,N$. {\it Then}
$$
D(f):=\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}\left( P_k(f)({\bf v},{\bf %
x})-S_k(f)({\bf v},{\bf x})\right) \log \left( C_kf_k({\bf v},{\bf x}%
)\,\right) d{\bf v\otimes }d{\bf x}\leq 0.
$$
{\it Moreover }$D(f)\equiv 0$ {\it iff for each} couple $(\alpha ,\beta )$
{\it such that} $K_{\alpha ,\beta }\not \equiv 0,$ {\it it follows that}
\begin{equation}
\label{34}C^\alpha f_\alpha ({\bf w},{\bf x})\equiv C^\beta f_\beta ({\bf u},%
{\bf x}),
\end{equation}
$\forall \,{\bf x}\in {\bf R}^3$, {\it provided that} ${\bf w,u}$ {\it %
satisfy (}\ref{15}{\it )}.
{\it b) Let $f_0$} {\it and $f$} be as in{\it \ Theorem 1. In addition,
suppose that } $f_{n,0}>0$ {\it and} $\sup (1+{\bf x}^2+{\bf v}%
^2)^{-1}\left| \log f_{n,0}({\bf v},{\bf x})\right| <\infty ${\it \ for all $%
n=1,...,N$. Then }$t\rightarrow H(f)$ {\it is of class }$C^1${\it and}%
$$
{\frac d{dt}}H(f)(t)=D(f(t)){\bf \,.}
$$
\
{\bf Proof.}a) In our case, $D(f)$ is well defined. Then, by Prop.1b),
taking $h=\log f_k$ and $f=(f_1,\ldots ,f_N)\in \dot C_\tau ^{+}$, with $%
f_n>0,$ for all $n=1,\ldots ,N$,
$$
\sum_{k=1}^N\int_{{\bf R}^3}P_k(f)({\bf v},{\bf x})\log \left( C_kf_k({\bf v}%
,{\bf x})\right) \,d{\bf v}=\lim \limits_{\eta \rightarrow 0}\lim
\limits_{\epsilon \rightarrow 0}\sum_{\alpha ,\beta \in {\cal M}}\int_{{\bf R%
}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid }}\,f_\beta ({\bf u},%
{\bf x})
$$
$$
\times \log \left( C^\alpha f_\alpha ({\bf w},{\bf x})\right) K_{\beta
,\alpha }({\bf u},{\bf w})\,\delta _\epsilon ^3(V_\beta ({\bf u})-V_\alpha (%
{\bf w}))\,\delta _\eta (W_\beta ({\bf u})-W_\alpha ({\bf w}))\,d{\bf %
u\otimes }d{\bf w\,}
$$
$$
=\lim \limits_{\eta \rightarrow 0}\lim \limits_{\epsilon \rightarrow
0}\sum_{\alpha ,\beta \in {\cal M}}\int_{{\bf R}^{3\mid \beta \mid }\times
{\bf R}^{3\mid \alpha \mid }}\,f_\alpha ({\bf w},{\bf x})\log \left( C^\beta
f_\beta ({\bf u},{\bf x})\right)
$$
\begin{equation}
\label{36}\times K_{\alpha ,\beta }({\bf w},{\bf u})\,\delta _\epsilon
^3(V_\beta ({\bf u})-V_\alpha ({\bf w}))\,\delta _\eta (W_\beta ({\bf u}%
)-W_\alpha ({\bf w}))d{\bf u\otimes }d{\bf w\,}
\end{equation}
where the last equality results by interchanging $\alpha $ and ${\bf w}$
with $\beta $ and ${\bf u}$ respectively, and using the symmetry of $\delta
_\epsilon ^3$ and $\delta _\eta $, respectively as well as the invariance of
$K_{\beta ,\alpha }$ at permutations
Similarly%
$$
\sum_{k=1}^N\int_{{\bf R}^3}S_k(f)({\bf v},{\bf x})\log \left( C_kf_k({\bf v}%
,{\bf x})\right) \,d{\bf v}=\lim _{\eta \rightarrow 0}\lim _{\epsilon
\rightarrow 0}\sum_{\alpha ,\beta \in {\cal M}}\int_{{\bf R}^{3\mid \beta
\mid }\times {\bf R}^{3\mid \alpha \mid }}\,f_\alpha ({\bf w},{\bf x})
$$
$$
\times \log \left( C^\alpha f_\alpha ({\bf w},{\bf x})\right) K_{\alpha
,\beta }({\bf w},{\bf u})\,\delta _\epsilon ^3\,(V_\beta ({\bf u})-V_\alpha (%
{\bf w}))\,\delta _\eta (W_\beta ({\bf u})-W_\alpha ({\bf w}))\,d{\bf %
u\otimes }d{\bf w\,}
$$
$$
=\lim \limits_{\eta \rightarrow 0}\lim \limits_{\epsilon \rightarrow
0}\sum_{\alpha ,\beta \in {\cal M}}\int_{{\bf R}^{3\mid \beta \mid }\times
{\bf R}^{3\mid \alpha \mid }}f_\beta ({\bf u},{\bf x})\log \left( C^\beta
f_\beta ({\bf u},{\bf x})\right)
$$
\begin{equation}
\label{38}\times K_{\beta ,\alpha }({\bf u},{\bf w})\,\,\delta _\epsilon
^3(V_\beta ({\bf u})-V_\alpha ({\bf w}))\,\delta _\eta (W_\beta ({\bf u}%
)-W_\alpha ({\bf w}))\,d{\bf u\otimes }d{\bf w\,}
\end{equation}
Then a few algebraic manipulations involving Rel. (\ref{36}), (\ref{38})
imply that%
$$
D(f)=2^{-1}\int_{{\bf R}^3}d{\bf x\,}\lim \limits_{\eta \rightarrow 0}\lim
\limits_{\epsilon \rightarrow 0}\sum_{\alpha ,\beta \in {\cal M}}\int_{{\bf R%
}^{3\mid \beta \mid }\times {\bf R}^{3\mid \alpha \mid }}H_{\alpha ,\beta }(%
{\bf w},{\bf u},{\bf x})
$$
\begin{equation}
\label{40}\times \delta _\epsilon ^3(V_\beta ({\bf u})-V_\alpha ({\bf w}%
))\,\delta _\eta (W_\beta ({\bf u})-W_\alpha ({\bf w}))\,d{\bf u\otimes }d%
{\bf w\,},
\end{equation}
with
$$
H_{\alpha ,\beta }({\bf w},{\bf u},{\bf x})=\left[ K_{\beta ,\alpha }({\bf u}%
,{\bf w})\,f_\beta ({\bf u},{\bf x})\,-K_{\alpha ,\beta }({\bf w},{\bf u}%
)\,f_\alpha ({\bf w},{\bf x})\right] \log {\frac{C^\alpha f_\alpha ({\bf w},%
{\bf x})}{C^\beta f_\beta ({\bf w},{\bf x})}.}
$$
Assuming that condition (\ref{33b}) is fulfilled, it follows that
$$
H_{\alpha ,\beta }({\bf w},{\bf u},{\bf x})=-K_{\beta ,\alpha }({\bf u},{\bf %
w})f_\beta ({\bf u},{\bf x})\left( {\frac{C^\alpha f_\alpha ({\bf w},{\bf x}%
) }{C^\beta f_\beta ({\bf w},{\bf x})}-1}\right) \log {\frac{C^\alpha
f_\alpha ({\bf w},{\bf x})}{C^\beta f_\beta ({\bf w},{\bf x})}}\le 0.
$$
hence $D(f)\le 0$. Rel.(\ref{34}) is now obvious.
b) Since $U_k^t$,$1\le k\le N$, is a positivity preserving isometry on $L^1(%
{\bf R}^3\times {\bf R}^3,\,d{\bf v\otimes }d{\bf x})$, using Rel.(\ref{31}%
), we obtain $H(f(t))=H(g(t))$. But $g(t)=U^{-t}f(t)$ is the unique solution
of Eq.(\ref{17}) and $g=U^{-t}f$ is of class $C^1$. From the definition of $%
I $ and the fact that $f_0$ satisfies the conditions of Prop.4 a) it follows
that $\forall \,t\in \left[ 0,T\right] $, $f(t)$ satisfies the conditions of
Prop.4 a), so that $D(f(t))$ is well defined. Moreover, using again the $L^1
$-properties of $U_k^t$,
$$
\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}d{\bf v\otimes }d{\bf x}\left(
P_k^{\#}(g)(t,{\bf v},{\bf x})-S_k^{\#}(g)(t,{\bf v},{\bf x})\right) \log
\left( C_kg_k(t,{\bf v},{\bf x})\right)
$$
$$
=\sum_{k=1}^N\int_{{\bf R}^3\times {\bf R}^3}d{\bf v\otimes }d{\bf x}\left(
P_k(f)(t,{\bf v},{\bf x})-S_k(f)(t,{\bf v},{\bf x})\right) \log C_kf_k(t,%
{\bf v},{\bf x}).
$$
Putting all these together and then using part a), it follows that $\frac
d{dt}H(f)(t)=\frac d{dt}H(g)(t)=$ $D(f)(t)$ $\leq 0$, concluding the proof.$%
\Box $
The main result of this section follows from Prop 4 using techniques similar
to those of \cite{be}.\ \
{\bf THEOREM 3.} {\it Let} $f_0=(f_{1,0},...,f_{N,0})$ and $%
\;f=(f_1,...,f_N) $ {\it be} {\it as in Theorem 1 with }$f_{n,0}>0$,$\;1\le
n\le N$. {\it Then under condition (}\ref{33b}),{\it \ $H(f(t_2))\leq
H(f(t_1))$} {\it for all} $t_2\geq t_1$.
{\bf Proof.} Define the sequence $%
f_0^{(l)}=(f_{1,0}^{(l)},...,f_{N,0}^{(l)}) $, $l=1,2,...$ by setting for
each $k=1,...,N$,%
$$
f_{k,0}^{(l)}({\bf v},{\bf x})=\max \left\{ f_{k,0}({\bf v},{\bf x}),\;\frac{%
\left| f_0\right| _\tau }{l(1+{\bf v}^2+{\bf x}^2)}\exp \left[ -\tau m_k(%
{\bf v}^2+{\bf x}^2)\right] \right\} .
$$
Let $f^{(l)}$and $f$ be the mild solutions of Eq.(\ref{11}), provided by
Theorem 1, for initial data $f_0^{(l)}$ and $f_0$, respectively. Obviously, $%
H(f(t))$ and $H(f^{(l)}(t))$ are well defined. Clearly $\left|
f_0^{(l)}-f_0\right| _\tau \rightarrow 0$, as $l\rightarrow \infty $. Then $%
\left\| f^{(l)}-f\right\| _0\rightarrow 0$, as $l\rightarrow \infty $, by
the continuity in initial datum, stated in Theorem 1. Moreover, for each $%
t\in \left[ 0,T\right] $, the sequence $\left( U^{-t}f^{(l)}(t)\right)
_{l\in {\bf N}}$ is bounded in $\dot C_\tau $. Using (\ref{tos}), it follows
that for each $t$, the sequence$(f^{(l)}(t)\log f^{(l)}(t))_{l\in {\bf N}}$
is bounded by some function in $L^1({\bf R}^3\times {\bf R}^3,\,d{\bf %
v\otimes }d{\bf x})$, hence $H(f^{(l)}(t))\rightarrow H(f(t))$, as $%
l\rightarrow \infty $, by the dominated convergence theorem. By (\ref{19}),
and the definition of $f_{k,0}^{(l)}$, the conditions of Prop.4b) are
fulfilled by $f_{k,0}^{(l)}$, for each $l$. Then the function $t\rightarrow
H(f^{(l)}(t))$ is non decreasing. Consequently, the same is true for $H(f)$,
concluding the proof. $\Box $.\
{\bf REMARK} Rel. (\ref{34}) is satisfied by local maxwellians (\cite{be})
and it provides a generalization of the law of the mass action.
Indeed, under condition{\it \ }(\ref{33b}), let the local maxwellians
solving Eq.(\ref{1}) be given by
\begin{equation}
\label{44}\omega _n=\omega _n(q_n,u,T):=q_n(m_n/2\pi kT)^{3/2}\exp [-m_n(%
{\bf v}-{\bf u})^2/2kT],
\end{equation}
for all $\;\;n=1,\ldots ,N$. Here $q_n=q_n(t,{\bf x})$ is the concentration
of species $n$, while ${\bf u}={\bf u}(t,{\bf x})$ and $T=T(t,{\bf x})$ are
the notations for the gas bulk velocity and the equilibrium temperature,
respectively ( $k$ denotes the Boltzmann constant).\ \
{\bf COROLLARY 1.}{\it \ The concentrations } $q_n$, $n=1,\ldots ,N$, {\it %
satisfy non trivially the law of the mass action, i.e., for all} $\alpha $, $%
\beta $,
\begin{equation}
\label{45}\sum_{n=1}^N(\alpha _n-\beta _n)[{\frac 32}\log (m_n/2kT)+\log
(C_nq_n)+E_n/kT]\equiv 0.
\end{equation}
\
{\bf Proof.} The result{\bf \ }is immediate from (\ref{34}), by the mass
conservation condition, namely $K_{\alpha ,\beta }=0$ when $\sum_n(\alpha
_n-\beta _n)m_n=0)$.$\Box $\
\section{Final remarks}
Theorem 1 is applicable to the reactive, expanding gas. Our global existence
results do not cover the case in which endo-energetic chemical reactions
(collisions) are present in the gas processes. In the latter case, one
should avoid possible pathologies, introduced by the particles which may
loose completely their relative kinetic energy during the endo-energetic
reactions.
The balance conditions (\ref{33b}) play no role in Theorems 1, 2, but it is
essential for the validity of the results stated in Theorem 3.
Theorem 1 can be analogously proved considering instead of $\dot C_0$ the
space of those $f=(f_1,...,f_N)$ with $f_n\in L^1({\bf R}^3\times {\bf R}^3;d%
{\bf v}\otimes d{\bf x)}$, equipped with the norm $\max _{1\leq k\leq
N}\left| f_k\right| _{L^1}$. In the latter case, the continuity condition on
$K_{\alpha ,\beta }$ can be replaced by measurability.\ Then $\dot C_\tau $
(with $\tau >0$) need be replaced by a space of measurable functions (e.g. $%
\dot C_{n,\tau }$ can be replaced by the space of those $h\in L^\infty ({\bf %
R}^3\times {\bf R}^3;d{\bf v}\otimes d{\bf x)}$, with $ess\sup \exp \left[
\tau ({\bf x}^2+m_k{\bf v}^2)\right] \left| h({\bf v},{\bf x})\right|
<\infty $ ).
Some of the results of this paper have been announced in \cite{gr}, (where
the main theorem is actually valid for $\tau =0$, since $\left\{ U^t\right\}
_{t\in {\bf R}}$ is not a continuous group of isometries on $\dot C_\tau $
for $\tau >0$).
\smallskip\
{\bf Acknowledgements:} One of us (C.P. Grunfeld) performed part of this
work at University Paris VII. He would like to thank the Laboratory of
Mathematical Physics and Geometry, particularly Prof. A. Boutet de Monvel
for the warm hospitality. It is a privilege to him to thank the French
C.N.R.S. for financial support.
\medskip\
{\Large {\bf Appendix A}}
\smallskip
Let $n$ be non-negative integer and $a_1,\ldots ,a_n>0$, constants. Consider
a positive quadratic form $T:=T({\bf v}_1,\ldots ,{\bf v}_n)=\sum_{i=1}^na_i%
{\bf v}_i^2$ on ${\bf R}^{3n}$ , $r_i\in {\bf R}^3,1\le i\le n$. Consider
the transformation%
$$
\qquad {\bf R}^{3n}\ni ({\bf v}_1,\ldots ,{\bf v}_n)\rightarrow (\underline{V%
},\zeta )\in {\bf R}^3\times {\bf R}^{3n-3}\;,\eqno (A.1)
$$
defined by%
$$
\underline{V}:=(\sum_{i=1}^na_i)^{-1}\sum_{i=1}^na_i{\bf v}_i,
$$
$$
\zeta :=(\zeta _1,\ldots ,\zeta _{n-1}),
$$
$$
\zeta _i:={\bf v}_{i+1}-(\sum_{j=1}^ia_j)^{-1}\sum_{j=1}^ia_j{\bf v}_j,\quad
\;i=1,\ldots ,n-1.
$$
By transformation (A.1), the form $T$ becomes%
$$
T=T(\underline{{\sl V}},\zeta )=(\sum_{i=1}^na_i)\underline{{\sl V}}%
^2+\sum_{i=1}^{n-1}\mu _i\zeta _i^2,
$$
with%
$$
\mu _i^{-1}=a_{i+1}^{-1}+(\sum_{j=1}^ia_j)^{-1},\qquad i=1,\ldots ,n-1.
$$
The system of coordinates on ${\bf R}^{3n}$ resulting from (A.1) will be
called a Jacobi system of coordinates associated to $T({\bf v}_1,\ldots ,%
{\bf v}_n)$. Moreover, the same term will designate the system of
coordinates obtained by the transformation%
$$
{\bf R}^{3n}\ni ({\bf v}_1,\ldots ,{\bf v}_n)\rightarrow (\underline{V},\xi
)\in {\bf R}^3\times {\bf R}^{3n-3},\eqno (A.2)
$$
where $\xi :=(\xi _1,\ldots ,\xi _{n-1})$ and $\xi _i:=\mu _i^{1/2}\zeta _i$
, with \underline{${\sl V}$} and $\zeta _i$ as in (A.1) ; $1\le i\le n-1$.
Obviously, by (A.2), $T=T(\underline{{\sl V}},\xi )=(\sum_{i=1}^na_i)
\underline{{\sl V}}^2+\xi ^2$. \
\medskip\
{\Large {\bf Appendix B}}
\smallskip
Part a) of Lemma 1 is straightforward. To prove b), we first consider $\mid
\gamma \mid =2$ (with $\gamma _k\ge 1$). Then, by (\ref{10}) clearly, there
exist two non-negative constants $c_0$ and $c$ such that%
$$
\Gamma _{k\gamma }(t,{\bf v},{\bf x})
$$
$$
\;\le c_0\int_{{\bf R}^3}d{\bf y}(1+\mid {\bf y}-{\bf v}\mid ^q)\exp (-c{\bf %
y}^2)\int_0^tds\exp \left\{ -c[{\bf x}-({\bf y}-{\bf v})s]^2\right\} \eqno %
(B.1)
$$
for all $t\ge 0,{\bf x},{\bf v}\in {\bf R}^3,q\in [0,1]$.
By (B.1), the $\sup $ estimation on $\Gamma $$_{k\gamma }(t,{\bf v},{\bf x})$
reduces to known inequalities of Lemma 2.5 in \cite{be}. Briefly, in this
case, for each $t\ge 0$,%
$$
\;\Gamma _0(t,{\bf v},{\bf x}):=\int_0^tds\exp \left\{ -c[{\bf x}-({\bf y}-%
{\bf v})s]^2\right\}
$$
$$
\leq \int_0^tds\exp \left\{ -c[\left| {\bf x}\right| -\left| {\bf y}-{\bf v}%
\right| s]^2\right\} \le \left| {\bf y}-{\bf v}\right| ^{-1}.\eqno (B.2)
$$
Introducing (B.2) in the right side of (B.1), and integrating with respect
to the reference frame with the{\bf \ }${\bf y}_3$ axis oriented in the
direction of ${\bf v}$, we get%
$$
\Gamma _{k\gamma }(t,{\bf v},{\bf x})\le \int_{-\infty }^{+\infty }d{\bf y}%
_3\exp (-c{\bf y}_3^2)\int_0^\infty \rho {\frac{1+[\rho ^2+({\bf y}_3-{\bf v}%
)^2]^{q/2}}{[\rho ^2+({{\bf y}_3-{\bf v}})^2]^{1/2}}}\exp [-c\rho ^2]\,d\rho
$$
$$
\le const.\int_{-\infty }^{+\infty }d{\bf y}_3\exp (-c{\bf y}%
_3^2)\int_0^\infty (1+\rho ^q)\exp [-c\rho ^2]d\rho \le const.
$$
The case $\mid \gamma \mid >2$ (with $\gamma _k\ge 1$) can be reduced to $%
\mid \gamma \mid =2$ as follows. With the notations of (\ref{14}), consider
the form $T_\gamma ({\bf \tilde w}_{(k)}):=W_\gamma ({\bf w}%
)-\sum_{n=1}^N\gamma _nE_n-2^{-1}m_k{\bf w}_{k,\gamma _k}^2$, representing
the kinetic energy of $\left| \gamma \right| -1$ particles in the channel $%
\gamma $ (more precisely, the kinetic energy of all the particles in channel
$\gamma $, except the particle with velocity ${\bf w}_{k,\gamma _k}$). To $%
T_\gamma ({\bf \tilde w}_{(k)})$, we associate a Jacobi system of
coordinates ${\bf R}^{3\mid \gamma \mid -3}\ni {\bf \tilde w}_{(k)}{\bf %
\rightarrow }(\underline{V},\xi )\in {\bf R}^3\times {\bf R}^{3\mid \gamma
\mid -6}$, of type (A.2) in Appendix A. Then, in the new variables, $%
m^{-1}(\sum_{n\in {\cal N}(\gamma )}\sum_{i=1}^{\gamma _n}m_n{\bf w}%
_{n,i}-m_k{\bf w}_{k,\gamma _k})=\underline{V}$, and $T_\gamma ({\bf \tilde w%
}_{(k)})=\frac m2\underline{V}^2+\xi ^2$ with $m:=\sum_{n=1}^N\gamma
_nm_n-m_k$. A few simple manipulations show that on $\{{\bf w\in {R}}^{3\mid
\gamma \mid }\mid {\bf w}_{k,\gamma _k}={\bf v}\}$, both $\Phi _\gamma $,
given by (\ref{13}), and the relative energy $W_{r,\gamma }$ of {\it all}
the particles in channel $\gamma $ (see Section 2), can be written in terms
of $(\underline{V},\xi )\in {\bf R}^3\times {\bf R}^{3(\mid \gamma \mid -2)}$
as
$$
\;\Phi _\gamma (t,{\bf w},{\bf x},{\bf v})_{\mid {\bf w}_{k,\gamma _k}={\bf v%
}}=m_k({\bf x}^2+{\bf v}^2)+m\underline{V}^2
$$
$$
+2(1+t^2)\xi ^2+m[{\bf x}-(\underline{V}-{\bf v})t]^2,\eqno (B.3)
$$
and%
$$
W_{r,\gamma }({\bf w})_{\mid {\bf w}_{k,\gamma _k}={\bf v}}=\xi
^2+2^{-1}(m_k^{-1}+m^{-1})^{-1}(\underline{V}-{\bf v})^2.\eqno (B.4)
$$
Then we choose $(\underline{V},\xi )$ as new integration variables for the
integral upon $d{\bf \tilde w}_{(k)}$ in (\ref{14}) and we introduce (B.3)
and (B.4) in (\ref{14}). One obtains that there exist two constants $c_0$
and $c$ such that%
$$
\;\quad \Gamma _{k,\gamma }(t,{\bf v},{\bf x})\le c_0\int_{{\bf R}^3}d
\underline{V}\exp (-c\underline{V}^2)\int_{{\bf R}^{3(\mid \gamma \mid
-2)}}d\xi \exp (-c\xi ^2)
$$
$$
\times \left\{ 1+[\xi ^2+(\underline{V}-{\bf v})^2]^{q/2}\right\}
\int_0^tds\exp \left[ -c[x-(\underline{V}-{\bf v})s]^2\right] ,\eqno (B.5)
$$
$t\ge 0,{\bf v},{\bf x}\in {\bf R}^3,q\in [0,1].$
Since for some constant $c_1>0$,%
$$
\quad 1+[\xi ^2+(\underline{V}-{\bf v})^2]^{q/2}\le c_1.\,(1+\mid \xi \mid
^q)\,(1+\mid \underline{V}-{\bf v}\mid ^q),\eqno (B.6)
$$
we introduce (B.6) in (B.5 ) and integrating with respect to $\xi $ we obtain%
$$
\Gamma _{k,\gamma }(t,{\bf v},{\bf x})\le c_2\int_{{\bf R}^3}d\underline{V}%
(1+\mid \underline{V}-{\bf v}\mid ^q)\exp (-c\underline{V}^2)\int_0^tds\exp
\{-c[{\bf x}-(\underline{V}-{\bf v})s]^2\}.
$$
with $c,c_2>0$, constants. This is exactly (B.1), concluding the proof. $%
\Box $.\
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\end{document}
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