\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 163, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/163\hfil The Goursat problem]
{Goursat problem for the Yang-Mills-Vlasov system
in temporal gauge}

\author[M. Dossa, M. Nanga \hfil EJDE-2011/163\hfilneg]
{Marcel Dossa, Marcel Nanga}  % in alphabetical order

\address{Marcel Dossa \newline
Universit\'e de Yaound\'e I, Facult\'e des Sciences, 
B.P. 812, Yaound\'e, Cameroun}
\email{marceldossa@yahoo.fr}

\address{Marcel Nanga \newline
Universit\'e de N'Djam\'ena, Facult\'e des Sciences Exactes
et Appliqu\'ees \newline
 B.P. 1027,  N'Djam\'ena, Tchad}
\email{marnanga@yahoo.fr}

\thanks{Submitted July 16, 2010. Published December 13, 2011.}
\subjclass[2000]{81Q13, 35L45,35L55}
\keywords{Goursat problem; characteristic initial
hypersurfaces; \hfill\break\indent
Yang-Mills-Vlasov system; temporal gauge; initial
constraints; evolution problem}

\begin{abstract}
 This article studies the characteristic Cauchy
 problem for the Yang-Mills-Vlasov (YMV) system in temporal gauge,
 where the initial data are specified on two intersecting
 smooth characteristic hypersurfaces of Minkowski spacetime
 $(\mathbb{R}^{4},\eta )$.
 Under a $\mathcal{C}^{\infty }$ hypothesis on the data,
 we solve the initial constraint problem and the evolution problem.
 Local in time existence and uniqueness results are established
 thanks to a suitable combination of the method of characteristics,
 Leray's Theory of hyperbolic systems and techniques
 developed by  Choquet-Bruhat for ordinary spatial Cauchy
 problems related to (YMV) systems.
\end{abstract}

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


%\newcommand{\va }{\varepsilon}

\section{Introduction}

The purpose of this article is to solve, locally in time in the
Minkowski spacetime $(\mathbb{R}^{4},\eta )$, the Cauchy problem
for the Yang-Mills-Vlasov (YMV) system, where the initial
data are prescribed on two intersecting smooth characteristic
hypersurfaces. Such problems are often referred to as non-linear
Goursat problems \cite{2,3,4,8,9,10,12,13,14,21,22,23,26,27,29,30}.

The Maxwell-Vlasov system and its generalization to the non
Abelian charge provided by the (YMV) system play a fundamental role
in numerous physical situations; possibly coupled with Einstein
Equations, they govern the dynamic of various species of plasma in
the absence of collisions. A plasma is a collection of charged
particles of various species moving at high speed under forces which
they have generated and maintained. The statistical distribution of
these particles is described by a density function of particles
subjected to the Vlasov equation. Here the particles have a non
Abelian charge called a Yang-Mills charge and are submitted to
forces generated by a Yang-Mills field which is solution of
Yang-Mills Equations with a current generated by the density of
particles. According to physicists, the matter which is lying in the
Universe is almost made up with plasmas such as the interior of a
star, reactor in fusion, ionosphere, solar winds, nebulous galaxies,
the plasmas of quarks and gluons of the primordial Universe.

Maxwell-Vlasov or Yang-Mills-Vlasov systems, possibly coupled with
Einstein Equations, are mostly studied by authors in the
setting of ordinary spatial Cauchy problems
\cite{1,7,15,16,17,18,19,20,24}. It is however important
and useful to consider the study of these same equations in the
setting of a Goursat problem; i.e.,  where the initial
hypersurface is a characteristic cone \cite{5}, or the union of two
intersecting smooth characteristic hypersurfaces. Indeed initial
data on a characteristic cone correspond to a ``physical'' Cauchy
problem more natural than the ordinary spatial Cauchy problem,
because they provide an ideal mathematical representation of the
measure at the present moment of the physical field studied.
Goursat problems arise also naturally in delicate physical
situations where radiation phenomena appear.
In this latter case the solutions considered must be global
or semi-global since they must be defined
in a neighborhood of null infinity.

The Goursat problem for the (YMV) system splits into two
sub-problems: the initial constraint problem and the evolution problem.
For a suitable choice of free data, we  solve globally the
initial constraint problem thanks to a hierarchy of
algebraic-integral-differential relations deduced from the (YMV)
system. Thereafter the solutions so obtained are used as initial
data for the evolution problem. The fundamental partial differential
equations (PDE) for the evolution problem consist of the Vlasov
equation (verified by the density of particles $f$) and a hyperbolic
symmetric system of first order extracted from the Yang-Mills
equations and the related Bianchi identities.

Thanks to domain of dependence arguments, we transform this problem
into an ordinary Cauchy problem with zero data on a spatial
hypersurface, which we solve using a suitable combination of the
classical method of characteristics, Leray's Theory of hyperbolic
systems \cite{25} and techniques  developed in \cite{7}
for the ordinary spatial Cauchy problems associated with (YMV)
systems.

For sake of clarity and simplicity, the Goursat problem considered
here for the (YMV) system in temporal gauge is studied under
$\mathcal{C}^{\infty }$ assumptions on the data. The study of solutions
of finite differentiability class would normally require the use of
a functional setting of non isotropic weighted Sobolev spaces
defined by cumbersome norms, which would considerably complicate the
analysis of this problem.

The work is subdivided in four sections. In section 2, we set the
geometrical framework and describe the PDE under consideration.
Section 3 is devoted to the resolution of the initial constraint problem. Section 4 is devoted to the determination of the
restrictions, to both initial characteristic hypersurfaces, of the
derivatives of all order of the possible $\mathcal{C}^{\infty }$
solution of the evolution problem; this is an important step towards
the transformation of the problem under consideration into an
ordinary Cauchy problem with zero initial data. The concern of
section 5 is the resolution of the evolution problem.

\section{Geometric setting, equations and mathematical formulation}

\subsection{Geometric setting and the unknown functions}

Throughout this article, we use the Einstein summation convention of
repeated indices, e. g.,
\begin{equation*}
a_{\alpha }b^{\alpha }=\sum_{\alpha }a_{\alpha }b^{\alpha
};\,\;\;\;\alpha =0,1,2,3.
\end{equation*}
Unless otherwise is specified, Greek indexes range from $0$ to $3$
and Latin ones from $1$ to $3$.

The fundamental geometric setting of this work is the
Minkowski spacetime $(\mathbb{R}^{4},\eta )$,
where the Minkowski metric $\eta $ is of
signature $( +,-,-,-)$. Let
$(x^{\alpha })=(x^0,x^i)$,
the global canonical coordinates system on $\mathbb{R}^{4} $,
where $x^0=t$ is the time coordinate and the $x^i$ are
the spatial coordinates. Let $\sqcup $ be a compact domain of
$\mathbb{R}^{4}$
with boundary $\partial \sqcup $ which is contained in the
half-space $x^0\geq 0 $. Let $H_1$ and $H_2$ be two hypersurfaces
defined as follows
\begin{gather*}
H_1=\{(x^0,x^1,x^2,x^3)\in \sqcup,\, x^0+x^1=0,\,x^0\geq 0\},
\\
H_2=\{(x^0,x^1,x^2,x^3)\in \sqcup
,\,x^0-x^1=0,\,x^0\geq 0\}.
\end{gather*}
Set $H=H_1\cup H_2$ and
\begin{equation*}
I=H_1\cap H_2=\{(x^0,x^1,x^2,x^3)\in \sqcup
,x^0=x^1=0\}.
\end{equation*}
Denote by $B$ the unique compact domain of $\mathbb{R}^2$ such
that
\begin{equation*}
I=\{0,0,x^2,x^3)\in \mathbb{R}^{4},\,(x^2,x^3)
\in B\}.
\end{equation*}
It is assumed that $H\subset \partial \sqcup $ and that $\partial
\sqcup \setminus H$ is a hypersurface of $\sqcup $, piecewise
smooth, spatial or null at each of its points, with unit normal
exterior to the domain $\sqcup $ which is future oriented. For every
$t\geq 0$, set
\begin{gather*}
\sqcup_{t} =\{(x^0,x^1,x^2,x^3)\in \sqcup : x^0\leq t\}; \\
\omega_{t} =\sqcup \cap \{x^0=t\};\quad
I_{t}^{r}=H_r\cap \{x^0=t\},\; r=1,2.
\end{gather*}
Denote by $A$ a Yang-Mills potential represented by a {1}-form on
$\sqcup $ which takes its values in an $N-$dimensional real Lie
algebra $\mathbf{K}$ of a Lie group $G$, endowed with an Ad-
invariant scalar product denoted by a dot (.).

In the global canonical coordinates
$(x^{\alpha })$ of $\mathbb{R}^{4}$ and an orthonormal basis
$(\varepsilon_{a})$ of $\mathbf{K}$, $A$ reads:
\[
A=A_{\alpha }dx^{\alpha },\text{ with }A_{\alpha }=A_{\alpha
}^{a}\cdot \varepsilon_{a},\ a=1,2,\dots ,N.
\]
We say that $A$ verifies the temporal gauge condition if
$A_0=0$ in $\sqcup $.
Denote by $F$ the Yang-Mills field associated to $A$, i. e., the
curvature of $A$. It is represented by a $\mathbf{K}$-valued
antisymmetric $2$-form of type $Ad$, defined on $\sqcup $ by
\begin{equation}
F=dA+\frac{1}{2}[A,A].  \label{eq0}
\end{equation}
The $2-$ form $F$ in $\sqcup $ is of type Ad, which means that,
if $F_{(i)}$ and $F_{(j)}$
are respectively the representatives in gauges $s_i$ and $s_j$ of
the $2-$ form $F$, then the relations between these two
representatives is $F_{(i)} = Ad(u^{-1}_{ij})F_{(j)}$ where
$u^{-1}_{ij}$ is roughly the transition function between the
two gauges $s_i$ and $s_j$.

In a global canonical coordinates $(x^{\alpha })$ and in the basis
$(\varepsilon_{a})$, \eqref{eq0} reads
\begin{equation}
F_{\alpha \beta }^{a}=\partial_{\alpha }A_{\beta }^{a}-\partial
_{\beta }A_{\alpha }^{a}+[A_{\alpha },A_{\beta }]^{a},
\label{eq1}
\end{equation}
with
\begin{equation*}
F=\frac{1}{2}F_{\alpha \beta }dx^{\alpha }\wedge dx^{\beta },\quad
[A_{\alpha },A_{\beta }]^{a}\equiv C_{bc}^{a}A_{\alpha
}^{b}A_{\beta }^{c}.
\end{equation*}
where the $C_{bc}^{a}$ are the structure constants of the Lie
group $G$ and $[,]$ denotes the Lie brackets of the Lie algebra
$\mathbf{K}$.

The {2}-form $F$ verifies Bianchi identities
\begin{equation}
\widehat{\nabla }_{\alpha }F_{\beta \mu }+\widehat{\nabla }_{\beta
}F_{\mu \alpha }+\widehat{\nabla }_{\mu }F_{\alpha \beta }\equiv 0,
\label{eq2}
\end{equation}
and the relation
\begin{equation*}
\widehat{\nabla }_{\alpha }\widehat{\nabla }_{\beta }F^{\alpha \beta
}\equiv 0,
\end{equation*}
where $\widehat{\nabla }_{\alpha }$ is the gauge covariant
derivative defined by
\begin{equation*}
\widehat{\nabla }_{\alpha }=\partial_{\alpha }+[A_{\alpha },]
\quad \text{with }\partial_{\alpha }=\frac{\partial }{\partial
{x}^{\alpha }}.
\end{equation*}
The trajectories of a particle with momentum $p$ and charge $q$ in a
Yang-Mills field $F$ defined in $(\sqcup ,\eta )$ verify the
differential system
\begin{equation}
\begin{gathered}
\frac{dx^{\alpha }}{ds}=p^{\alpha } \\
\frac{dp^{\alpha }}{ds}=p^{\beta }q.F_{\beta }^{\alpha } \\
\frac{dq^{a}}{ds}=-p^{\alpha }[A_{\alpha },q]^{a}.
\end{gathered}  \label{eq3}
\end{equation}
This system expresses the fact that tangent vector $Y$ to the
trajectory of a particle in $\mathbf{P}=T\sqcup \times K$ is
$Y=(p,P,Q)$, with
\begin{equation*}
p=(p^{\alpha }),\quad
P=(P^{\alpha })\equiv (p^{\beta }q.F_{\beta }^{\alpha }),\quad
Q=(Q^{a})\equiv (-p^{\alpha }C_{bc}^{a}q^{c}A_{\alpha }^{b}).
\end{equation*}
If the particles have the same rest mass $m$ and a charge $q$
of given size $e$, then their phase space, that is the domain
described by their trajectories is a subset $\mathbf{P}_{m,e}$
of $T\sqcup \times K$ with equations
\begin{equation}
p^0=\Big(m^2+\sum_{i=1}^3(p^i)^2\Big)^{1/2},\quad
q.q=e. \label{eq4}
\end{equation}
Coordinates system on $\mathbf{P}_{m,e}$ is given by
\begin{equation*}
x^0\equiv t,x^i,p^i,q^L,\quad  i=1,2,3;\; L=1,2,\dots ,N-1.
\end{equation*}
$\mathbf{P}_{m,e}$ can then be identified with $\sqcup \times
\mathbb{R}^3\times O\equiv \widehat{\sqcup }$.

Set
\begin{equation*}
P_{x}=\{x\}\times \mathbb{R}^3\times O\simeq \mathbb{R}^3\times
O.
\end{equation*}
Let $f$ be a distribution (or density) function for charged
particles, that is a positive scalar function defined on
$\mathbf{P}_{m,e}$. $f$ satisfies the Vlasov equation if in the
coordinates $(x^0,x^i,p^i,q^L)$ considered on
$\mathbf{P}_{m,e}$, it holds that
\begin{equation}
p^{\alpha }\frac{\partial f}{\partial x^{\alpha }}+P^i\frac{\partial f}{
\partial p^i}+Q^L\frac{\partial f}{\partial q^L}=0,\quad
L=1,2,\dots ,N-1.
\label{eq5}
\end{equation}
The current generated by the distribution function $f$ of particles
is represented by a $\mathbf{K}$- valued vector field $J$,
which is of type $Ad$ by gauge transformation, defined at a point
$x\in \sqcup $ by
\begin{equation}
J^{\beta }(x)=\int_{\mathbf{P}_{x}}p^{\beta }qf\omega_{p}\omega
_{q}, \label{eq6}
\end{equation}
where ${\omega_{p}=\frac{1}{p_0}dp^1dp^2dp^3}$ is the Leray
form induced by the volume element \\ $dp^0dp^1dp^2dp^3$ on
$T_{x}\sqcup $ and $\omega_{q}$ is the Leray form induced on $O$ by
an Ad-invariant volume element on $\mathbf{K}$.

\begin{remark} \label{rmk2.1} \rm
As the Lie algebra $\mathbf{K}$ has an Ad-invariant scalar
product, if $f$ satisfies the Vlasov equation in $\sqcup$, then we
have
\begin{equation*}
\widehat{\nabla }_{\alpha }J^\alpha=0\quad \text{in }\sqcup.
\end{equation*}
\end{remark}

\subsection{Definition of Yang-Mills-Vlasov system}

The Yang-Mills equations read
\begin{equation*}
\widehat{\nabla }_{\alpha }F^{\alpha \beta }=J^{\beta }\quad
 \text{in }\sqcup ,
\end{equation*}
where
\begin{equation*}
\widehat{\nabla }_{\alpha }F^{\alpha \beta }\equiv \partial_{\alpha
}F^{\alpha \beta }+[A_{\alpha },F^{\alpha \beta }].
\end{equation*}
By definition, the ``complete Yang-Mills-Vlasov
system'' is the following system defined in
$\widehat{\sqcup }$ and with unknown function $(A,F,f)$,
\begin{equation}\label{YMVsyst}
\begin{gathered}
\widehat{\nabla }_{\alpha }F^{\alpha \beta }=J^{\beta } \\
\widehat{\nabla }_{\alpha }F_{\beta \mu }
 +\widehat{\nabla }_{\beta }F_{\mu
\alpha }+\widehat{\nabla }_{\mu }F_{\alpha \beta }=0 \\
p^{\alpha }\frac{\partial f}{\partial x^{\alpha }}
 +P^i\frac{\partial f}{
\partial p^i}+Q^L\frac{\partial f}{\partial q^L}=0
\end{gathered}
\end{equation}
By definition, the reduced system, in temporal gauge, extracted from
\eqref{YMVsyst}
is the system of unknown function $(A,F,f)$, defined in
$\widehat{\sqcup }$ by
\begin{equation}\label{reducedsyst}
\begin{gathered}
A_0=0 \\
\widehat{\nabla }_{\alpha }F^{\alpha i}=J^i \\
\widehat{\nabla }_0F_{ij}+\widehat{\nabla }_iF_{j0}+\widehat{\nabla }
_jF_{0i}=0 \\
p^{\alpha }\frac{\partial f}{\partial x^{\alpha }}+P^i\frac{\partial f}{
\partial p^i}+Q^L\frac{\partial f}{\partial q^L}=0 \\
\partial_0A_i=F_{0i},
\end{gathered}
\end{equation}
where
$i,j=1,2,3$; $\alpha =0,1,2,3$; $L=1,2,\dots ,N-1$.

The evolution problem for the (YMV) system in temporal gauge with
initial data prescribed on the two intersecting smooth
characteristic hypersurfaces $H_1$ and $H_2$ consists in
solving the reduced system
\eqref{reducedsyst} under the initial conditions:
\begin{equation}\label{initialcond}
A_i\big|_H=a_i,\quad F^{0i}\big|_H=b^i, \quad
F_{ij}\big|_H=\Phi_{ij}, \quad
f\big|_{\widehat{H}}=\varphi
\end{equation}
where $\widehat{H}=H\times \mathbb{R}^3\times O$.

The initial constraint problem consists in studying how to
generally prescribe the initial data of the conditions
\eqref{initialcond} such that the unique solution of the evolution problem \eqref{reducedsyst}, \eqref{initialcond} is also solution
of the complete system \eqref{YMVsyst} of the Yang-Mills-Vlasov
Equations (and satisfies the temporal gauge condition).

\section{The initial constraint problem}

The following useful notation will be needed. For every function
(or tensor field) $v$ defined in the domain $\sqcup $, we denote by
$[ v]_r$ the restriction to $H_r$ of $v$, $r=1,2$,
and $[v] $ the restriction to $H$ of $v$; i.e,
\begin{gather*}
[ v]_1( x^1,x^2,x^3) = v(-x^1,x^1,x^2,x^3) \quad\text{on }H_1,\\
[ v]_2( x^1,x^2,x^3) = v(x^1,x^1,x^2,x^3) \quad\text{on }H_2,\\
[ v] ( x^1,x^2,x^3 ) = v (|x^1|, x^1,x^2,x^3)\quad\text{on }H.
\end{gather*}

\begin{theorem}\label{theo1.1}
Consider $V=(A_0\equiv 0,A_i,F^{0i},F_{ij},f)$ a
$\mathcal{C}^{\infty }$ solution, defined in a neighborhood
$\widetilde{\sqcup }$ of $\widehat{H}$ in $\widehat{\sqcup }$,
of the reduced system \eqref{reducedsyst} defined above.

(1) Set
\begin{equation}
\begin{gathered}
A_i\big|_H =a_i=\begin{cases}
\overline{a}_i&\text{on }H_1 \\
\widetilde{a}_i&\text{on }H_2;
\end{cases}\quad
F^{0i}\big|_H=b^i=\begin{cases}
\overline{b}^i&\text{on }H_1 \\
\widetilde{b}^i&\text{on }H_2;
\end{cases} \\
F_{ij}\big|_H=\Phi_{ij}=\begin{cases}
\overline{\Phi }_{ij}&\text{on }H_1 \\
\widetilde{\Phi }_{ij}&\text{on }H_2;
\end{cases}\quad
f\big|_{\widehat{H}} =\varphi =\begin{cases}
\overline{\varphi }&\text{on}\widehat{H}_1 \\
\widetilde{\varphi }&\text{on }\widehat{H}_2,
\end{cases}
\end{gathered}  \label{eq12}
\end{equation}
where $i,j=1,2,3$. Then the functions
$\overline{a}_i,\overline{b}^i$ and $\overline{\Phi }_{ij}$
(resp. $\widetilde{a}_i,\widetilde{b}^i$ and
$\widetilde{\Phi }_{ij}$) are $\mathcal{C}^{\infty }$ on $H_1$
(resp. $H_2$ ) and $\overline{\varphi }$
(resp. $\widetilde{\varphi }$) is  $\mathcal{C}^{\infty }$ on
$\widehat{H}_1$ (resp. $\widehat{H}_2$). Furthermore,
these functions satisfy the following compatibility conditions:
\begin{equation}
\begin{gathered}
\overline{a}_i=\widetilde{a}_i,\overline{b}^i=\widetilde{b}^i
\quad\text{on } H_1\cap H_2,\; i=1,2,3  \\
\overline{\varphi }=\widetilde{\varphi }\quad
\text{on }\widehat{H}_1\cap \widehat{H}_2\\
\partial_1\overline{a}_i-\partial_1\widetilde{a}_i=2\overline{b}
^i=2\widetilde{b}^i\quad \text{on }H_1\cap H_2,\;i=1,2,3
\end{gathered}  \label{eq13}
\end{equation}
The functions $a_i$, $b^i$ and $\varphi $ are continuous on
$\widehat{H}= \widehat{H}_1\cup \widehat{H}_2$, and the functions
$\Phi_{ij}$ verify $\Phi_{ij}=-\Phi_{ji}$. The functions
$\Phi_{ij},a_i,b^i$ are linked by the  algebraic
differential relations
\begin{equation}
\begin{gathered}
\Phi_{1j}=\begin{cases}
\overline{\Phi }_{1j}=\partial_1\overline{a}_j-\partial_j\overline{a}
_1+[\overline{a}_1,\overline{a}_j]-\overline{b}^{j}\quad\text{on }
H_1,\,j=2,3; \\
\widetilde{\Phi }_{1j}=\partial_1\widetilde{a}_j-\partial_j
\widetilde{a}_1+[\widetilde{a}_1,\widetilde{a}_j]+\widetilde{b}
^{j}\quad \text{on }H_2.
\end{cases}
\\
\Phi_{23}=\begin{cases}
\overline{\Phi }_{23}=\partial_2\overline{a}_{3}
-\partial_{3}\overline{a}_2+[\overline{a}_2,\overline{a}_{3}]\quad
\text{on }H_1 \\
\widetilde{\Phi }_{23}=\partial_2\widetilde{a}_{3}-\partial_{3}
\widetilde{a}_2+[\widetilde{a}_2,\widetilde{a}_{3}]\quad
\text{on } H_2.
\end{cases}
\end{gathered}  \label{eq14}
\end{equation}

(2) Furthermore $V=(A_0\equiv 0,A_i,F^{0i},F_{ij},f)$, with
$i,j=1,2,3$, verifies the equation
$\widehat{\nabla }_{\alpha}F^{\alpha0}=J^0$ on $\widehat{H}$,
if and only if the function $\overline{b}^1$
(resp. $\widetilde{b}^1$) is solution of the following Cauchy
problem \eqref{eq15} (resp. \eqref{eq16}),
\begin{equation}
\begin{gathered}
\partial_1\overline{b}^1+[\overline{a}_1,\overline{b}^1]
=-\left( \overline{J}^0+\overline{J}^1+\partial_2\overline{\Psi }
_{12}+\partial_{3}\overline{\Psi }_{13}+[ \overline{a}_2,\overline{
\Psi }_{12}] +[ \overline{a}_{3},\overline{\Psi
}_{13}]
\right)\\
 \text{on } \widehat{H}_1; \\
\overline{b}^1(0,x^2,x^3)=\frac{1}{2}(\partial_1
\overline{a}_1-\partial_1\widetilde{a}_1)(0,x^2,x^3)
,\quad \forall ( x^2,x^3) \in B
\end{gathered}  \label{eq15}
\end{equation}
and
\begin{equation}  \label{eq16}
\begin{gathered}
\partial_1\widetilde{b}^1+[ \widetilde{a}_1,\widetilde{b}^1
] =\Big( \widetilde{J}^0-\widetilde{J}^1+\partial
_2\widetilde{ \Psi }_{12}+\partial_{3}\widetilde{\Psi
}_{13}+[ \widetilde{a}_2,
\widetilde{\Psi }_{12}] +[ \widetilde{a}_{3},\widetilde{\Psi }_{13}
] \Big) \\
\text{on }  \widehat{H}_2; \\
\widetilde{b}^1( 0,x^2,x^3) =\frac{1}{2}( \partial_1
\overline{a}_1-\partial_1\widetilde{a}_1) (
0,x^2,x^3), \quad \forall ( x^2,x^3)\in B
\end{gathered}
\end{equation}
with
\begin{gather*}
\overline{\Psi }_{12}=\partial_1\overline{a}_2-\partial_2\overline{
a }_1+[ \overline{a}_1,\overline{a}_2], \quad
\overline{\Psi }_{13}=\partial_1\overline{a}_{3}-\partial_{3}
\overline{a}_1+[ \overline{a}_1,\overline{a}_{3}], \\
\overline{J}^\alpha=\int_{\mathbb{R}^3\times O}
p^\alpha q\overline{\varphi} \omega_{p}\omega_{q},\quad
\alpha=0,1;\\
\widetilde{\Psi }_{12}=\partial_1\widetilde{a}_2-\partial_2
\widetilde{a}_1+[ \widetilde{a}_1,\widetilde{a}_2] ,\quad
\widetilde{\Psi }_{13}=\partial
_1\widetilde{a}_{3}-\partial_{3} \widetilde{a}_1+[ \widetilde{a}
_1,\widetilde{a}_{3}], \\
\widetilde{J}^\alpha=\int_{\mathbb{R}^3\times O}p^\alpha q\widetilde{
\varphi }\omega_{p}\omega_{q},\quad \alpha=0,1.
\end{gather*}
\end{theorem}

\begin{remark} \label{rmk3.2} \rm
 We observe that the PDE appearing in \eqref{eq15} and \eqref{eq16}
are in fact linear ordinary differential equations of the scalar
variable $x^1$, smoothly depending on the parameters $x^2$ and
$x^3$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{theo1.1}]
(1) As restrictions to the smooth surface $H_1$ (resp $H_2$) of the
$\mathcal{C}^{\infty }$ functions $A_i,F^{0i},F_{ij}$, the functions
$\overline{a}_i,\overline{b}^{j}$ $\overline{\Phi }_{ij}$
(resp. $\widetilde{a}_i,\widetilde{b}^{j}$, $\widetilde{\Phi }_{ij}$)
are $\mathcal{C}^{\infty }$ on $H_1$ (resp. $H_2$). Likewise, as
restrictions to the smooth surface $\widehat{H}_1$ (resp.
$\widehat{H}_2$) of the function $f$ which is $\mathcal{C}^{\infty }$
on $\hat{\sqcup}$, $\overline{\varphi }$
(resp. $\ \widetilde{\varphi }$) is $\mathcal{C}^{\infty }$ on
$\widehat{H}_1$\ (resp. $\widehat{H}_2$).

The first and the second compatibility conditions  of \eqref{eq13}
are obviously satisfied.
Moreover, by considering the restrictions to $H_r$ $(r=1,2)$ of
 $A_i$ $(i=1,2,3)$, we have:
\begin{equation}\label{restriction of A}
\begin{gathered}
 \partial_1[A_i]_1=-[\partial_0A_i]_1+
[\partial_1A_i]_1\quad\text{on }H_1,\\
\partial_1[A_i]_2=[\partial_0A_i]_2+
[\partial_1A_i]_2\quad\text{on }H_2,\\
 \partial_j[A_i]_r=[\partial
_jA_i]_r\quad\text{on }H_r,\; r=1,2,\; j=2,3,\; i=1,2,3.
\end{gathered}
\end{equation}
To show the third compatibility condition  of
\eqref{eq13}, we observe that on $H_1\cap H_2$ the following
relations are valid:
$[\partial_1A_i]_1=[\partial _1A_i]_2$ and
$[\partial _0A_i]_1=[\partial_0A_i]_2$. We then deduce
in view of first and second relations of \eqref{restriction of A}
that
\begin{equation*}
2[\partial_0A_i]_r=\partial_1[A_i]_2-\partial_1[A_i]_1
\quad\text{on }H_1\cap H_2,\quad r=1,2.
\end{equation*}
Then, using the fact that
$[\partial_0A_i]_1=-F^{0i}\big|_{H_1}=-\overline{b}^i$ and
$[\partial_0A_i] _2=-F^{0i}\big|_{H_2}=-\widetilde{b}^i$,
we deduce the third relation  of \eqref{eq13},
\begin{equation*}
\partial_1\overline{a}_i-\partial_1\widetilde{a}_i=2\overline{b}
^i=2\widetilde{b}^i\quad\text{on }H_1\cap H_2,\;i=1,2,3.
\end{equation*}
The algebraic differential relations stated in \eqref{eq14} follow
directly from definitions \eqref{eq1}, \eqref{eq12} and relations
\eqref{restriction of A}.

(2) To show that $\overline{b}^1$ is solution of
\eqref{eq15} if $\widehat{\nabla }_{\alpha }F^{\alpha 0}=J^0$ on
$\hat{H}$, we consider the restrictions to $\widehat{H}_1$
 of equations $\widehat{\nabla }_{\alpha }F^{\alpha 1}=J^1$
 and $\widehat{\nabla}_{\alpha }F^{\alpha 0}=J^0$. By adding
these restrictions and in view of definitions \eqref{eq12}
and the relations \eqref{eq14}, we obtain
\begin{equation*}
\partial_1\overline{b}^1+[\overline{a}_1,\overline{b}^1]
=-\left( \overline{J}^0+\overline{J}^1+\partial_2\overline{\Psi }
_{12}+\partial_{3}\overline{\Psi }_{13}+[ \overline{a}_2,\overline{
\Psi }_{12}] +[ \overline{a}_{3},\overline{\Psi}_{13}] \right) \quad\text{on }\widehat{H}_1.
\end{equation*}
Moreover, we know that the third relation  of \eqref{eq13}, for
$i=1$, gives
\begin{equation*}
\overline{b}^1(0,x^2,x^3)=\frac{1}{2}(\partial_1
\overline{a}_1-\partial_1\widetilde{a}_1)(0,x^2,x^3) ,\quad \forall (x^2,x^3)\in B.
\end{equation*}
We then deduce that $\overline{b}^1$ solves \eqref{eq15}.

By the same process, subtracting the restriction to
$\widehat{H}_2$ of the equation $\widehat{\nabla }_{\alpha
}F^{\alpha 1}=J^1$ from that of the equation $\widehat{\nabla
}_{\alpha }F^{\alpha 0}=J^0$, we obtain in view
of the third relation  of \eqref{eq13} that $\widetilde{b}^1$
solves problem \eqref{eq16}. Conversely,  if $\overline{b}^1$ (resp
$\widetilde{b}^1$ )is solution of the problem  \eqref{eq15} (resp
\eqref{eq16}), then it is obvious that we have
$\widehat{\nabla}_{\alpha }F^{\alpha 0}=J^0$ on $\widehat{H}$.
\end{proof}

\subsection{Precise statement of the initial constraint problem:
the choice of the free data}

Let $T\in \mathbb{R}_{+}^{\ast }$ such that
$T\leq \sup \{x^0=t,( x^0,x^i) \in \sqcup\}$, $T$
given. We assume the temporal gauge condition $A_0=0$ in $\sqcup
_T$.

(a) \textbf{Free data of the initial constraint problem}:
Consider the arbitrary functions:
\begin{equation}
\begin{gathered}
\overline{a}_i(x^1,x^2,x^3) \;\mathcal{C}^{\infty} \text{ function, }
(-x^1,x^1,x^2,x^3) \in H_1,\; i=1,2,3;
\\
\widetilde{a}_i(x^1,x^2,x^3) \; \mathcal{C}^{\infty}
\text{ function, }(x^1,x^1,x^2,x^3) \in H_2;
\\
\overline{b}^{j}(x^1,x^2,x^3) \; \mathcal{C}^{\infty }
\text{ function, } (-x^1,x^1,x^2,x^3) \in H_1,j=2,3;
\\
\widetilde{b}^{j}(x^1,x^2,x^3) \; \mathcal{C}^{\infty}
\text{ function, } (x^1,x^1,x^2,x^3) \in H_2,j=2,3;
\\
\overline{\varphi } (x^1,x^2,x^3,p^i,q^L)
\mathcal{C}^{\infty } \text{ function, }
(-x^1,x^1,x^2,x^3,p^i,q^L) \in \widehat{H}_1,L=1,\dots N-1;
\\
\widetilde{\varphi }( x^1,x^2,x^3,p^i,q^L)
\mathcal{C}^{\infty }\text{ function, }
(x^1,x^2,x^3,p^i,q^L) \in \widehat{H}_2,\; L=1,\dots ,N-1;
\\
\text{with  $\overline{\varphi}$ and $\widetilde{\varphi}$ having
 compact support.}
\end{gathered}  \label{eq17}
\end{equation}

These functions satisfy the compatibility conditions:
\begin{equation}  \label{eq18}
\begin{gathered}
\overline{a}_i(0,x^2,x^3)
=\widetilde{a}_i(0,x^2,x^3),\quad \text{where }(x^2,x^3)
 \in B,\; i=1,2,3; \\
\overline{b}^{j}(0,x^2,x^3)
=\widetilde{b}^{j}\left(0,x^2,x^3\right) ,\;j=2,3; \\
\left( \partial_1\overline{a}_j-\partial
_1\widetilde{a}_j\right) (0,x^2,x^3)
=2\overline{b}^{j}(0,x^2,x^3) =2
\widetilde{b}^{j}(0,x^2,x^3); \\
\overline{\varphi }\left( 0,x^2,x^3,p^i,q^L\right)
=\widetilde{
\varphi }\left( 0,x^2,x^3,p^i,q^L\right),\; L=1,\dots ,N-1.
\end{gathered}
\end{equation}

(b) For the reduced system \eqref{reducedsyst}, as
initial conditions, we consider
\begin{equation}  \label{eq19}
\begin{gathered}
A_i\big|_H =a_i=\begin{cases}
\overline{a}_i&\text{on }H_1 \\
\widetilde{a}_i&\text{on }H_2;
\end{cases} \quad
F^{0i}\big|_H=b^i=\begin{cases}
\overline{b}^i&\text{on}H_1 \\
\widetilde{b}^i&\text{on }H_2;
\end{cases}
\\
F_{ij}\big|_H=\Phi_{ij}=\begin{cases}
\overline{\Phi }_{ij}&\text{on }H_1 \\
\widetilde{\Phi }_{ij}&\text{on }H_2;
\end{cases} \quad
f\big|_{\widehat{H}}=\varphi =\begin{cases}
\overline{\varphi }&\text{on }\widehat{H}_1 \\
\widetilde{\varphi}&\text{on }\widehat{H}_2;
\end{cases}
\quad i,j=1,2,3
\end{gathered}
\end{equation}
where:
$ \Phi_{ij}$  are given by relation
\eqref{eq14} of theorem~\ref{theo1.1} and
$b^1$ is such that $\overline{b}^1$ (resp. $\widetilde{b}^1)$
 is the unique  solution of  \eqref{eq15}  (resp. \eqref{eq16})
 of theorem~\ref{theo1.1}, with
\begin{equation}
\overline{b}^1(0,x^2,x^3) =\widetilde{b}^1(0,x^2,x^3)
 =\frac{1}{2}(\partial _1\overline{a}_1-\partial
_1\widetilde{a}_1)(0,x^2,x^3) .
  \label{eq20}
\end{equation}

Our goal is now to show that every $\mathcal{C}^\infty $ solution of
the reduced system \eqref{reducedsyst} subjected to initial
conditions \eqref{eq19}, \eqref{eq20} and compatibility conditions
\eqref{eq18} is in fact $\mathcal{C}^\infty $ solution of the
complete system \eqref{YMVsyst} of
Yang-Mills-Vlasov Equations.

\begin{theorem}\label{theo1.2}
 Every $\mathcal{C}^{\infty }$ solution
 $V=(A_0\equiv 0,A_i,F^{0i},F_{ij},f)$,
defined in a neighborhood $\widetilde{\sqcup }$ of $\widehat{H}$
in $\widehat{\sqcup }$, for the reduced system \eqref{reducedsyst}
with initial conditions and compatibility conditions
\eqref{eq19}, \eqref{eq20} and \eqref{eq18} associated to free
data \eqref{eq17} and such that the support of
$\varphi $ is compact, is solution of the complete system
\eqref{YMVsyst} of Yang-Mills-Vlasov Equations in the domain
$\widehat{\sqcup }$.
\end{theorem}

\begin{proof}
 Let $V=( A_0\equiv 0,A_i,F^{0i},F_{ij},f)$ be
a $\mathcal{C}^{\infty }$ solution, defined in a neighborhood
$\widetilde{\sqcup }$ of $\widehat{H}$, for the reduced system
\eqref{reducedsyst} with the initial conditions described as above.
 We give the proof in three steps:


\textbf{Step 1.}
We  show that $F$ is the curvature of $A$. Set $\Omega =dA$,
that is the curvature of $A$. In the global canonical coordinates
system on $\sqcup $ it holds that
\begin{equation*}
\Omega_{\alpha \beta }=\partial_{\alpha }A_{\beta }-\partial
_{\beta }A_{\alpha }+[ A_{\alpha },A_{\beta }] ,\quad
\alpha ,\beta =0,1,2,3.
\end{equation*}
In view of the temporal gauge condition $A_0=0$
in $\widetilde{\sqcup }$ and the reduced system \eqref{reducedsyst}
which is verified in $\widetilde{\sqcup }$, this implies that
\begin{equation*}
\Omega_{0i}=\partial_0A_i=F_{0i}\quad \text{in }
 \widetilde{\sqcup },\; i=1,2,3.
\end{equation*}
It remains to prove that $\Omega_{ij}=F_{ij}$ in $\widetilde{
\sqcup}$, $i,j=1,2,3$. Since $V$ solves the reduced system
\eqref{reducedsyst}, it holds that
\begin{equation*}
\partial_0F_{ij}+\partial_iF_{j0}+\partial_jF_{0i}+[A_i,F_{j0}
]+[A_j,F_{0i}]=0\ \ \text{in }\widetilde{\sqcup
},\quad  i,j=1,2,3.
\end{equation*}
In view of \eqref{reducedsyst}, it holds that $F_{0i}=\partial
_0A_i$. Inserting this latter relation in the previous one and
integrating with respect to the variable $x^0$ on
$[\vert x^1\vert ,t] $, we obtain
\begin{align*}
&\Big(F_{ij}-\partial_iA_j+\partial_jA_i-[A_i,A_j]\Big)
(t,x^1,x^2,x^3) \\
&-\Big(F_{ij}-\partial_iA_j+\partial_jA_i-[A_i,A_j]\Big)
(\vert x^1\vert ,x^1,x^2,x^3)=0.
\end{align*}
Now the initial conditions \eqref{eq19} and \eqref{eq20} imply that
\begin{equation*}
\big(-F_{ij}+\partial_iA_j-\partial_jA_i+\ [A_i,A_j]
\big)( | x^1| ,x^1,x^2,x^3)=0.
\end{equation*}
Hence, we obtain
\begin{equation}
\begin{split}
F_{ij}\left( x^0,x^1,x^2,x^3\right)
&=\Big(\partial _iA_j-\partial_jA_i+[A_i,A_j]\Big)
(x^0,x^1,x^2,x^3) \\
&=\Omega_{ij}(x^0,x^1,x^2,x^3) .
\end{split}  \label{eqI.10}
\end{equation}

\textbf{Step 2.}
 We must verify that
\begin{equation*}
\widehat{\nabla }_{k}F_{ij}+\widehat{\nabla }_iF_{jk}+\widehat{\nabla }
_jF_{ki}=0\quad k,i,j=1,2,3.
\end{equation*}
This is obvious since the first step provides
\begin{equation*}
F_{\alpha \beta }=\partial_{\alpha }A_{\beta }-\partial_{\beta
}A_{\alpha }+[ A_{\alpha },A_{\beta }] ,\quad \forall
\alpha ,\beta =0,1,2,3.
\end{equation*}
This implies Bianchi identities
\begin{equation*}
\widehat{\nabla }_{\alpha }F_{\beta \gamma }+\widehat{\nabla
}_{\beta }F_{\gamma \alpha }+\widehat{\nabla }_{\gamma }F_{\alpha
\beta }=0,\quad  \alpha ,\beta ,\gamma =0,1,2,3.
\end{equation*}
The result then follows.

\textbf{Step 3.}
We  show that $V$ satisfies the equation $\widehat{\nabla
}_{\alpha }F^{\alpha 0}=J^0$ at each point $(x^0,x^i)\in
\widetilde{\sqcup }$ with $x^1\neq 0$. Indeed, the equation
$\widehat{\nabla }_{\alpha }F^{\alpha i}=J^i$, which is verified
according to \eqref{reducedsyst}, can be written as follows
\begin{equation*}
\partial_0F^{0i}=J^i-\partial_jF^{ji}-[A_j,F^{ji}],\quad
i,j=1,2,3.
\end{equation*}
By integrating this latter expression with respect to $x^0$ on
$[| x^1| ,t] $ and differentiating with respect to $x^i$, we obtain
\begin{equation}
\begin{split}
&\partial_iF^{0i}(t,.,x^3)\\
&=\partial_i\{F^{0i}(| x^1| ,.,x^3)\}
+\partial _i\big\{ \int_{| x^1|
}^{t}\{J^i-\partial_jF^{ji}-[ A_j,F^{ji}]\}
(\tau ,.,x^3) \big\} d\tau .
\end{split} \label{eq21}
\end{equation}
Direct calculations give
\begin{equation}
\partial_i\big\{F^{0i}\left( | x^1|
,x^1,x^2,x^3\right) \big\}=\partial_0F^{0i}\left(
| x^1| ,x^1,x^2,x^3\right) .\varepsilon
_i+\partial
_iF^{0i}\left( | x^1| ,x^1,x^2,x^3\right) ,
\label{eq22}
\end{equation}
with
\[
\varepsilon_i=\begin{cases}
\varepsilon &\text{if }i=1 \\
0&\text{if }i=2,3;
\end{cases} \quad
\varepsilon =\begin{cases}
1 & \text{if }  x^1>0 \\
-1 & \text{if }  x^1<0;
\end{cases}
\]
and
\begin{equation}  \label{deq}
\begin{split}
&\partial_i\big\{ \int_{| x^1| }^{t}\{
J^i-\partial_jF^{ji}-[A_j,F^{ji}]\}\left( \tau
,x^1,x^2,x^3\right) d\tau \big\} \\
&=\int_{| x^1| }^{t}\{\partial
_iJ^i-\partial_i\partial_jF^{ji}-\partial
_i[A_j,F^{ji}]\}\left( \tau
,x^1,x^2,x^3\right) d\tau \\
&\quad -\varepsilon\{J^1-\partial_jF^{j1}-[ A_j,F^{j1}]\}\left(
| x^1| ,x^1,x^2,x^3\right) .
\end{split}
\end{equation}
Now, as $V$ is solution of the system \eqref{reducedsyst}
in $\widetilde{\sqcup }$, $f$ satisfies Vlasov equation in
$\widetilde{\sqcup }$ and this implies
$\widehat{\nabla }_{\beta }J^{\beta }=0$ in
$\widetilde{\sqcup }$; i. e.,
\begin{equation*}
\partial_iJ^i=-\partial_0J^0-[A_i,J^i] ,\quad i=1,2,3.
\end{equation*}
By inserting this latter relation into \eqref{deq} and in view of
the identity $( \partial_i\partial_jF^{ji}=0) $ we
gain
\begin{equation}
\partial_i\Big\{ \int_{| x^1| }^{t}\{
J^i-\partial_{l}F^{li}-[ A_{l},F^{li}] \}\left(
\tau ,x^1,x^2,x^3\right) d\tau \Big\} =Z,  \label{eq24}
\end{equation}
where
\begin{align*}
Z&=\int_{| x^1| }^{t}\{-\partial
_0J^0-[ A_i,J^i] -\partial_i[A_{l},F^{li}] \}\left(
\tau ,x^1,x^2,x^3\right) d\tau \\
&\quad -\{J^1-\partial_jF^{j1}-[ A_j,F^{j1}]\}\left(
| x^1| ,x^1,x^2,x^3\right) .\varepsilon .
\end{align*}
Equation \eqref{eq21} then becomes
\begin{equation}\label{deq1}
\begin{split}
&\partial_iF^{0i}\left( t,x^1,x^2,x^3\right)\\
& =\partial _0F^{01}\left( | x^1|,x^1,x^2,x^3\right)\varepsilon
+\partial_iF^{0i}\left( | x^1|,x^1,x^2,x^3\right)
-J^0\left( t,x^1,x^2,x^3\right)\\
&\quad +J^0\left( | x^1| ,x^1,x^2,x^3\right)
-\big\{J^1-\partial_jF^{j1}-[ A_j,F^{j1}]\big\}
\left(| x^1| ,x^1,x^2,x^3\right) \varepsilon \\
&\quad -\int_{| x^1| }^{t}\big\{[
A_i,J^i] +\partial_i[ A_{l},F^{li}]
\big\}\left( \tau,x^1,x^2,x^3\right) d\tau .
\end{split}
\end{equation}
The relation $\widehat{\nabla }_{\alpha }F^{\alpha 1}=J^1$ in
$\widetilde{\sqcup }$, which follows from \eqref{reducedsyst}
can be written as
\begin{equation}
\partial_0F^{01}=J^1-\partial_jF^{j1}-[ A_j,F^{j1}] .
\label{eq26}
\end{equation}
Inserting \eqref{eq26} into \eqref{deq1}, we obtain
\begin{equation}\label{deq2}
\begin{split}
\partial_iF^{0i}\left( x^0,x^1,x^2,x^3\right)
&=\partial _iF^{0i}\left( | x^1| ,x^1,x^2,x^3\right) \\
&\quad -\int_{| x^1| }^{t}\Big\{[A_i,J^i]
+\partial_i[ A_{l},F^{li}] \Big\}
\left( \tau,x^1,x^2,x^3\right) d\tau \\
&\quad -J^0\left( t,x^1,x^2,x^3\right)
+J^0\left( |x^1| ,x^1,x^2,x^3\right) .
\end{split}
\end{equation}
Likewise, the relation $\widehat{\nabla }_{\alpha }F^{\alpha i}=J^i$
 reads
\begin{equation*}
\partial_0F^{0i}+\partial_{l}F^{li}+[ A_{l},F^{li}]=J^i,\quad
i,l=1,2,3.
\end{equation*}
Setting
\begin{equation*}
L=\int_{| x^1| }^{t}[ A_i,J^i]d\tau ,
\end{equation*}
it follows that
\begin{equation}\label{deq3}
\begin{split}
L&=\int_{| x^1| }^{t}\big\{ [ A_i,\partial_0 F^{0i}]
+[ A_i,\partial_{l}F^{li}] +[ A_i,[ A_{l},F^{li}] ] \big\}d\tau \\
&=\int_{| x^1| }^{t}\Big\{\partial_0[
A_i,F^{0i}] -[ \partial_0A_i,F^{0i}] +[
A_i,\partial_{l}F^{li}] +[ A_i,[ A_{l},F^{li}] ] \Big\}d\tau \\
&=[ A_i,F^{0i}] \left( t,x^1,x^2,x^3\right)
  -[ A_i,F^{0i}] \left( | x^1|,x^1,x^2,x^3\right)\\
&\quad +\int_{| x^1| }^{t}\Big\{
 [A_i,\partial_{l}F^{li}] +[ A_i,[ A_{l},F^{li}]] \Big\}d\tau ,
\end{split}
\end{equation}
thanks to
\begin{equation*}
[ \partial_0A_i,F^{0i}] =[ F_{0i},F^{0i}]
=\sum_{i=1}^3[ -F^{0i},F^{0i}] =0.
\end{equation*}
The insertion of \eqref{deq3} into \eqref{deq2} gives
\begin{equation}
\begin{split}
&\partial_iF^{0i}\left( t,x^1,x^2,x^3\right)\\
&=-[ A_i,F^{0i}] (t,.,x^3) +[ A_i,F^{0i}] \left(| x^1| ,.,x^3\right) \\
&\quad +\partial_iF^{0i}\left( | x^1|,.,x^3\right)
 -J^0(t,.,x^3) +J^0\left(| x^1|,.,x^3\right) -R,
\end{split} \label{eq29}
\end{equation}
where
\begin{equation*}
R=\int_{| x^1| }^{t}\Big\{[ A_i,\partial
_{l}F^{li}] +\partial_i[ A_{l},\partial_{l}F^{li}] +
[ A_i,[ A_{l},F^{li}] ] \Big\}d\tau .
\end{equation*}
By permuting indexes $l$ and $i$ in $[ A_i,\partial_{l}F^{li}] $
and using the fact that $F$ is antisymmetric, we obtain
\begin{equation*}
[ A_i,\partial_{l}F^{li}] +\partial_i[ A_{l},F^{li}
] =[ \partial_iA_{l},F^{li}] .
\end{equation*}
Thus
\begin{equation}
R=\int_{| x^1| }^{t}\Big\{[ \partial
_iA_{l},F^{li}] +[ A_i,[ A_{l},F^{li}] ]
\Big\}d\tau .  \label{eq30}
\end{equation}
Moreover it holds that
\begin{equation*}
F^{li}=F_{li}=\partial_{l}A_i-\partial_iA_{l}+[A_{l},A_i] .
\end{equation*}
By inserting this latter relation in \eqref{eq30}, using several
times Jacobi identity and the fact that Lie bracket $[,]$ is
antisymmetric, we obtain $R=0$. But in view of theorem
~\ref{theo1.1}, $V$ satisfies the equation
\begin{equation*}
\widehat{\nabla }_{\alpha }F^{\alpha 0}\big|_{\widehat{H}}
=J^0\big|_{\widehat{H}}.
\end{equation*}
This implies
\begin{equation}
\big(\partial_iF^{0i}+[ A_i,F^{0i}]+J^0\big)\left( | x^1|
,x^1,x^2,x^3\right) =0.  \label{eq36}
\end{equation}
Thus \eqref{eq29} becomes
\begin{equation*}
\widehat{\nabla }_{\alpha }F^{\alpha 0}(t,x^1,x^2,x^3)
=J^0(t,x^1,x^2,x^3)\quad \text{with } x^1\neq 0,
\end{equation*}
which implies, by an obvious continuity argument,
\begin{equation*}
\widehat{\nabla }_{\alpha }F^{\alpha 0}\left(
t,x^1,x^2,x^3\right) =J^0\left( t,x^1,x^2,x^3\right)\quad
\text{with } x^1=0.
\end{equation*}
Consequently
\begin{equation*}
\widehat{\nabla }_{\alpha }F^{\alpha 0}=J^0\quad  \text{in }
 \widetilde{\sqcup }.
\end{equation*}
This completes the proof.
\end{proof}

\section{Determination of restrictions}
In this section we determine  restrictions to the initial
characteristic hypersurfaces $\widehat{H}_1$ and $\widehat{H}_2$
of derivatives of all order of a possible $\mathcal{C}^\infty $
solution of the evolution problem \eqref{reducedsyst}
subjected to initial conditions
\eqref{eq18},  \eqref{eq19}, \eqref{eq20}.
associated with free data. We will use Rendall's method \cite{29}
which consists in transforming the Goursat problem under
consideration into an ordinary Cauchy problem with zero data
specified on the spatial hypersurface $x^0=0$.
An important step is the determination of
the restrictions to $\widehat{H}_1$
and $\widehat{H}_2$ of the derivatives of all order of the possible
$\mathcal{C}^{\infty }$ solutions of the evolution problem
\eqref{reducedsyst} subject to \eqref{eq18}, \eqref{eq19},
\eqref{eq20}, which is the goal of the present section.
To reach it, it will be useful to reinforce
hypothesis of free data $\overline{\varphi }$ and
$\widetilde{\varphi }$

\subsection*{Hypothesis I}
If $m>0$, we assume:
\begin{itemize}
\item[(i)] $\overline{\varphi }$
(resp $\widetilde{\varphi }$) is smooth on
$\widehat{H}_1$ (resp $\widehat{H}_2$) and $\operatorname{supp}
(\overline{\varphi })$ (resp
$\operatorname{supp}(\widetilde{\varphi })$) is compact.

\item[(ii)] $\operatorname{supp}(\overline{\varphi })\cap
\{ \widehat{H}_1 \cap \widehat{H}_2\} =\emptyset $ and
$\operatorname{supp}(\widetilde{\varphi })\cap
\{\widehat{H}_1\cap\widehat{H}_2\} =\emptyset$.

\end{itemize}
If $m=0$, we add (i) and (ii) the following hypothesis
\begin{itemize}
\item[(iii)] $\operatorname{supp}(\overline{\varphi })\subset
\{X=(-x^1,x^1,x^2,x^3,p^i,q^L) :(p^2) ^2+( p^3) ^2>0\}$
and
$\operatorname{supp}(\widetilde{\varphi })\subset
\{ X=( x^1,x^1,x^2,x^3,p^i,q^L) :(p^2) ^2+( p^3) ^2>0\}$.
\end{itemize}

\begin{remark}\label{rem2}\rm
From  hypothesis (i), (ii), (iii), we have the following:

 If $m\neq 0$, for every $X=(-x^1,x^1,x^2,x^3,p^i,q^L) \in
\operatorname{supp}(\overline{\varphi })$, we have
$p^0+p^1\neq 0$;  and for every
$X=(x^1,x^1,x^2,x^3,p^i,q^L) \in \operatorname{supp}
(\widetilde{\varphi })$ we have
$p^0-p^1\neq 0$.

If Hypothesis (i) holds, that is the compactness of the supports
of $\overline{\varphi }$ and $\widetilde{\varphi }$, then
\begin{gather*}
\inf_{X \in \operatorname{supp} (\overline{\varphi})} | p^0+p^1|
=\min_{X \in \operatorname{supp}(\overline{\varphi})}| p^0+p^1|
=C_1>0,
\\
\inf_{X \in \operatorname{supp}(\widetilde{\varphi})}
 |p^0-p^1|
=\min_{X \in \operatorname{supp}(\widetilde{\varphi })} | p^0-p^1|
=C_2>0.
\end{gather*}
Analogously, if $m=0$ hypothesis (i) and (iii) again imply
\begin{gather*}
\inf_{X\in \operatorname{supp}(\overline{\varphi})}|p^0+p^1|
=\min_{X\in \operatorname{supp}(\overline{\varphi })} |p^0+p^1| =C_1>0
\\
\inf_{X\in \operatorname{supp}(\widetilde{\varphi})} | p^0-p^1|
=\min_{X\in \operatorname{supp}(\widetilde{\varphi })} |p^0-p^1| =C_2>0.
\end{gather*}
These remarks are crucial to establish the unique
determination of the restrictions to
$\widehat{H}_r$ $(r=1,2)$ of the derivatives of all order
of a possible $\mathcal{C}^\infty $ solution $f$ of the Vlasov
equation.

Hypothesis (ii) is a sufficient condition for showing that
\begin{equation*}
[ \frac{\partial ^{l}f}{( \partial x^0) ^{l}}]_1=
[ \frac{\partial ^{l}f}{( \partial x^0) ^{l}}]_2\quad
\text{on }\widehat{\mathit{H}}_1\cap
\widehat{\mathit{H}}_2,\;\forall l\in \mathbb{N},
\end{equation*}
which are necessary conditions for a function
$\mathcal{C}^{\infty}$ in a neighborhood of
$\widehat{H}_1\cap \widehat{H}_2$
\end{remark}

\subsection{Determination of the restrictions}
In this section we determine the restriciotns to $\widehat{H}$
of the first derivatives of any possible $\mathcal{C}^\infty $
solution of the evolution problem.
Let $V=\left( A_0\equiv 0,A_i,F^{0i},F_{ij},f\right) $ be a
$\mathcal{C}^\infty $ solution, defined in the neighborhood
 $\widetilde{\sqcup}$ of $\widehat{H}$ in $\widehat{\sqcup}$,
of the evolution problem. We will use the equations of system
\eqref{reducedsyst} to determine the restrictions to
$\widehat{H} _r(r=1, 2)$ of the first order derivatives of $V$.

(i) To determine $[\partial_0A_i]_1$, we use the equation
$\partial_0A_{l}=F_{0l}$ of \eqref{reducedsyst}, to obtain
\begin{equation}
[\partial_0A_i]_1=-\overline{b}^i,\quad  i=1,2,3.\label{eq37}
\end{equation}

(ii) To determine $[ \frac{\partial f}{\partial x^0}]_1$, we use
the fourth equation of \eqref{reducedsyst} which
implies in view of Remark \ref{rem2},
\begin{equation}  \label{eq39}
[ \frac{\partial f}{\partial x^0}]_1
=\frac{-1}{p^0+p^1}
\big\{ p^{i_0}\frac{\partial \overline{\varphi }}{\partial x^{i_0}}+
\overline{P}^{j_0}\frac{\partial \overline{\varphi }}{\partial p^{j_0}}
+\overline{Q}^{L_0}\frac{\partial \overline{\varphi }}{\partial q^{L_0}}
\big\} ,
\end{equation}

(iii) To determine $[ \partial_0F^{01}]_1$, we consider
the restriction to $\widehat{H}_1$ of the equation
$\widehat{\nabla }_{\alpha }F^{\alpha 1}=J^1$ which implies
\begin{equation}
[ \partial_0F^{01}]_1=\overline{J}^1+\sum
_{j=1}^3\partial_j\overline{\Psi }_{1j}-\partial_j\overline{b}
^{j}+\sum_{j=1}^3[ \overline{a}_j,\overline{\Psi }_{1j}-
\overline{b}^{j}] ,\;\;j\neq 1,  \label{eq40}
\end{equation}
with
\begin{equation*}
\overline{\Psi }_{1j}=\partial_1\overline{a}_j-\partial_j\overline{a}
_1+[ \overline{a}_1,\overline{a}_j] ,\quad
\bar{J}^1=\int_{\mathbb{R}^3\times O}p^1q\overline{\varphi }\omega
_{p}\omega_{q}.
\end{equation*}

(iv) To determine $[\partial_0F_{ij}]_1$ with
$i,j\neq 1$, we consider the restriction to $H_1$
of Bianchi identities
(third equation of \eqref{reducedsyst}) which leads to
\begin{equation}
[\partial_0F_{ij}]_1
=-\partial_i\overline{b}^{j}-[\overline{a}_i,\overline{b}^{j}]
+\partial_j\overline{b}^i+ [ \overline{a}_j,\overline{b}^i] ,\quad
i,j=2,3;\;i\neq j. \label{eq41}
\end{equation}

(v) It remains to determine $[ \partial_0F^{1i}]_1$ and
$[ \partial_0F^{0i}]_1$ for $i\neq 1$. Consider the
following equations extracted from system \eqref{reducedsyst},
\begin{equation*}
\widehat{\nabla }_{\alpha }F^{\alpha i}=J^i, \quad
\widehat{\nabla } _0F_{i1}+\widehat{\nabla }_iF_{10}+\widehat{\nabla
}_1F_{0i}=0,\quad \text{for }i\neq 1.
\end{equation*}
Differentiating the sum of the latter equalities with respect to
$x^0$ and taking the restriction to $H_1$ of the relation
obtained, we gain
\begin{equation}
\partial_1\Big\{[ \partial_0F^{1i}]_1-[ \partial
_0F^{0i}]_1\Big\}+[ \overline{a}_1,[ \partial
_0F^{1i}]_1-[ \partial_0F^{0i}]_1] =
\overline{D},  \label{eq44}
\end{equation}
with $\overline{D}$ a known function on $\widehat{H}_1$. Moreover,
it holds that
\begin{gather*}
F^{1i}(-x^1,x^1,x^2,x^3)
= ( \overline{\Psi }_{1i}- \overline{b}^i) (x^1,x^2,x^3)
\quad\text{on }H_1
\\
F^{1i}(x^1,x^1,x^2,x^3) =\big( \widetilde{\Psi }_{1i}+
\widetilde{b}^i\big) (x^1,x^2,x^3) \quad\text{on } H_2,\;i\neq 1,
\end{gather*}
By differentiating with respect to $x^1$ the above relations, and
using the fact that the $F^{1i}$ and their derivatives are continuous
on $I=H_1\cap H_2$, for $i\neq 1$,
we have
\begin{equation}
[ \partial_0F^{1i}] \left( 0,0,x^2,x^3\right)
=\frac{1}{2} \left\{ \big( \partial_1\widetilde{\Psi }_{1i}
+\partial_1\widetilde{b} ^i\big)
-\big( \partial_1\overline{\Psi }_{1i}-\partial_1
\overline{b}^i\big) \right\} \left( 0,0,x^2,x^3\right), \label{eq46}
\end{equation}
Analogously, we obtain
\begin{equation}
[ \partial_0F^{0i}] \left( 0,0,x^2,x^3\right) =\frac{1}{2}
\left\{ \partial_1\widetilde{b}^i-\partial
_1\overline{b}^i\right\} \left( 0,0,x^2,x^3\right) ,i\neq 1,
\label{eq47}
\end{equation}
From \eqref{eq46} and \eqref{eq47}, for $i\neq 1$, we have
\begin{equation}
\{[ \partial_0F^{1i}]_1-[ \partial_0F^{0i}]_1\}(0,x^2,x^3)
=\frac{1}{2}\{\partial_1\widetilde{\Psi }_{1i}
-\partial_1\overline{\Psi }_{1i}+2\partial_1
\overline{b}^i \} (0,x^2,x^3) ,
\label{eq48}
\end{equation}
We then deduce $[ \partial_0F^{1i}]_1-[
\partial_0F^{0i}]_1$ on $H_1$ as the unique solution
of the Cauchy problem \eqref{eq44}, \eqref{eq48}. We can then set
\begin{equation}
[ \partial_0F^{1i}]_1-[ \partial
_0F^{0i}]_1=\overline{C}  \label{eqII.13}
\end{equation}
where $\overline{C}$ is now a known smooth function on $H_1$. To
determine
$\big([\partial_0F^{1i}]_1,[\partial_0F^{0i}]_1 \big)$
on $H_1$, considering now the restriction to $H_1$ of the
equation $\widehat{\nabla }_{\alpha }F^{\alpha i}=J^i$,
$i\neq 1$, we obtain
\begin{equation}
[\partial_0F^{0i}]_1+[ \partial_0F^{1i}]
_1=[ J^i]_1-\partial_j[ F^{ji}]_1-[
[ A_j,F^{ji}] ]_1,
\label{eqII.14}
\end{equation}
for $i=2,3$, $j=1,2,3$.
The relations \eqref{eqII.13} and \eqref{eqII.14} then determine
$[\partial_0F^{1i}]_1$ and $[\partial_0F^{0i}]_1$. By
the same process, we can uniquely determine $[ \partial
_0A_i]_2$ (for $i=1,2,3$),
$[ \partial_0f]_2$, $[ \partial_0F^{01}]_2,[\partial_0F_{ij}]_2$,
 for $i,j=2,3$, $[\partial_0F^{1i}]_2$,
$[ \partial _0F^{0i}]_2$ for $i=2,3$ and these functions are
$\mathcal{C} ^{\infty }$ on $\widehat{H}_2$.

We have then proved the following proposition.

\begin{proposition}\label{proprestriction1}
Let $V=( A_0\equiv 0,A_i,F_{ij},F^{0i},f)$ a
$\mathcal{C}^{\infty }$ solution, defined in a neighborhood
 $\widetilde{\sqcup }$ of $\widehat{H}$ in $ \widehat{\sqcup}$
of the evolution problem \eqref{reducedsyst} subject to
\eqref{eq18}, \eqref{eq19}, \eqref{eq20}
 such that initial datum $\varphi $
satisfies hypothesis (I).
Then the restrictions to $\widehat{H}$ of
all first order derivatives of $V$, that is
$[ \partial _0A_i] $,
$[ \partial_0F_{ij}]$, $[ \partial_0F^{0i}] $ and
$[ \partial_0f]$, are uniquely determined on
$\widehat{H}$. These functions are continuous on $\widehat{H}$and
are $\mathcal{C}^{\infty} $ on $\widehat{H}_r,(r=1,2)$.
Moreover, $\operatorname{supp}[ \partial_0f] $ is compact
and contained in the support of $\varphi $.
\end{proposition}

\subsection{Determination of derivatives}
In this section we determine derivatives all order of any possible
$\mathcal{C}^\infty $ solution of the evolution problem
\eqref{reducedsyst} subject to \eqref{eq18},
\eqref{eq19},  \eqref{eq20}.

Let $V=\left( A_0\equiv 0,A_i,F^{0i},F_{ij},f\right) $
be a $\mathcal{C}^{\infty }$ solution, defined on a neighborhood
$\widetilde{\sqcup }$ of $\widehat{H}$ in $\widehat{\sqcup }$,
of the evolution problem. We want to show, for every
$k\in \mathbb{N}$, that $[ \partial_0^{k}V] $ is uniquely determined,
is continuous on $\widehat{H}$ and that
$[\partial_0^{k}V]_r$ is $\mathcal{C}^{\infty }$
on $\widehat{H} _r,\;r=1,2$. This can obviously be done,
by induction on $k$, by considering suitable combinations of $k$
order derivatives with respect to $ x^0$ or $x^1$ of equations
of the reduced system \eqref{reducedsyst} and by using continuity
of $V$ and its derivatives of all order in the neighborhood
of $I=H_1\cap H_2$.
We then obtain the following proposition which generalizes
proposition \ref{proprestriction1}.

\begin{proposition}\label{proprestriction2}
(a) Let $V=\left( A_0\equiv
0,A_i,F^{0i},F_{ij},f\right) $ be a $\mathcal{C}^{\infty }$
solution, defined in a neighborhood  $\widetilde{\sqcup }$ of
$\widehat{H}$ in $ \widehat{\sqcup }$, of the evolution problem
\eqref{reducedsyst} subject to \eqref{eq18}, \eqref{eq19}, \eqref{eq20}
defined in a neighborhood such that initial data $\varphi $
satisfies hypothesis (I).
Then the restrictions to $\widehat{H}$ of derivatives of order
$l$ of $V$, that is $[\partial_0^{l}A_i] $,
$[ \partial_0^{l}F_{ij}] $, $[ \partial_0^{l}F^{0i}] $ and
$[ \partial _0^{l}f] $, $l\in \mathbb{N}$, are uniquely
determined on $\widehat{H}$. These functions are continuous
on $\widehat{H}$ and are $\mathcal{C}^{\infty }$ on
$\widehat{H}_r,r=1,2$. Moreover, for every $l\in \mathbb{N}$,
$\operatorname{supp}[\partial_0^{l}f] $ is compact and contained
in $\operatorname{supp}\varphi $.

(b) Moreover, if $W$ is a $\mathcal{C}^{\infty }$ function defined in
a neighborhood of $\widehat{H}$ in $\widehat{\sqcup }$ such that
for every $l\in \mathbb{N}$,
$[ \partial_0^{l}W] =[ \partial _0^{l}V] $ on $\widehat{H}$,
then $W$ satisfies on $\widehat{H}$ the reduced
system \eqref{reducedsyst} and its
derivatives of all orders.
\end{proposition}

In the next section, we will use the following convenient notation.
\begin{equation*}
[ \partial_0^{k}V] =\big( \Lambda_i^{(k) },\Xi_i^{(k) },
\Omega_{ij}^{(k)},f^{(k) }\big) ,k\in \mathbb{N},
\end{equation*}
with
\begin{gather*}
\Lambda_i^{(k) } =\begin{cases}
\overline{\Lambda }_i^{(k) } &\text{on } H_1 \\
\widetilde{\Lambda }_i^{(k) }&\text{on }H_2;
\end{cases}\quad
E_i^{(k) }=\begin{cases}
\overline{E}_i^{(k) }&\text{on } H_1 \\
\widetilde{E}_i^{(k) }&\text{on } H_2;
\end{cases}
\\
\Omega_{ij}^{(k) }=\begin{cases}
\overline{\Omega }_{ij}^{(k) }&\text{on } H_1 \\
\widetilde{\Omega }_{ij}^{(k) }&\text{on } H_2;
\end{cases} \quad
f^{(k) }=\begin{cases}
\overline{f}^{(k) }&\text{on } \widehat{H}_1 \\
\widetilde{f}^{(k) }&\text{on } \widehat{H}_2;
\end{cases}
\\
\overline{\Lambda }_i^{(k) }=[ \partial_0^{k}A_i]_1,\quad
\overline{E}_i^{(k) }=[ \partial_0^{k}F^{0i}]_1,\quad
\overline{\Omega }_{ij}^{(k) }=[ \partial_0^{k}F{ij}]_1,\quad
\overline{f}^{(k) }=[ \partial_0^{k}f]_1,
\\
\widetilde{\Lambda }_i^{(k) }=[ \partial_0^{k}A_i]_2,\quad
\widetilde{E}_i^{(k) }=[\partial_0^{k}F^{0i}]_2,\quad
\widetilde{\Omega }_{ij}^{(k) }=[\partial_0^{k}F_{ij}]_2,\quad
\widetilde{f}^{(k) }=[ \partial_0^{k}f]_2.
\end{gather*}
We also use the notation
\begin{equation}
[ \partial_0^{k}V] =\Phi ^{(k) }
=\begin{cases}
\overline{\Phi }^{(k) }(x^1,x^2,x^3)&\text{on }\widehat{H}_1 \\
\widetilde{\Phi }^{(k) }(x^1,x^2,x^3)&\text{on }\widehat{H}_2,
\end{cases}  \label{eq58}
\end{equation}
where
\begin{equation*}
\overline{\Phi }^{(k) }\equiv \big( \overline{\Lambda }
_i^{(k) },\overline{E}_i^{(k)
},\overline{\Omega}_{ij}^{(k) },\overline{f}^{(k) }\big),\quad
\widetilde{\Phi }^{(k) }\equiv \big( \widetilde{\Lambda }
_i^{(k) },\widetilde{E}_i^{(k) },\widetilde{
\Omega }_{ij}^{(k) },\widetilde{f}^{(k)
}\big),\quad \forall k\in \mathbb{N}.
\end{equation*}

\section{Resolution of problem \eqref{reducedsyst}, \eqref{eq18},
\eqref{eq19}, \eqref{eq20}}

The goal of this section is to solve the evolution problem
\eqref{reducedsyst},
\eqref{eq18}, \eqref{eq19}, \eqref{eq20}, where the initial data
$\varphi $ satisfies the support condition of
hypothesis (I). The method used consists in reducing this problem
into an ordinary Cauchy problem with zero data assigned on
the spatial hypersurface $x^0=0$, which we solve thanks to a
suitable combination of the
classical characteristics method, the Leray's theory \cite{25} of
hyperbolic systems and techniques of solution developed in \cite{7}
for the ordinary Cauchy problem associated to Yang-Mills-Vlasov
equations. According to section $4$, the evolution problem at hand,
of unknown $V=\left( A_0\equiv 0,A_i,F^{0i},F_{ij},f\right) $,
is equivalent to the following Goursat
problem defined in $\widehat{\sqcup }_T=\sqcup_T\times \mathbb{R}
^3\times O$,
\begin{equation}\label{goursatsyst}
\begin{gathered}
\widehat{\nabla }_{\alpha }F^{\alpha i}=J^i \\
\widehat{\nabla }_0F_{ij}+\widehat{\nabla }_iF_{j0}+\widehat{\nabla }
_jF_{0i}=0 \\
p^{\alpha }\frac{\partial f}{\partial x^{\alpha }}+P^i\frac{\partial f}{
\partial p^i}+Q^L\frac{\partial f}{\partial q^L}=0, \\
\partial_0A_i=F_{0i}, \\
[ \partial_0^{k}V] =\Phi ^{(k) }
=\begin{cases}
\overline{\Phi }^{(k) }&\text{on } \widehat{H}_1 \\
\widetilde{\Phi }^{(k) }&\text{on } \widehat{H}_2,
\end{cases}
\end{gathered}
\end{equation}
where
$L=1,\dots,N-1$; $\alpha =0,1,2,3$;
$i,j=1,2,3$; $k\in \mathbb{N}$.

Proceeding as in \cite{11} and \cite{29}, we transform problem
\eqref{goursatsyst}
into a Goursat problem defined in
$\widehat{\sqcup } _T $ with zero initial data on $\widehat{H}$,
by introducing a new unknown function
$V_1=\left( C_i,D^{0i},D_{ij},v\right) $ such
that $V=W+V_1$, where the auxiliary function
$W=\left( B_i,G^{0i},G_{ij},h\right) $ must
be a $\mathcal{C}^{\infty }$  function on
$\widehat{\sqcup }$ such that
\begin{equation}
[ \partial_0^{l}W] =\Phi ^{(l)},\quad \forall
l\in \mathbb{N};  \label{eq59}
\end{equation}
i. e., for all $l\in \mathbb{N}$,
\begin{equation}
[\partial_0^{l}B_i] =\Lambda_i^{(l) },\quad
[ \partial_0^{l}G^{0i}] =E_i^{(l) },\quad
[ \partial_0^{l}G_{ij}] =\Omega_{ij}^{(l) },\quad
[ \partial_0^{l}h] =f^{(l) }\,. \label{eq60}
\end{equation}
As in \cite{11} and \cite{21} the construction of the function $W$
is made thanks to some variants of classical Borel lemma
\cite{28}. The function $W$ so constructed is defined not only in
the domain $\widehat{\sqcup }_T$, but also in the whole domain
$\widehat{\Omega }_T^{\varepsilon }=\Omega_T^{\varepsilon
}\times \mathbb{R}^3\times O$, where $\Omega_T^{\varepsilon }$
is the maximal subdomain of
\begin{equation*}
\mathcal{L}=\{(x^0,x^1,x^2,x^3)\in \mathbb{R}^{4},0\leq x^0\leq
T,-T\leq x^1\leq T,(x^2,x^3)\in B\}
\end{equation*}
containing $\sqcup_T$, of which the future boundary $\partial
\Omega_T^{(f)}$ contains $\partial \sqcup_T\setminus H$ and
of which the past boundary is equal to $\sqcup_T\cap
\{x^0=0\}$. The component $h$ of $W$ is $\mathcal{C}^{\infty }$
\ with a compact support, since the ${f}^{(l)}$ have their
supports contained in $\operatorname{supp}\varphi $ which is
compact.
The transformed problem of unknown $V_1$ can also be written in
the domain $\widehat{\sqcup }_T$ as follows:
\begin{equation}\label{eq66}
\begin{gathered}
\partial_0D^{0i}+\partial_jD^{ji}+[ B_j+C_j,D^{ji}] +
[ C_j,G^{ji}] -\int_{\mathbb{R}^3\times
O} \,qp^ivw_{p}w_{q}=X^i
\\
\begin{aligned}
&\partial_0D^{ij}-\partial_iD^{j0}-\partial_jD^{0i}-[
B_i+C_i,D^{j0}] -[ B_j+C_j,D^{0i}] \\
&+[C_i,G_{j0}] +[ C_j,G_{0i}] =Z_{0ij}, \end{aligned}
\\
\begin{aligned}
&p^{\alpha }\frac{\partial v}{\partial x^{\alpha }}+p^{\mu }q\left(
G_{\mu }^i+D_{\mu }^i\right) \frac{\partial v}{\partial
p^i}-p^{\alpha }\left( [ B_{\alpha },q] ^L
+[C_{\alpha },q]
^L\right) \frac{\partial v}{\partial q^L}\\
&+p^{\mu }qD_{\mu }^i\frac{\partial h}{\partial p^i}-p^{\alpha }
[ C_{\alpha },q] ^L\frac{\partial h}{\partial q^L}=\omega,
\end{aligned}
\\
\partial_0C_i-D_{0i}=E_{0i} \\
[ \partial_0^{l}V_1]_{\widehat{H}_T}=0,\quad \forall
l\in \mathbb{N},
\end{gathered}
\end{equation}
where
\begin{gather*}
X^i\equiv \int_{\mathbb{R}^3\times O}qp^ihw_{p}w_{q}-\partial
_0G^{0i}-\partial_jG^{ji}-[ B_j,G^{ji}] ,
\\
Z_{0ij}\equiv -\partial_0G_{0i}-\partial_iG_{j0}-\partial_jG_{0i}-
[B_i,G_{j0}]-[B_j,G_{0i}],
\\
\omega \equiv -p^{\alpha }\frac{\partial h}{\partial x^{\alpha
}}-p^{\mu
}qG_{\mu }^i\frac{\partial h}{\partial p^i}+p^{\alpha }[B_{\alpha },q
]^L\frac{\partial h}{\partial q^L},
\\
E_{0i}\equiv G_{0i}-\partial_0B_i.
\end{gather*}

\begin{remark} \label{rem3}\rm
 According to \eqref{eq59}, and in view of proposition
\ref{proprestriction2} the terms of the right hand side of the system
\eqref{eq66}; i.e., $X^i$, $Z_{0ij}$, $\omega $ and $E_{0i}$ and
their derivatives of all order vanish on $\widehat{H}$.
\end{remark}

Consider now the functions $\overline{X}^i$, $\overline{Z}_{0ij}$,
$\overline{\omega }$ and $\overline{E}_{0i}$ which are the following
continuations on $\widehat{\Omega }_T$ of the functions $X^i$,
$Z_{0ij}$, $\omega$ and $E_{0i}$:
\begin{gather*}
\overline{X}^i=\begin{cases}
X^i&\text{on }\widehat{\sqcup }_T \\
0&\text{on }\widehat{\Omega }_T\backslash
\widehat{\sqcup }_T;
\end{cases} \quad
\overline{Z}_{0ij}=\begin{cases}
Z_{0ij}&\text{on }\widehat{\sqcup }_T \\
0&\text{on }\widehat{\Omega }_T\backslash \widehat{\sqcup }_T;
\end{cases}
\\
\overline{\omega }=\begin{cases}
\omega &\text{on }\widehat{\sqcup }_T \\
0&\text{on }\widehat{\Omega }_T\backslash \widehat{\sqcup }_T;
\end{cases}\quad
\overline{E}_{0i}=\begin{cases}
E_{0i}&\text{on }\widehat{\sqcup }_T \\
0&\text{on }\widehat{\Omega }_T\backslash \widehat{\sqcup }_T.
\end{cases}
\end{gather*}

\begin{remark} \label{rem4}\rm
 In view of Remark \ref{rem3}, the functions $\overline{X}^i$,
$\overline{Z}_{0ij}$, $\overline{\omega }$ and
$\overline{E}_{0i}$ are $\mathcal{C}^{\infty }$ on
$\widehat{\Omega }_T$.
\end{remark}

To study  problem \eqref{eq66}, we will first consider the
following ordinary Cauchy
problem  of unknown $\overline{V}_1=\big( \overline{C}_i,
\overline{D}^{0i},\overline{D}^{ij},\overline{v}\big) $,
defined on $\widehat{\Omega }_T$, with zero initial data
given on the spatial hyperplane $x^0=0$:
\begin{equation}\label{ordinarycp}
\begin{gathered}
\partial_0\overline{D}^{0i}+\partial_j\overline{D}^{ji}+[ B_j+
\overline{C}_j,\overline{D}^{ji}] +[ \overline{C}_j,G^{ji} ]
-\int_{\mathbb{R}^3\times
O}qp^i\overline{v}w_{p}w_{q}=\overline{X}^i,
\\
\begin{aligned}
&\partial_0\overline{D}^{ij}-\partial_i\overline{D}^{j0}-\partial_j
\overline{D}^{0i}-[ B_i+\overline{C}_i,\overline{D}^{j0}]\\
&-[ B_j+\overline{C}_j,\overline{D}^{0i}]
+[\overline{C}_i,G_{j0}]+ [ \overline{C}_j,G_{0i}]
=\overline{Z}_{0ij},
\end{aligned}
\\
\begin{aligned}
&p^{\alpha }\frac{\partial \overline{v}}{\partial x^{\alpha }}
+p^{\mu}q\big( G_{\mu }^i+\overline{D}_{\mu }^i\big)
\frac{\partial \overline{v}}{\partial p^i}-p^{\alpha }
\left( [ B_{\alpha },q]^L
+[ \overline{C}_{\alpha },q] ^L\right) \frac{\partial
\overline{v}}{\partial q^L}\\
&+p^{\mu }q\overline{D}_{\mu }^i\frac{\partial h}{\partial p^i}-
p^{\alpha }[ \overline{C}_{\alpha },q] ^L\frac{\partial
h}{\partial q^L}=\overline{\omega },
\end{aligned}
\\
\partial_0\overline{C}_i-\overline{D}_{0i}=\overline{E}_{0i},
\\
\overline{V}_1=0 \text{ on }x^0=0.
\end{gathered}
\end{equation}

We will show, in the last step, by using mostly the techniques of
solution of \cite{7}, that problem \eqref{ordinarycp} admits
in a domain
$\widehat{\Omega }_{T_1}=\widehat{\Omega }_T\cap \{x^0\leq T_1\}$,
$T_1\in]0,T_0] $, small enough, a unique
$\mathcal{C}^{\infty }$ solution
$\overline{V}_1$ such that the support of $\overline{V}_1$ is
contained in $\widehat{\sqcup }_{T_1}$ with the
$\operatorname{supp}\overline{v}$ compact.

We will then deduce that $V_1=\overline{V}_1 \big|_{\widehat{\sqcup }
_{T_1}}$
is the unique solution of problem \eqref{eq66}, consequently, the
$\mathcal{C}^{\infty }$ function $V=W+V_1$ will be the unique
solution of the evolution problem
\eqref{reducedsyst} subject to (\eqref{eq18}, \eqref{eq19},
\eqref{eq20} in the domain $\widehat{\sqcup }_{T_1}$.

\subsection{Resolution of problem \eqref{ordinarycp}}

We write problem \eqref{ordinarycp} defined in
$\widehat{\Omega}_T$ in the following appropriate form:
\begin{equation}\label{ordinarycpbis}
\begin{gathered}
\partial_0\overline{D}^{0i}+\partial_j\overline{D}^{ji}=F_1\big( X,
\overline{C}_j,\overline{D}^{0i},\overline{D}^{ji},\overline{v}\big),
\\
\partial_0\overline{D}^{ij}-\partial_i\overline{D}^{j0}-\partial_j
\overline{D}^{0i}=F_2\big( X,\overline{C}_j,\overline{D}^{0i},\overline{
D}^{ji},\overline{v}\big),
\\
\begin{aligned}
&p^{\alpha }\frac{\partial \overline{v}}{\partial x^{\alpha }}+p^{\mu
}q\big( G_{\mu }^i+\overline{D}_{\mu }^i\big) \frac{\partial
\overline{v}}{\partial p^i}-p^{\alpha }\left( [B_{\alpha },q]^L+
[ \overline{C}_{\alpha },q] ^L\right) \frac{\partial \overline{v
}}{\partial q^L}\\
&= F_{3}\big( X,\overline{C}_j,\overline{D}^{0i},\overline{D}^{ji},
\overline{v}\big),
\end{aligned}\\
\partial_0\overline{C}_i=F_{4}\big( X,\overline{C}_j,\overline{D}
^{0i},\overline{D}^{ji},\overline{v}\big) \quad
\text{on }\widehat{\Omega }_T,
\\
\overline{V}_1=0\quad \text{on }x^0=0,
\quad\text{where }X=(t,x^1,x^2,x^3,p^i,q^L)\in \widehat{\Omega }_T\,.
\end{gathered}
\end{equation}

\begin{definition} \label{def5.3} \rm
We will call linearized problem associated to problem
\eqref{ordinarycpbis},
to the given $\mathcal{C}^{\infty }$ functions $
\widehat{C}_j$, $\widehat{D}^{0i}$, $\widehat{D}^{ji}$,
$\widehat{v}$ defined on $\widehat{\Omega }_T$, the following
linear problem  defined in $\widehat{\Omega }_T$, of unknown
$\overline{V}_1=\big(
\overline{C}_i,\overline{D}^{0i},\overline{D}^{ij},\overline{v}\big)$:
\begin{gather}
\partial_0\overline{D}^{0i}+\partial_j\overline{D}^{ji}
= F_1\big( X,\widehat{C}_j,\widehat{D}^{0i},\widehat{D}^{ji},
\widehat{v}\big),  \label{linearcp1}
\\
  \partial_0\overline{D}^{ij}-\partial_i\overline{D}^{j0}-\partial_j
\overline{D}^{0i}
= F_2\big( X,\widehat{C}_j,\widehat{D}^{0i},\widehat{D}
^{ji},\widehat{v}\big), \label{linearcp2}
\\
\partial_0\overline{C}_i= F_{4}\big( X,\widehat{C}_j,\widehat{D}^{0i},
\widehat{D}^{ji},\widehat{v}\big) \quad\text{on }\widehat{\Omega
}_T, \label{linearcp3}
\\  \label{vlasoveq}
\begin{aligned}
&p^{\alpha }\frac{\partial \overline{v}}{\partial x^{\alpha }}+p^{\mu
}q\big( G_{\mu }^i+\overline{D}_{\mu }^i\big) \frac{\partial
\overline{v}}{\partial p^i}-p^{\alpha }\left( [B_{\alpha },q]^L+
[ \overline{C}_{\alpha },q] ^L\right) \frac{\partial \overline{v
}}{\partial q^L}\\
&= F_{3}\big( X,\widehat{C}_j,\widehat{D}^{0i},\widehat{D}^{ji},
\widehat{v}\big),
\end{aligned}
\\ \label{charactcondit}
\overline{V}_1=0\quad \text{on }x^0=0,\quad
\text{where }X=(t,x^1,x^2,x^3,p^i,q^L)\in \widehat{\Omega }_T\,.
\end{gather}
\end{definition}

\begin{proposition}\label{prop1}
If  $\widehat{v}\in \mathcal{C_0}^{\infty }( \widehat{
\Omega }_T)$ and $\widehat{C},\widehat{D} \in
\mathcal{C}^{\infty }({\Omega }_T)$, then the linear
system \eqref{linearcp1}, \eqref{linearcp2},
\eqref{linearcp3} under initial condition
$\overline{C}=0$, $\overline{D}=0$, on $x^0=0$,
admits in the domain ${\Omega}_T$ a unique
$\mathcal{C}^\infty $ solution with support contained in
${\sqcup}_T$.
\end{proposition}

\begin{proof}
 The subsystem \eqref{linearcp1}, \eqref{linearcp2},
\eqref{linearcp3} is a linear symmetric hyperbolic system of first
order with unknown $\overline{C}$ and $\overline{D}$ of which the
right hand side is $\mathcal{C}^{\infty }$ with support contained
in $\widehat{\sqcup }_T$. Thanks to Leray's theory \cite{25} of
hyperbolic systems, this system possesses in the domain
${\Omega }_T$ a unique $\mathcal{C}^{\infty }$ solution with
support contained in the future emission of
$\widehat{\sqcup }_T$ which is equal to $\widehat{\sqcup }_T$.
Hence $\operatorname{supp}\overline{C}
\subset \widehat{\sqcup }_T$, and
$\operatorname{supp}\overline{D}\subset \widehat{\sqcup }_T$.
\end{proof}

\begin{proposition}\label{prop2}
If $\widehat{C}$ and $\widehat{D}$ are
$\mathcal{C}^\infty $ on ${\Omega }_T$, then the linear equation
\eqref{vlasoveq} under the initial condition $\overline{v}={0}$ on
$x^0=0$ admits in $\widehat{\Omega }_T$ a unique
$\mathcal{C}^\infty $ solution $\overline{v}$ with compact support
contained in $\widehat{\sqcup }_T$.
\end{proposition}

We will mostly use the classical method of characteristics to solve
the problem considered in proposition \ref{prop2}.
The differential characteristic system associated to the first order
PDE \eqref{vlasoveq} is in fact
\begin{equation}\label{charact}
d\overline{x}^0=\frac{\overline{p}^0d\overline{x}^i}{
\overline{p}^i}=\frac{\overline{p}^0d\overline{p}^i}{\overline{p}^0
\overline{q}\big( G_{\mu }^i+\overline{D}_{\mu }^i\big) }
=\frac{d \overline{q}^L}{-\overline{p}^i[ B_i+\overline{C}_i,\overline{q}
] ^L}=d\tau ,
\end{equation}
with $\overline{p}^0=\big( m^2+\sum (\overline{p}^i) ^2\big) ^{1/2}$.
The solutions of
\eqref{charact} will be called characteristic curves.
The proof of proposition \ref{prop2} is based on the following
lemma.

\begin{lemma}\label{lem1}
(1) If $m > 0$ and $\widehat{D}$ is differentiable in
${\Omega }_T$, and $\widehat{D}$ and its derivatives are
bounded on ${\Omega}_T$, then the solution
$\tau \mapsto X (x_0^0,x_0^i,p_0^i,q_0^L,\tau )$ of the
differential characteristic system \eqref{charact} such that
$X\left( x_0^0,x_0^i,p_0^i,q_0^L,0\right)
 = X\left( x_0^0,x_0^i,p_0^i,q_0^L\right)$
is defined in the interval $]-x_0^0,T-x_0^0[$.

If moreover $\widehat{C}$ and $\widehat{D}$ are
$\mathcal{C}^\infty $ on ${\Omega }_T$, the function:
$X \left(x_0^0,x_0^i,p_0^i,q_0^L,0\right) \mapsto X \left(
x_0^0,x_0^i,p_0^i,q_0^L,\tau \right)$ is
$\mathcal{C}^\infty $.

(2) If $m =0$, $\operatorname{supp} \varphi \subset \{p^0>0 \}$ and
$p_0^0>0$, then the
solution of the differential system \eqref{charact} is defined in
the interval $]-x_0^0,\min(T-x_0^0,\varepsilon)[$, where
$\varepsilon$ is a strictly positive real number depending only on
the bounds $\widehat{D}$ on ${\Omega }_T$ and on $p_0^0$.
\end{lemma}

For a proof of the above lemma, see
\cite[Theorem of section 2]{7}.

\begin{proof}[Proof of Proposition \ref{prop2}]
From the methods of characteristics and Lemma
\ref{lem1}, the linear problem under consideration has,
in the domain $\widehat{\Omega }_T$, a unique
$\mathcal{C}^{\infty }$\ solution given by
\begin{equation}\label{obj0}
\begin{aligned}
\overline{v}\left( x_0^0,x_0^i,p_0^i,q_0^L\right)
&=\overline{v}\left( \overline{x}^0\left( 0\right) ,
\overline{x}^i\left( 0\right) ,
\overline{p}^i\left( 0\right) ,\overline{q}^L\left( 0\right) \right) \\
&=\int_{-x_0^0}^0[ \frac{1}{\overline{p}^0(s) }
G_{3}\left( \overline{x}^0(s) ,\overline{x}^i(s),\overline{p}^i(s) ,\overline{q}^L
(s) \right)] ds,
\end{aligned}
\end{equation}
with
\begin{equation}\label{obj1}
G_{3}\left( X\right)
=\begin{cases}
-p^{\alpha }(X)\frac{\partial h(X)}{\partial x^{\alpha }}
-(p^0qG^{i0}(X)-p^{j}qG_{ij}(X))\frac{\partial h(X)}{\partial p^i}
 \\
+p^{\alpha }[B_{\alpha },q]^L(X)\frac{\partial h(X)}{\partial q^L}
-(p^0q\widehat{D}^{i0}(X)
\\
-p^{j}q\widehat{D}_{ij}(X))  \frac{\partial h(X)}{\partial p^i}
+p^{\alpha }[\widehat{C}_{\alpha },q]^L(X)\frac{
\partial h(X)}{\partial q^L},
&\text{in }\widehat{\sqcup }_T, \\[3pt]
0&\text{in }\widehat{\Omega }_T\backslash \widehat{\sqcup }_T.
\end{cases}
\end{equation}
The support of $G_{3}$ is compact, contained in $\widehat{\sqcup }_T$,
since the support of $h$ is compact.

We will now show that the support of $\overline{v}$ is contained in
$\widehat{\sqcup }_T$. Let
$(x_0^0,x_0^i,p_0^i,q_0^L) \in \widehat{\Omega }_T\setminus
\widehat{\sqcup }_T$. It suffices, in view of the fact that
$\operatorname{supp} G_{3}$ is contained in $\widehat{\sqcup }_T$
(see \ref{obj1}), to show that the part of the characteristic
curve $\tau \mapsto X\left(x_0^0,x_0^i,p_0^i,q_0^L,\tau \right) $
originating from $\left( x_0^0,x_0^i,p_0^i,q_0^L\right) $ and
corresponding to parameters $s\in ] -x_0^0,0] $
is entirely contained in
$\widehat{\Omega }_T\setminus \widehat{\sqcup }_T$. For so doing,
it suffices to show, by setting
\begin{equation*}
X\left( x_0^0,x_0^i,p_0^i,q_0^L,\tau \right) \equiv
\left( x^0(\tau ),x^1(\tau ),p^i(\tau ),q^L(\tau )\right) ,
\end{equation*}
that $| \overline{x}^1(\tau) | >\overline{x
}^0(\tau) $ for all $\tau \in ]-x_0^0,0]$; this is an obvious
consequence of the following relations:
\[
x_0^0<| x_0^1|,\quad
\overline{x}^1(\tau)\tau) =x_0^1+\int_{\tau }^0
( \frac{\overline{p}^1}{\overline{p}^0}) (s) ds,
\quad
| \frac{\overline{p}^1}{\overline{p}^0}| \leq
1,\forall \tau \in] -x_0^0,0] .
\]
We will now show that $\operatorname{supp}\overline{v}$ is
compact.

According to the relations \eqref{obj0} and \eqref{obj1},
$\operatorname{supp}\overline{v}$ is contained in an
$\varepsilon $ -closed neighborhood of $\operatorname{supp} h$,
where $\varepsilon $ is a positive real number depending only
on $T$ and the bounds of the continuous and bounded
functions
$\overline{q}\big( G^{i0}+\overline{D}^{i0}\big)
-\frac{\overline{P}^{j}}{\overline{p}^0}\overline{q}
\left( G_{ij}+\overline{D}_{ij}\right) $ on
${\Omega }_T$. We then deduce that $\operatorname{supp}\overline{v}$
is compact, as support of $h$ is compact.
\end{proof}

\subsection{Functional spaces used for the resolution of \eqref{eq66}}

Let $s$ be an integer and $k$ a given real number such that $s>4$
and $k>3/2$.

\begin{definition}[\cite{7}] \label{def5.7} \rm
Let $\bar{D}=( \bar{D}_{\lambda \mu }) $ denote a
$2$-form defined on $\sqcup $.
$E^{s}(\Omega_T) $ is the closure of $\mathcal{C}^{\infty }
(\Omega_T) $ with respect to the norm
\begin{equation*}
\|\bar{D}\|_{E^{s}(\Omega_T)}=\sup_{0\leq \tau \leq T}
\|\bar{D}\|_{s}^{\tau },
\end{equation*}
with
\begin{equation*}
\|\overline{D}\|_{s}^{\tau }
=\Big\{ \int_{\omega_{\tau
}}\sum_{|r|\leq s}\big|\partial ^{r}\overline{D}\big|^2\mu_{\tau
}\Big\} ^{1/2},\quad |\partial ^{r}\overline{D}|
^2=\sum_{\lambda \leq \mu }(\partial ^{r}\overline{D}_{\lambda \mu }
)^2,
\end{equation*}
where
\begin{gather*}
\mu_{\tau } = dx^1dx^2dx^3,\quad
\partial ^{r}=\sum_{\mid \alpha \mid \leq r}
\frac{\partial ^{|\alpha|}}{(\partial x^0)^{\alpha_0}
(\partial x^1)^{\alpha_1}(\partial x^2)^{\alpha_2}(\partial
x^3)^{\alpha_{3}}}, \\
\alpha = (\alpha_0,\alpha_1,\alpha_2,\alpha_{3}),\quad
|\alpha |=\alpha_0+\alpha_1+\alpha_2+\alpha_{3}\,.
\end{gather*}
$E^{s,k}(\widehat{\Omega_T})$ is the closure of
$\mathcal{C}_0^{\infty}(\widehat{\Omega }_T) $
with respect to the norm
\begin{equation*}
\|\overline{v}\|_{E^{s,k}(\Omega_T)}=\sup_{0\leq
\tau \leq T}\|\overline{v}\|_{s,k}^{\tau },
\end{equation*}
with
\begin{equation*}
\|\overline{v}\|_{s,k}^{\tau }
=\Big\{ \sum_{|l|\leq s}\int_{
\widehat{\omega }_{\tau }}(p^0)^{2k+2(\widehat{l}+\widetilde{l})+1}
(D^{l}\overline{v})^2\beta_{\tau }\Big\}^{1/2},
\end{equation*}
where
\[
\widehat{\omega }_{\tau }
=w_{\tau }\times \mathbb{R}^3\times \mathcal{O},\quad
\beta_{\tau}=dx^1dx^2dx^3dp^1dp^2dp^3dq^1dq^2\dots dq^{N-1},
\]
\begin{align*}
&D^{l} \\
&=\sum_{\mid \alpha \mid \leq l}\frac{\partial ^{|\alpha |}}{
(\partial x^0)^{\alpha_0}(\partial x^1)^{\alpha
_1}(\partial x^2)^{\alpha_2}(\partial x^3)^{\alpha
_{3}}(\partial p^1)^{\alpha_{4}}\dots (\partial p^3)^{\alpha
_{6}}(\partial q^1)^{\alpha_{7}}\dots (\partial q^{N-1})^{\alpha
_{N+6}}}.
\end{align*}
\end{definition}

\begin{remark} \label{rem5} \rm
The functional spaces defined above are the same as
those defined in \cite{7}.
\end{remark}

\subsection{Energy inequalities for the linearized problems}

Problem \eqref{ordinarycp} is an ordinary Cauchy problem
defined in $\widehat{\Omega}_T$ with zero data specified on
the spatial hyperplane $x^0=0$. We can establish for the smooth
solution of this problem the same energy inequalities as those
given in \cite{7}. This energy
inequalities will be expressed in the functional spaces
$$
H^{s,k} ( \widehat{\Omega}_T ) \equiv ( E^{s}({\Omega}
_T))^{9}\times E^{s,k}(\widehat{\Omega}_T).
$$

\begin{proposition}\label{prop3}
If $\overline{C}$ and $\overline{D}\in \mathcal{C}^{\infty}
({\Omega }_T)$ satisfy the linearized Yang-Mills problem
\eqref{linearcp1}, \eqref{linearcp2}, \eqref{linearcp3} under
the initial condition $\overline{C}_i=0$, $\overline{D}^{0i}=0$,
$\overline{D}_{ij}=0$ on $x^0=0$, then for every $t\in ] 0,T] $,
the following inequality is satisfied
\begin{equation}
\|\overline{C}\|_{E^{s}(\Omega_{t})}+\|\overline{D}\|
_{E^{s}(\Omega_{t})}
\leq Ct\big[\|F(X,\widehat{C},\widehat{D},
\widehat{v})\|_{\mathcal{H}^{s,k}\left( \widehat{\Omega
}_{t}\right) }\big],  \label{eq3.0.25}
\end{equation}
where $C$ is a positive constant depending only on $T$.
\end{proposition}

For a proof of the above proposition see \cite{7}.

\begin{proposition}\label{prop4}
If $\widehat{v}\in \mathcal{C}_0^{\infty }( \widehat{\Omega }_T) $
satisfies the linearized Vlasov equation \eqref{vlasoveq}
with initial condition $\overline{v}=0$ on $x^0=0$, then, for all
$t\in ] 0,T] $, the following inequality is valid
\begin{equation}
\|\overline{v}\|_{E^{s,k}(\widehat{\Omega }_{t})}
\leq Ct\big[\|F (X,\widehat{C},\widehat{D})\|_{E^{s}(\Omega
_{t})}\big], \label{eq3.0.26}
\end{equation}
where $C$ is a positive constant depending only on $T$.
\end{proposition}

For a proof of the above proposition see \cite{7}.
To complete the resolution of problem \eqref{eq66},
let us consider
\begin{gather*}
g_0:\left( \mathcal{C}^{\infty }({\Omega }_T)
\right) ^{9}\times \mathcal{C}_0^{\infty }( {\widehat{\Omega
}}_T) \longrightarrow \left( \mathcal{C}^{\infty }({\Omega }_T)
\right) ^{9}\times \mathcal{C}_0^{\infty }( {\widehat{\Omega }}
_T)  \\
\widehat{V}=\big( \widehat{C}_j,\widehat{D}^{0i},\widehat{D}_{ij},
\widehat{v}\big) \longmapsto \overline{V}_1=\big( \overline{C}_j,
\overline{D}^{0i},\overline{D}_{ij},\overline{v}\big) ,
\end{gather*}
where $\overline{V}_1$ is the unique solution of the linearized
problem \eqref{linearcp1}, \eqref{linearcp2}, \eqref{linearcp3},
\eqref{vlasoveq}, \eqref{charactcondit}.

By using the denseness of
$\left( \mathcal{C}^{\infty }({\Omega }_T) \right) ^{9}
\times \mathcal{C}_0^{\infty }( { \widehat{\Omega }}_T) $
in $H^{s,k}(\widehat{\Omega }_T) $, and propositions
\ref{prop1} and \ref{prop2}, we can
obviously show that $g_0$ can be extended to a function:
\begin{align*}
g:H^{s,k}(\widehat{\Omega }_T) &\to
H^{s,k}(\widehat{\Omega }_T) \\
\widehat{V} &\mapsto \overline{V_1},
\end{align*}
where $\overline{V_1}$ is now the unique solution of the
linearized problem \eqref{linearcp1}, \eqref{linearcp2},
\eqref{linearcp3}, \eqref{vlasoveq}, \eqref{charactcondit}, with
$\widehat{V}$ belonging to $H^{s,k}(\widehat{\Omega }_T) $.
Then we show using again propositions \ref{prop1} and
\ref{prop2} that there exist some constants $R>0$, large enough,
$T_0\in] 0,T] $, small enough, such that $g$ is a contraction
from the closed ball $B( 0,R)$ of the Banach space
$H^{s,k}(\widehat{\Omega }_T) $ into itself; $g$ then
has a unique fixed
point $\overline{V}_1=\big( \overline{C}_i,\overline{D}^{0i},\overline{D
}_{ij},\overline{v}\big) $,
$\operatorname{supp}\overline{V}_1\subset \widehat{\sqcup }_{T_1}$
with $\operatorname{supp}\overline{v}$ compact. We can
also show by a classical argument \cite{6} that
$\overline{V}_1\in \left( \mathcal{C}^{\infty }\left( {\Omega
}_{T_0}\right) \right) ^{9}\times \mathcal{C}_0^{\infty
}( {\widehat{\Omega }}_{T_0}) $. $\overline{V}_1$
is then the unique solution of the problem \eqref{ordinarycp}

We have then proved the following theorem.

\begin{theorem} \label{thm5.11}
There exists $T_0\in ] 0,T] $, small enough, such that
the
evolution problem \eqref{reducedsyst}, (\eqref{eq18}, \eqref{eq19},
\eqref{eq20}, with initial data $\varphi $ satisfying hypothesis
$(I)$, admits in the domain $\widehat{\sqcup }_{T_0}$ a unique
$\mathcal{C}^{\infty }$  solution.
\end{theorem}

We sum up the whole work in the following Theorem.

\begin{theorem} \label{thm5.12}
For any free data $\overline{a}_i,\overline{b}^{j}$,
$(i=1,2,3;\,j=2,3)$ $\mathcal{C}^{\infty }$  on $H_1$,
$\overline{\varphi }$ $\mathcal{C}^{\infty }$ on
$\widehat{H}_1$, and $\widetilde{a}_i,\widetilde{b}^{j}$
$\mathcal{C}^{\infty }$ on $H_2$, $\widetilde{\varphi }$
$\mathcal{C}^{\infty }$ on $\widehat{H}_2$
and satisfying the following compatibility conditions
\begin{gather*}
\overline{a}_i(0,x^2,x^3) =\widetilde{a}_i(0,x^2,x^3) ,\quad
\text{where }(x^2,x^3) \in B,\,i=1,2,3;
\\
\overline{b}^{j}(0,x^2,x^3) =\widetilde{b}^{j}(0,x^2,x^3) ,\quad
j=2,3; \\
\big(\partial_1\overline{a}_j-\partial
_1\widetilde{a}_j\big)(0,x^2,x^3) =2\overline{b}^{j}(0,x^2,x^3) =2
\widetilde{b}^{j}(0,x^2,x^3); \\
\overline{\varphi }( 0,x^2,x^3,p^i,q^L)
=\widetilde{ \varphi }\left( 0,x^2,x^3,p^i,q^L\right) ,\quad
(p^i,q^L) \in \mathbb{R}^3\times O, \;
L=1,\dots ,N-1.
\end{gather*}
there exists $T_0\in ] 0,T] $, small enough, such that
the complete system \eqref{YMVsyst} of Yang-Mills-Vlasov equations
admits, in the domain $\widehat{\sqcup }_{T_0}$, a unique
$\mathcal{C}^{\infty }$\ solution $V=(A_0 = 0, A_i, F^{0i},
F_{ij}, f) $ satisfying the following conditions:
\begin{gather*}
A_i\big|_H =a_i=\begin{cases}
\overline{a}_i&\text{on} H_1 \\
\widetilde{a}_i&\text{on} H_2,
\end{cases}
\quad i=1,2,3;
\\
F^{0j}\big|_{H}=b^{j}=\begin{cases}
\overline{b}^{j}&\text{on} H_1 \\
\widetilde{b}^{j}&\text{on} H_2,
\end{cases}
\quad j=2,3;
\\
f\big|_{\widehat{H}}=\varphi =\begin{cases}
\overline{\varphi }&\text{on }\widehat{H}_1 \\
\widetilde{\varphi }&\text{on }\widehat{H}_2.
\end{cases}
\end{gather*}
\end{theorem}

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\end{thebibliography}

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