# The Maxwell-Boltzmann-Euler System with a Massive Scalar Field in All Bianchi Spacetimes

Authors: Raoul Domingo Ayissi; Norbert Noutchegueme; Hugues Paulin Mbeutcha Tchagna
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

We prove the existence and uniqueness of regular solution to the coupled Maxwell-Boltzmann-Euler system, which governs the collisional evolution of a kind of fast moving, massive, and charged particles, globally in time, in a Bianchi of types I to VIII spacetimes. We clearly define function spaces, and we establish all the essential energy inequalities leading to the global existence theorem.


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## Body

## 1. Introduction

In this paper, we study the coupled Maxwell-Boltzmann-Euler system which governs the collisional evolution of a kind of fast moving, massive, and charged particles and which is one of the basic systems of the kinetic theory.The spacetimes considered here are the Bianchi of types I to VIII spacetimes where homogeneous phenomena such as the one we consider here are relevant. Note that the whole universe is modeled and particles in the kinetic theory may be particles of ionized gas as nebular galaxies or even cluster of galaxies, burning reactors, and solar wind, for which only the evolution in time is really significant, showing thereafter the importance of homogeneous phenomena.The relativistic Boltzmann equation rules the dynamics of a kind of particles subject to mutual collisions, by determining theirdistribution function, which is a nonnegative real-valued function of  both the position and the momentum of the particles. Physically, this function is interpreted as the probability of the presence density of the particles in a given volume, during their collisional evolution. We consider the case of instantaneous, localized, binary, and elastic collisions. Here the distribution function is determined by the Boltzmann equation through a nonlinear operator called the collision operator. The operator acts only on the momentum of the particles and describes, at any time, at each point where two particles collide with each other, the effects of the behaviour imposed by the collision on the distribution function, also taking in account the fact that the momentum of each particle is not the same, before and after the collision, with only the sum of their two momenta being preserved.The Maxwell equations are the basic equations of  electromagnetism and determine the electromagnetic fieldF created by the fast moving charged particles. We consider the case where the electromagnetic field F is generated, through the Maxwell equations by the Maxwell current defined by the distribution function f of the colliding particles, a charge density e, and a future pointing unit vector u, tangent at any point to the temporal axis.The matter and energy content of the spacetime is represented by the energy-momentum tensor which is a function of the distribution functionf, the electromagnetic field F, and a massive scalar field Φ, which depends only on the time t.The Euler equations simply express the conservation of the energy-momentum tensor.The system is coupled in the sense thatf, which is subject to the Boltzmann equation, generates the Maxwell current in the Maxwell equations and is also present in the Euler equations, whereas the electromagnetic field F, which is subject to the Maxwell equations, is in the Lie derivative of f with respect to the vectors field tangent to the trajectories of the particles. F also figures in the Euler equations.We consider for the study all the Bianchi of types I to VIII spacetimes, excluding thereby the Bianchi type IX spacetime also called the Kantowski-Sachs spacetime which has the flaw to develop singularities in peculiar finite time and is not convenient for the investigation of global existence of solutions.The main objective of the present work is to extend the result obtained in [1–3] where the particular case of the Bianchi type I spacetime is investigated. The choice of function spaces and the process of establishing the energy inequalities are highly improved.The paper is organized as follows.In Section2, we introduce the spacetime and we give the unknowns.In Section3, we describe the Maxwell-Boltzmann-Euler system.In Section4, we define the function spaces and we establish the energy inequalities.In Section5, we study the local existence of the solution.In Section6, we prove the global existence of the solution.


## 2. The Spacetime and the Unknowns

Greek indexesα,β,γ,… range from 0 to 3, and Latin indexes i,j,k,… from 1 to 3. We adopt the Einstein summation convention:
(1)AαBα=∑α‍AαBα.We consider the collisional evolution of a kind of fast moving, massive, and charged particles in the time-oriented Bianchi types 1 to 8 spacetimes(ℝ4,g~) and denote by xα=(x0,xi)=(t,xi) the usual coordinates in ℝ4, where x0=t represents the time and (xi) the space; g~ stands for the given metric tensor of  Lorentzian signature (-,+,+,+) which writes
(2)g~=-(dt)2+gijdxidxj,
where gij are continuously differentiable functions on ℝ, components of a 3-symmetric metric tensor g=(gij), whose variable is denoted by t.The expression of the Levi-Civita connection∇ associated with g~, which is
(3)Γαβλ=12g~λμ[∂αg~μβ+∂βg~αμ-∂μg~αβ],gives directly(4)Γij0=12∂0gij,Γ0ji=12gil∂0gjl,i,j∈{1,2,3}Γαβλ=0otherwise.Recall thatΓαβλ=Γβαλ.We require the assumption that∂0gij/gij are bounded. This implies that there exists a constant C>0 such that
(5)|∂0gijgij|≤C.As a direct consequence, we have, fort∈ℝ+,
(6)|gij(t)|≤gij0eCt,|1gij(t)|≤1gij0eCt,
where gij0=gij(0).The massive particles have a rest massm>0, normalized to the unity, that is, m=1. We denote by T(ℝ4) the tangent bundle of ℝ4 with coordinates (xα,pβ), where p=(pβ)=(p0,p¯) stands for the momentum of each particle and p¯=(pi),i=1,2,3. Really the charged particles move on the future sheet of the mass shell or the mass hyperboloid P(ℝ4)⊂T(ℝ4), whose equation is Px(p):g~x(p,p)-1 or, equivalently, using expression (2) of g~:
(7)p0=1+gijpipj,
where the choice p0>0 symbolizes the fact that, naturally, the particles eject towards the future.Setting(8)ϱ=∑i=13‍(pi)2=ϱ(p¯),
if ϱ>1, the relations (6) and (7) also show that in any interval [0,T], T>0:
(9)Ap0≤ϱ≤Bp0,
where A=A(T)>0, B=B(T)>0 are constants.The invariant volume element inPx(p) reads
(10)ωp=|g~|1/2dp1dp2dp3p0,
where
(11)|g~|=|detgαβ|.We denote byf the distribution function which measures the probability of the presence of particles in the plasma. f is a nonnegative unknown real-valued function of both the position (xα) and the 4-momentum of the particles p=(pα), so:
(12)f:T(ℝ4)≈ℝ4×ℝ4⟶ℝ+,(xα,pα)↦f(xα,pα)∈ℝ+.We define a scalar product onℝ3 by setting for p=(p0,p¯)=(p0,pi) and q=(q0,q¯)=(q0,qi):
(13)p¯·q¯=gijpiqj.In this paper we consider the homogeneous case for whichf depends only on the time x0=t and p¯. According to the Laplace law, the fast moving and charged particles create an unknown electromagnetic field F which is a 2-closed antisymmetric form and locally writes
(14)F=Fαβdxα∧dxβ.So in the homogeneous case we consider(15)Fαβ:ℝ⟶ℝ,t↦Fαβ(t)∈ℝ.In the presence of the electromagnetic fieldF, the trajectories s↦(xα(s),pα(s)) of the charged particles are no longer the geodesics of spacetime (ℝ4,g) but the solutions of the differential system:
(16)dxαds=pα;dpαds=Pα,
where
(17)Pα=P(F,f)=-Γλμαpλpμ+epβFβα,
where e=e(t) denotes the charge density of particles.Notice that the differential system (16) shows that the vectors field X(F) defined locally by
(18)X(F)=(pα,Pα(F)),
where Pα is given by (17), is tangent to the trajectories.The charged particles also create a currentJ=(Jβ), β=0,1,2,3, called the Maxwell current which we take in the form
(19)Jβ=∫ℝ3‍pβfωp-euβ
in which u=(uβ) is a unit future pointing timelike vector, tangent to the time axis at any point, which means that u0=1, ui=ui=0, and i=1,2,3. The particles are then supposed to be spatially at rest.The electromagnetic fieldF=(F0i,Fij), where F0i and Fij stand for the electric and magnetic parts, respectively, is subject to the Maxwell equations.


## 3. The Maxwell-Boltzmann-Euler System inF, f, and Φ

### 3.1. The Maxwell Equations inF

The Maxwell system inF can be written, using the covariant notation:
(20)∇αFαβ=Jβ(21)∇αFβγ+∇βFγα+∇γFαβ=0.Equations (20) and (21) are, respectively, the first and second groups of the Maxwell equations, and ∇α stands for the convariant derivative in g. In (20), Jβ represents the Maxwell current we take in the form (19). Now the well-known identity ∇α∇βFαβ=0 imposes, given (20), that the current jβ is always subject to the conservation law:
(22)∇βJβ=0.However usingβ=0 in (20), we obtain since F=F(t), Fαλ=-Fλα and by (4) that
(23)J0=0.By (23), the expression (19) of Jβ in which we set β=0 then allows to compute e and gives, since u0=1,
(24)e(t)=∫ℝ3‍f(t,p¯)(detg)1/2dp¯,
which shows that f determines e.The second set (21) of the Maxwell equations is identically satisfied since F=F(t), and the first set reduces to ∂Fij=0. Then Fij is constant and
(25)Fij=Fij(0)=φij.This physically shows that the magnetic part ofF does not evolve and stays in its primitive state. It remains to determine the electric part F0i.Writing (19) for β=i, using (4), ωp=|g|1/2(dp1dp2dp3/p0), and ui=0, implies that
(26)Ji=∫ℝ3‍pif(t,p¯)(detg)1/2p0dp¯.By (20), we obtain the linear o.d.e in F0i which writes
(27)∂0F0i-HF0i=∫ℝ3‍pif(t,p¯)(detg)1/2p0dp¯.Remark 1.
In (27), the expression
(28)H=-12gkl∂0gkl
represents the second fundamental form in (ℝ3,g).  Really H is the trace of the 2-symmetric tensor K=(Kij) where Kij=-(1/2)∂0gij. H is called themiddle curvature of (ℝ4,g~). Since g=(gij) is given, so is H.


### 3.2. The Relativistic Boltzmann Equation inf

The relativistic Boltzmann equation inf, for charged particles in the Bianchi types 1 to 8 spacetimes, can be written:
(29)LXf=Q(f,f),
where LX is the Lie derivative of f with respect to the vectors field X(F) defined by (18) and Q(f,f), the collision operator we now introduce.According to Lichnerowicz and Chernikov, we consider a scheme, in which, at a given position(t,xi), only two particles collide with each other, without destroying each other, with the collision affecting only the momentum of each particle, which changes after shock, only the sum of the two momenta being preserved. If p,q stand for the two momenta before the shock and p′, q′ for the two momenta after the shock, then we have
(30)p+q=p′+q′.The collision operatorQ is then defined, using functions f and g on ℝ3, and the previous notations by
(31)Q(f,g)=Q+(f,g)-Q-(f,g),
where
(32)Q+(f,g)=∫ℝ3‍ωq¯∫S2‍f(p¯′)g(q¯′)σ(t,p¯,q¯,p¯′,q¯′,Ω)dΩQ-(f,g)=∫ℝ3‍ωq¯∫S2‍f(p¯)g(q¯)σ(t,p¯,q¯,p¯′,q¯′,Ω)dΩ
whose elements we now introduce step by step, specifying properties and hypotheses we adopt:(i)
S2 is the unit sphere of ℝ3, whose area element is denoted by dΩ;(ii)
σ is a nonnegative continuous real-valued function of all its arguments, called the collision kernel or the cross-section of the collisions, on which we require the boundedness and Lipschitz continuity assumptions, in which C1>0 is a constant:
(33)0≤σ(t,p¯,q¯,Ω)≤C1|σ(t,p¯1,q¯,p¯′,q¯′,Ω)-σ(t,p¯2,q¯,p¯′,q¯′,Ω)|≤C1∥p¯1-p¯2∥,
where ∥p¯∥=(∑i=13‍(pi)2)1/2=ϱ is the norm in ℝ3.(iii)
The conservation lawp+q=p′+q′ splits into
(34)p0+q0=p′0+q′0,(35)p¯+q¯=p¯′+q¯′.Equation (34) expresses, using (7), the conservation of the quantity:
(36)e~=1+gijpipj+1+gijqiqj
called the elementary energy of the unit rest mass particles; we can interpret (35) by setting, following Glassey and Strauss in [4, equation (42)],
(37)p¯′=p¯+a(p¯,q¯,Ω)Ωq¯′=q¯-a(p¯,q¯,Ω)Ω(Ω∈S2)
in which a(p¯,q¯,Ω) is a real-valued function. Using (7) to express p′0, q′0 in terms of   p¯′, q¯′ and next (37) to express p¯′, q¯′ in terms of  p¯,q¯, we prove that (34) leads to a quadratic equation in a, which solves to give the only nontrivial solution:
(38)a(p¯,q¯,Ω)=2p0q0e~Ω·(q¯^-p¯^)(e~)2-[Ω·(p¯+q¯)]2
in which p¯^=p¯/p0, e~ is given by (36), and the dot (·) is the scalar product defined by (13).It consequently appears, using (37), that the functions in the integrals (32) depend only on p¯, q¯, Ω and that these integrals with respect to q¯ and Ω give functions Q+(f,g) and Q-(f,g) of the single variable p¯.Using now the usual properties of the determinants, we compute the Jacobian of the change of variables(p¯,q¯)↦(p¯′,q¯′) defined by (37) and find
(39)∂(p′,q′)∂(p,q)=-p′0q′0p0q0.Butf=f(t,p¯),  so using (7), the Boltzmann equation (29) leads to the following form:
(40)∂f∂t+Pip0∂f∂pi=1p0Q(f,f).


### 3.3. The Euler Equations

The Euler equations only express the conservation of the energy-momentum tensorTαβ and write
(41)∇αTαβ=0.In (41),
(42)Tαβ=Tαβ1+ταβ+Tαβ2,
where
(43)Tαβ1=∫ℝ3‍pαpβf(t,p¯)(detg)1/2dp¯p0,(44)ταβ=-g~αβ4FλμFλμ+FβλFαλ,(45)Tαβ2=∇αΦ∇βΦ-g~αβ2[∇λΦ∇λΦ+m02Φ2].(i)
Tαβ1 is the energy-momentum tensor associated with f;(ii)
ταβ is the Maxwell tensor associated with F;(iii)
Tαβ2 is the energy-momentum tensor associated with the scalar field Φ whose mass is denoted by m0, with m0>0.Equation (42) shows that (41) writes
(46)∇αT1,αβ+∇αταβ+∇αT2,αβ=0.But it is proved in [5] that if f verifies the Boltzmann equation (40), then T1,αβ defined by (43) verifies ∇αT1,αβ=0; (46) reduces then to
(47)∇αταβ+∇αT2,αβ=0.Now, using (21), we have
(48)∇αταβ=Fλβ∇αFαλ,and using (45),
(49)∇αT2,αβ=∇βΦ(□g~Φ-m02),
where □g~=∇α∇α is the D'Alembertian.We deduce from (20), (48), and (49) that the Euler equations (41) are satisfied if Φ verifies the second-order differential equation:
(50)∇βΦ(□g~Φ-m02)+Fλβ∇αFαλ=0.Forβ=i, (50) leads to the constraints system:
(51)Fji∫ℝ3‍pjp0f(t,p¯)(detg)1/2dp¯=0,i=1,2,3,between the unknown functionsF and f, constraints which we have to solve in what is to follow.Forβ=0, (50) leads to a nonlinear differential equation of second order:
(52)Φ˙(Φ¨-HΦ˙+m02Φ)=gijF0j∫ℝ3‍pjp0f(t,p¯)(detg)1/2dp¯,
where H is defined in (27).Setting in (52)
(53)U=12(Φ˙)2,it comes that(54)U≥0Φ˙=±2U.One supposes in what follows that
Φ
is continuously differentiable, is not a constant, and is decreasing. This implies that
(55)
Φ
˙
=
-
2
U
Φ
(
t
)
≤
Φ
(
0
)
,
t
∈
ℝ
+
.Equation (52) is then equivalent to the nonlinear first-order differential system given as follows:
(56)Φ˙=-2U,(57)U˙=2HU+m02Φ2U+gijEi∫ℝ3‍pjp0f(t,p¯)(detg)1/2dp¯,
where Ei=F0i.


### 3.4. The Coupled System

From (17), using (4), we obtain
(58)Pip0=-2Γ0jipj+e[-Ei+gijpkFkjp0],i=1,2,3.Using (24), (27), (40), (56), and (57), the Maxwell-Boltzmann-Euler system in (F,f,Φ,U) reduces to the following form:
(59)E˙i=HEi+∫ℝ3‍pif(t,p¯)(detg)1/2p0dp¯,(60)∂f∂t+[-2Γ0jipj+(-Ei+gijpkφkjp0)×∫ℝ3‍f(t,p¯)(detg)1/2dp¯(-Ei+gijpkφkjp0)]∂f∂pi=1p0Q(f,f),(61)Φ˙=-2U,(62)U˙=2HU+m02Φ2U+gijEj∫ℝ3‍pip0f(t,p¯)(detg)1/2dp¯
which is an integrodifferential system to solve in what is to follow.We are searching a solution(F,f,Φ,U) of the Cauchy problem (59)-(60)-(61)-(62) globally in time on [0,+∞[ for the initial data:
(63)Ei(0)=E0i;Fij(0)=φij,f(0,p¯)=f0(p¯),p¯∈ℝ3;Φ(0)=Φ0;U(0)=U0.


### 3.5. The Problem of Constraints

We must find a nontrivial solution(F,f,Φ,U) of the Cauchy problem (59)-(60)-(61)-(62) satisfying the system (51) of constraints which writes after computation
(64)(g12J1-g11J2)φ21+(g13J1-g11J3)φ31+(g13J2-g12J3)φ32=0(g22J1-g21J2)φ21+(g23J1-g21J3)φ31+(g23J2-g22J3)φ32=0(g32J1-g31J2)φ21+(g33J1-g31J3)φ31+(g33J2-g32J3)φ32=0.


## 3.1. The Maxwell Equations inF

The Maxwell system inF can be written, using the covariant notation:
(20)∇αFαβ=Jβ(21)∇αFβγ+∇βFγα+∇γFαβ=0.Equations (20) and (21) are, respectively, the first and second groups of the Maxwell equations, and ∇α stands for the convariant derivative in g. In (20), Jβ represents the Maxwell current we take in the form (19). Now the well-known identity ∇α∇βFαβ=0 imposes, given (20), that the current jβ is always subject to the conservation law:
(22)∇βJβ=0.However usingβ=0 in (20), we obtain since F=F(t), Fαλ=-Fλα and by (4) that
(23)J0=0.By (23), the expression (19) of Jβ in which we set β=0 then allows to compute e and gives, since u0=1,
(24)e(t)=∫ℝ3‍f(t,p¯)(detg)1/2dp¯,
which shows that f determines e.The second set (21) of the Maxwell equations is identically satisfied since F=F(t), and the first set reduces to ∂Fij=0. Then Fij is constant and
(25)Fij=Fij(0)=φij.This physically shows that the magnetic part ofF does not evolve and stays in its primitive state. It remains to determine the electric part F0i.Writing (19) for β=i, using (4), ωp=|g|1/2(dp1dp2dp3/p0), and ui=0, implies that
(26)Ji=∫ℝ3‍pif(t,p¯)(detg)1/2p0dp¯.By (20), we obtain the linear o.d.e in F0i which writes
(27)∂0F0i-HF0i=∫ℝ3‍pif(t,p¯)(detg)1/2p0dp¯.Remark 1.
In (27), the expression
(28)H=-12gkl∂0gkl
represents the second fundamental form in (ℝ3,g).  Really H is the trace of the 2-symmetric tensor K=(Kij) where Kij=-(1/2)∂0gij. H is called themiddle curvature of (ℝ4,g~). Since g=(gij) is given, so is H.


## 3.2. The Relativistic Boltzmann Equation inf

The relativistic Boltzmann equation inf, for charged particles in the Bianchi types 1 to 8 spacetimes, can be written:
(29)LXf=Q(f,f),
where LX is the Lie derivative of f with respect to the vectors field X(F) defined by (18) and Q(f,f), the collision operator we now introduce.According to Lichnerowicz and Chernikov, we consider a scheme, in which, at a given position(t,xi), only two particles collide with each other, without destroying each other, with the collision affecting only the momentum of each particle, which changes after shock, only the sum of the two momenta being preserved. If p,q stand for the two momenta before the shock and p′, q′ for the two momenta after the shock, then we have
(30)p+q=p′+q′.The collision operatorQ is then defined, using functions f and g on ℝ3, and the previous notations by
(31)Q(f,g)=Q+(f,g)-Q-(f,g),
where
(32)Q+(f,g)=∫ℝ3‍ωq¯∫S2‍f(p¯′)g(q¯′)σ(t,p¯,q¯,p¯′,q¯′,Ω)dΩQ-(f,g)=∫ℝ3‍ωq¯∫S2‍f(p¯)g(q¯)σ(t,p¯,q¯,p¯′,q¯′,Ω)dΩ
whose elements we now introduce step by step, specifying properties and hypotheses we adopt:(i)
S2 is the unit sphere of ℝ3, whose area element is denoted by dΩ;(ii)
σ is a nonnegative continuous real-valued function of all its arguments, called the collision kernel or the cross-section of the collisions, on which we require the boundedness and Lipschitz continuity assumptions, in which C1>0 is a constant:
(33)0≤σ(t,p¯,q¯,Ω)≤C1|σ(t,p¯1,q¯,p¯′,q¯′,Ω)-σ(t,p¯2,q¯,p¯′,q¯′,Ω)|≤C1∥p¯1-p¯2∥,
where ∥p¯∥=(∑i=13‍(pi)2)1/2=ϱ is the norm in ℝ3.(iii)
The conservation lawp+q=p′+q′ splits into
(34)p0+q0=p′0+q′0,(35)p¯+q¯=p¯′+q¯′.Equation (34) expresses, using (7), the conservation of the quantity:
(36)e~=1+gijpipj+1+gijqiqj
called the elementary energy of the unit rest mass particles; we can interpret (35) by setting, following Glassey and Strauss in [4, equation (42)],
(37)p¯′=p¯+a(p¯,q¯,Ω)Ωq¯′=q¯-a(p¯,q¯,Ω)Ω(Ω∈S2)
in which a(p¯,q¯,Ω) is a real-valued function. Using (7) to express p′0, q′0 in terms of   p¯′, q¯′ and next (37) to express p¯′, q¯′ in terms of  p¯,q¯, we prove that (34) leads to a quadratic equation in a, which solves to give the only nontrivial solution:
(38)a(p¯,q¯,Ω)=2p0q0e~Ω·(q¯^-p¯^)(e~)2-[Ω·(p¯+q¯)]2
in which p¯^=p¯/p0, e~ is given by (36), and the dot (·) is the scalar product defined by (13).It consequently appears, using (37), that the functions in the integrals (32) depend only on p¯, q¯, Ω and that these integrals with respect to q¯ and Ω give functions Q+(f,g) and Q-(f,g) of the single variable p¯.Using now the usual properties of the determinants, we compute the Jacobian of the change of variables(p¯,q¯)↦(p¯′,q¯′) defined by (37) and find
(39)∂(p′,q′)∂(p,q)=-p′0q′0p0q0.Butf=f(t,p¯),  so using (7), the Boltzmann equation (29) leads to the following form:
(40)∂f∂t+Pip0∂f∂pi=1p0Q(f,f).


## 3.3. The Euler Equations

The Euler equations only express the conservation of the energy-momentum tensorTαβ and write
(41)∇αTαβ=0.In (41),
(42)Tαβ=Tαβ1+ταβ+Tαβ2,
where
(43)Tαβ1=∫ℝ3‍pαpβf(t,p¯)(detg)1/2dp¯p0,(44)ταβ=-g~αβ4FλμFλμ+FβλFαλ,(45)Tαβ2=∇αΦ∇βΦ-g~αβ2[∇λΦ∇λΦ+m02Φ2].(i)
Tαβ1 is the energy-momentum tensor associated with f;(ii)
ταβ is the Maxwell tensor associated with F;(iii)
Tαβ2 is the energy-momentum tensor associated with the scalar field Φ whose mass is denoted by m0, with m0>0.Equation (42) shows that (41) writes
(46)∇αT1,αβ+∇αταβ+∇αT2,αβ=0.But it is proved in [5] that if f verifies the Boltzmann equation (40), then T1,αβ defined by (43) verifies ∇αT1,αβ=0; (46) reduces then to
(47)∇αταβ+∇αT2,αβ=0.Now, using (21), we have
(48)∇αταβ=Fλβ∇αFαλ,and using (45),
(49)∇αT2,αβ=∇βΦ(□g~Φ-m02),
where □g~=∇α∇α is the D'Alembertian.We deduce from (20), (48), and (49) that the Euler equations (41) are satisfied if Φ verifies the second-order differential equation:
(50)∇βΦ(□g~Φ-m02)+Fλβ∇αFαλ=0.Forβ=i, (50) leads to the constraints system:
(51)Fji∫ℝ3‍pjp0f(t,p¯)(detg)1/2dp¯=0,i=1,2,3,between the unknown functionsF and f, constraints which we have to solve in what is to follow.Forβ=0, (50) leads to a nonlinear differential equation of second order:
(52)Φ˙(Φ¨-HΦ˙+m02Φ)=gijF0j∫ℝ3‍pjp0f(t,p¯)(detg)1/2dp¯,
where H is defined in (27).Setting in (52)
(53)U=12(Φ˙)2,it comes that(54)U≥0Φ˙=±2U.One supposes in what follows that
Φ
is continuously differentiable, is not a constant, and is decreasing. This implies that
(55)
Φ
˙
=
-
2
U
Φ
(
t
)
≤
Φ
(
0
)
,
t
∈
ℝ
+
.Equation (52) is then equivalent to the nonlinear first-order differential system given as follows:
(56)Φ˙=-2U,(57)U˙=2HU+m02Φ2U+gijEi∫ℝ3‍pjp0f(t,p¯)(detg)1/2dp¯,
where Ei=F0i.


## 3.4. The Coupled System

From (17), using (4), we obtain
(58)Pip0=-2Γ0jipj+e[-Ei+gijpkFkjp0],i=1,2,3.Using (24), (27), (40), (56), and (57), the Maxwell-Boltzmann-Euler system in (F,f,Φ,U) reduces to the following form:
(59)E˙i=HEi+∫ℝ3‍pif(t,p¯)(detg)1/2p0dp¯,(60)∂f∂t+[-2Γ0jipj+(-Ei+gijpkφkjp0)×∫ℝ3‍f(t,p¯)(detg)1/2dp¯(-Ei+gijpkφkjp0)]∂f∂pi=1p0Q(f,f),(61)Φ˙=-2U,(62)U˙=2HU+m02Φ2U+gijEj∫ℝ3‍pip0f(t,p¯)(detg)1/2dp¯
which is an integrodifferential system to solve in what is to follow.We are searching a solution(F,f,Φ,U) of the Cauchy problem (59)-(60)-(61)-(62) globally in time on [0,+∞[ for the initial data:
(63)Ei(0)=E0i;Fij(0)=φij,f(0,p¯)=f0(p¯),p¯∈ℝ3;Φ(0)=Φ0;U(0)=U0.


## 3.5. The Problem of Constraints

We must find a nontrivial solution(F,f,Φ,U) of the Cauchy problem (59)-(60)-(61)-(62) satisfying the system (51) of constraints which writes after computation
(64)(g12J1-g11J2)φ21+(g13J1-g11J3)φ31+(g13J2-g12J3)φ32=0(g22J1-g21J2)φ21+(g23J1-g21J3)φ31+(g23J2-g22J3)φ32=0(g32J1-g31J2)φ21+(g33J1-g31J3)φ31+(g33J2-g32J3)φ32=0.


## 4. Function Spaces and Energy Inequalities

We define now the function spaces in which we are searching the solution to the Maxwell-Boltzmann-Euler system. We also establish some useful energy estimations.Definition 2 (Esm(0,T,ℝ3)).
LetT>0, m∈ℕ, s∈ℝ be given.
We defineEsm(ℝ3) as
(65)Esm(ℝ3)={h:ℝ3⟶ℝ,(1+ϱ)s+|β|∂p¯βh∈L2(ℝ3),|β|≤m(1+ϱ)s+|β|∂p¯β}.
E
s
m
(
ℝ
3
) will be endowed with the norm
(66)∥h∥Esm(ℝ3)=max0≤|β|≤m∥(1+ϱ)s+|β|∂p¯βh∥L2(ℝ3).
E
¯
s
m
(
ℝ
3
) will be the completion of Esm(ℝ3) in the norm ∥·∥Esm(ℝ3).
We also define(67)Esm(0,T,ℝp¯3)={(1+ϱ)s+|β|∂p¯βy∈𝒞([0,T];ℰ(ℝp¯3)),(1+ϱ)s+|β|∂p¯βy(t,·)∈L2(ℝp¯3),|β|≤m,0≤t≤T}.Endowed with the norm(68)|∥y∥|Esm(0,T,ℝp¯3)=max0≤|β|≤msupt∈[0,T]∥(1+ϱ)s+|β|∂p¯βy(t,·)∥L2(ℝp¯3),E
s
m
(
0
,
T
,
ℝ
3
) is a Banach space.H
s
m
(
0
,
T
,
ℝ
p
¯
3
) will be the completion of Esm(0,T,ℝp¯3) for the norm |∥·∥|Esm(0,T,ℝp¯3).Forr>0 to be given, we define
(69)Es,rm={y∈Esm(0,T,ℝ3),y≥0a.e,|∥y∥|Esm(0,T,ℝp¯3)≤r}.Endowed with the induced distance by the norm|∥·∥|Esm(0,T,ℝp¯3), Es,rm is a complete metric subspace of Esm(0,T,ℝp¯3).Remark 3.
Ifm=0, then y∈Esm(ℝ3)⇔(1+ϱ)sy∈L2(ℝ3), so Es0(ℝ3) will be denoted by Ls2(ℝ3).Remark 4.
The reasons for the choice of the function spaceHsm(0,T,ℝp¯3) for m=3 and d>5/2.
With the objective of the present work being the existence of solution to the Maxwell-Boltzmann-Euler system, and particularly the Boltzmann equation (40), we are searching a function f=f(t,p¯) which is continuously differentiable; in particular we can search f=f(t,·) belonging to the space 𝒞b1(ℝp¯3).
We want to use the Faedo-Galerkin method which is applied for separable Hilbert spaces. That is the case for the Sobolev spacesHsm(0,T,ℝp¯3), m∈ℕ.We need then to find an integerm such that
(70)Hm(ℝp¯3)↪𝒞b1(ℝp¯3).But we know by the Sobolev theorems that(71)Wpm(ℝn)↪ℰk(ℝn),m>k+np.Since in our case we haven=3, p=2, and k=1(W2m=Hm), we must choose m such that
(72)m>1+32=52.The smallest integerm satisfying m>5/2 is naturally m=3.Consequently we have(73)Hd3(ℝp¯3)↪H3(ℝ3)↪𝒞b1(ℝp¯3).Furthermore if(74)d>52,then(75)Hdm(ℝp¯3)↪Ld2(ℝp¯3)↪L21(ℝp¯3),
where L21(ℝp¯3) is defined in [1].It then results that(76)Hdm(ℝp¯3)∩L21(ℝp¯3)=Hdm(ℝp¯3).We can now state the following results which will be fundamental.Lemma 5.
There exists a real numberT>0 such that
(77)(e~)2-[Ω·(p¯+q¯)]2>2.
Furthermore, one has(78)(e~)2-[Ω·(p¯+q¯)]2≥(1-∑i,j‍gij)(p0)2(e~)2-[Ω·(p¯+q¯)]2≥(1-∑i,j‍gij)(q0)2(e~)2-[Ω·(p¯+q¯)]2≥(1-∑i,j‍gij)p0q0,
and the function (p¯,q¯,Ω)↦Dp¯βc(p¯,q¯,Ω), 1≤|β|≤3 is bounded.Proof.
See [2].Proposition 6.
Letd>5/2, ∥σ∥L1(ℝ3×S2)∈L∞(ℝ3) and (∂βσ)(1+|p¯|)|β|-1∈L∞(ℝ3×ℝ3×S2), |β|≤3 be given.
Iff,g∈Hd3(ℝ3), then (1/p0)Q(f,g)∈Hd3(ℝ3), and one has
(79)∥1p0Q(f,g)∥Hd3(ℝ3)≤C∥f∥Hd3(ℝ3)∥g∥Hd3(ℝ3),
where C=C(T)>0.
Moreover(80)∥1p0Q(f,f)-1p0Q(g,g)∥Hd3(ℝ3)≤C(∥f∥Hd3(ℝ3)+∥g∥Hd3(ℝ3))∥f-g∥Hd3(ℝ3).Proof.
We simply use Lemma5. For the details, see [2].Proposition 7.
Letd>5/2,f∈Hd0(ℝ3) be given. Then |Ji|≤C(detg)1/2∥f∥Hd0(ℝ3), i=1,1,3.Proof.
See [2].Remark 8.
The hypothesis of Proposition6 concerning the collision kernel σ is a supplementary hypothesis for the investigation of the solution to the Boltzmann equation.In what is to follow, we are searching the local existence and the uniqueness of the solution to the Cauchy problem (59)-(60)-(61)-(62) in a function space which we will precise, applying the standard theory of first-order differential systems.The framework we will refer to forf is Hd3(0,T,ℝp¯3).The framework we will refer to forE¯ is ℝ3, whose norm is denoted by ∥·∥ or ∥·∥ℝ3:
(81)𝒞([[0,T];ℝ3])={h:[0,T]⟶ℝ3continuousandbounded}.𝒞
(
[
[
0
,
T
]
;
ℝ
3
]
) is a Banach space for the norm:
(82)|∥h∥|=sup{∥h(t)∥,t∈[0,T]}.The framework we will refer to forΦ and U is ℝ, whose norm is denoted by |·|(or∥·∥ℝ):
(83)𝒞([[0,T];ℝ])={h:[0,T]⟶ℝcontinuousandbounded}.𝒞
(
[
[
0
,
T
]
;
ℝ
]
) is a Banach space for the norm:
(84)|∥h∥|=sup{|h(t)|,t∈[0,T]}.(i)
We consider onℝ3×Hdm(0,T,ℝp¯3)×ℝ×ℝ the norm
(85)∥(E¯,f,Φ,U)∥=∥E¯∥+∥f∥+|Φ|+|U|.(ii)
We consider on𝒞([[0,T];ℝ3])×Hd3(0,T,ℝp¯3)×𝒞([[0,T];ℝ])×𝒞([[0,T];ℝ]) the norm:
(86)|∥(E¯,f,Φ,U)∥|=∥|E¯|∥+|∥f∥|+|∥Φ∥|+|∥U∥|.(iii)
We will consider the Cauchy problem (59)-(60)-(61)-(62) for the initial data:
(87)Ei(0)=E0i,Fij(0)=φij,f(0,p¯)=f0(p¯),p¯∈ℝ3;Φ(0)=Φ0;U(0)=U0,
where f0 is given in Hd,r3(0,T,ℝp¯3), E0i,φij, Φ0∈ℝ, i,j=1,2,3, and U0∈ℝ+.


## 5. The Local Existence of Solution

Theorem 9.
Letf~∈Hd,r3(0,T,ℝp¯3) be given, and let E¯~=(Ei~)∈ℝ3 be fixed. Then the linearized partial differential equation
(88)∂f∂t+[-2Γ0jipj+(-Ei~+pkp0gijφkj)×∫ℝ3‍f~(t,p¯)(detg)1/2dp¯(-Ei~+pkp0gijφkj)]∂f∂pi=1p0Q(f~,f~)
whose unknown is f, with f(0,p¯)=f0, has in Hd3(0,T,ℝp¯3) a local unique and bounded ⋆-weak solution.Proof.
We use the Faedo-Galerkin method in the function spaceHd3(0,T,ℝp¯3). For the other details, see [1].Theorem 10.
LetE¯~=(Ei~)∈ℝ3, i=1,2,3,  be fixed. Then the Boltzmann equation,
(89)∂f∂t+[-2Γ0jipj+(-Ei~+pkp0gijφkj)×∫ℝ3‍f(t,p¯)(detg)1/2dp¯(-Ei~+pkp0gijφkj)]∂f∂pi=1p0Q(f,f),
has in Hd3(0,T,ℝp¯3) a local unique ⋆-weak solution f such that f(0)=f0.Proof.
We use the Banach fixed point theorem inHd3(0,T,ℝp¯3) for the map:
(90)f~∈Hd,r3(0,T,ℝp¯3)↦Ξ(f~)=f,
where f satisfies (88).(i)
We firstly prove, using a sequence of approximations(fN) of f, the Banach-Alaoglu theorem and the fact that Hd3(0,T,ℝp¯3) is a reflexive space (see [1]) that we can choose ∥f0∥Hd,r3(0,T,ℝp¯3) and T>0 such that
(91)f~∈Hd,r3(0,T,ℝp¯3)⟹Ξ(f~)=f∈Hd,r3(0,T,ℝp¯3).(ii)
Let nowf1~,f2~∈Hd,r3(0,T,ℝp¯3) be given, and let f1,f2 be two solutions of (88). Then
(92)∂f1∂t+[-2Γ0jipj+(-Ei~+pkp0gijφkj)×∫ℝ3‍f1~(t,p¯)(detg)1/2dp¯pkp0]∂f1∂pi=1p0Q(f1~,f1~)∂f2∂t+[-2Γ0jipj+(-Ei~+pkp0gijφkj)×∫ℝ3‍f2~(t,p¯)(detg)1/2dp¯pkp0]∂f2∂pi=1p0Q(f2~,f2~).
LetG=f1-f2 and G~=f1~-f2~.
Then we get(93)∂G∂t+P~i(G~,E¯~)p0∂G∂pi=1p0Q(f1~,G~)-1p0Q(G~,f2~).
Conveniently using energy inequalities established in [1], the system (92), and remembering that G(0,p¯)=0, we obtain
(94)∥(1+|p¯|)d+|β|∂p¯βG(t)∥L2≤C2∫0t‍∥(1p0Q(f1~,G~)-1p0Q(G~,f2~))(1+|p¯|)d+|β|∂p¯β×(1p0Q(f1~,G~)-1p0Q(G~,f2~))∥L2(τ)dτ,
where C2=C(gij0,r,T,|E0i|,|φij|) is a positive constant.
Then taking thesup in (94), for t∈[0,T] and |β|≤3, we get
(95)∥G∥Hd3(0,T,ℝp¯3)≤C∫0T‍supt∈[0,T],|β|≤3(∥-1p0Q(G~,f2~))∥L2(1+|p¯|)d+|β|∂p¯β×(-1p0Q(G~,f2~))1p0Q(f1~,G~)-1p0Q(G~,f2~))∥L2)(τ)dτ
which implies(96)∥G∥Hd3(0,T,ℝp¯3)≤CT∥G~∥Hd3(0,T,ℝp¯3).
The relations (91), (96) show clearly that Hd3(0,T,ℝp¯3)→Hd3(0,T,ℝp¯3), f~↦Ξ(f~)=f is a contracting map, so by the Banach theorem Ξ has a unique fixed point f=f~ and the proof of Theorem 10 is complete.Next, let us introduce the subgroupG of 𝒪3 defined by
(97)G={(ɛ000cosθ-sinθ0sinθcosθ)Nɛ,θ∈𝒪3,Nɛ,θ=(ɛ000cosθ-sinθ0sinθcosθ),ɛ,θ∈ℝ,ɛ2=1(ɛ000cosθ-sinθ0sinθcosθ)}.A functiong on ℝ3 is said to be invariant under G if
(98)g(Np¯)=g(p¯);∀N∈G,∀p¯∈ℝ3.Using the observation thatp0  is invariant underG, it is proved in [6] that if f0 is invariant under G, then so will be the solution f of the Boltzmann equation satisfying f0(p¯)=f(0,p¯). It is also proved in [7] that Ji=∫ℝ3‍(pif(t,p¯)(detg)1/2/p0)dp¯=0, i=1,2,3 if and only if f0 is invariant under G.One requires in all what follows that the initial datum
f
0
=
f
(
0
;
·
)
of the distribution function
f
is not invariant under
G. 					The immediate consequence is that(99)Ji=∫ℝ3‍pif(t,p¯)(detg)1/2p0dp¯≠0,i=1,2,3.Now, computing the determinant of the system (64), we conclude that, under our requirement, the problem of constraints (64) admits on [0,T] a nontrivial solution:
(100)f≠0;φ12=φ13=φ23=0,
where f is the unique solution to the Boltzmann equation (60) on [0,T] in which E¯~ is given.Let us now state the following result which shows helpful in what is to follow.Proposition 11.
The Cauchy problem (59)-(60)-(61)-(62) is equivalent to the following problem, for i=1,2,3:
(101)(S){dEidt=HEi+∫ℝ3‍qif(detg)1/2q0dq¯(E1)dpidt=-2Γ0jipj+[-Ei+gijpkφkjp0]∫ℝ3‍f(detg)1/2dq¯(E2)dfdt=1p0(p¯)Q(f,f)(E3)dΦdt=-2U(E4)dUdt=2HU+m02Φ2U+gijEj∫ℝ3‍qif(detg)1/2q0dq¯(E5)Ei(0)=E0i;Fij(0)=φij,p¯(0)=p0¯,f(0,p¯)=f0(p¯),p¯∈ℝ3;Φ(0)=Φ0;U(0)=U0.Proof.
See [1].The framework we will refer to forp¯ is ℝ3, whose norm is denoted by ∥·∥ or ∥·∥ℝ3.LetI=(I1,I2,I3,I4,I5) denote the r.h.s. of (E1)-(E2)-(E3)-(E4)-(E5), that is,
(102)I1(t,E,¯p¯,f,Φ,U)=(HEi+∫ℝ3‍qif(detg)1/2q0dq¯)I2(t,E,¯p¯,f,Φ,U)=(-2Γ0jipj+[-Ei+gijpkφkjp0]∫ℝ3‍f(detg)1/2dq¯)I3(t,E,¯p¯,f,Φ,U)=1p0(p¯)Q(f,f,p¯)I4(t,E,¯p¯,f,Φ,U)=2UI5(t,E,¯p¯,f,Φ,U)=2HU+m02Φ2U+gijEj∫ℝ3‍qif(detg)1/2q0dq¯,i=1,2,3.It then appears that, on the contrary to the uncharged case studied in [6, 8], the momentum p=(p0,p¯) also becomes an unknown in the charged case. Note that f and p¯ are now independent variables for the system (E1)-(E2)-(E3)-(E4)-(E5). In this context, the collision operator Q defined by (31) will depend on p¯ only through the collision kernel σ, and we show it by writing now Q(f,f,p¯) instead of  Q(f,f)(p¯). One must from now be careful in order to avoid any confusion between the unknown p¯ of the system (E1)-(E2)-(E3)-(E4)-(E5) and the variable p¯ in (26), (27), (57), and (62), for example. For this reason, we denoted the variable in the integrals in (E1), (E2), and (E5) by q¯ instead of   p¯.Proposition 12.
Letp¯=(pi), pj¯=(pji)∈ℝ3, j=1,2, f∈Hd3(ℝ3) be given. Then
(103)p0≥gii|pi|,i=1,2,3|p1kp10-p2kp20|≤B1∥p1¯-p2¯∥|1p10-1p20|≤B2∥p1¯-p2¯∥pj0∥1pj0Q(f,f,p1¯)-1pj0Q(f,f,p2¯)∥≤B3∥f∥∥p1¯-p2¯∥,
where
(104)B1=2∑i‍gii∑j‍1gjj+2∑i,j‍gij(∑k‍1gkk)2+1,B2=2∑i‍gii+2∑i,j‍gij∑k‍1gkk,B3=B3(T;gij0).Proof.
See [7].We prove the following.Proposition 13.
LetE1¯,E2¯,p1¯,p2¯∈ℝ3,  Φ1,Φ2,U1,U2∈ℝ, f1,f2∈Hd3(0,T,ℝp¯3) be given. Then
(105)∥I1(t,E1¯,p1¯,f1,Φ1,U1)-I1(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ3≤C3(∥E1¯-E2¯∥ℝ3+∥f1-f2∥Hd3(0,T,ℝp¯3))(106)∥I2(t,E1¯,p1¯,f1,Φ1,U1)-I2(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ3≤C4(∥f1-f2∥Hd3(0,T,ℝp¯3)∥E1¯-E2¯∥ℝ3+∥p1¯-p2¯∥ℝ3+∥f1-f2∥Hd3(0,T,ℝp¯3))(107)∥I3(t,E1¯,p1¯,f1,Φ1,U1)-I3(t,E2¯,p2¯,f2,Φ2,U2)∥Hd3(0,T,ℝp¯3)≤C5(∥p1¯-p2¯∥ℝ3+∥f1-f2∥Hd3(0,T,ℝp¯3))(108)∥I4(t,E1¯,p1¯,f1,Φ1,U1)-I4(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ≤C6∥U1-U2∥ℝ(109)∥I5(t,E1¯,p1¯,f1,Φ1,U1)-I5(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ≤C7(∥E¯1-E¯2∥ℝ3+∥f1-f2∥Hd3(0,T,ℝp¯3)+∥Φ1-Φ2∥ℝ+∥U1-U2∥ℝ∥f1-f2∥Hd3(0,T,ℝp¯3)),
where
(110)C3=C(|H|+(detg)1/2),C4=C(∑i,j‍gij)2+C(detg)1/2(∥f1∥+∥E2¯∥)+C(detg)1/2∑i,j‍gij∑k,j‍|φkj|(B1+∑k‍1gkk),C5=B3+C(T,gij0)(detg)1/2(∥f1∥+∥f2∥)+CB2,C6=2U1+U2,C7=C|H|+m02|Φ2|2U1+U2+m022U1+C(detg)1/2∑i,j‍gij(∥E¯2∥+∥f1∥).Proof.
(a) We have, using (102),
(111)I1(t,E1¯,p1¯,f1,Φ1,U1)-I1(t,E2¯,p2¯,f2,Φ2,U2)=(qi(f1(t,p¯)-f2(t,p¯))(detg)1/2q0dq¯H(E1i-E2i)+∫ℝ3‍qi(f1(t,p¯)-f2(t,p¯))(detg)1/2q0dq¯).
So by (5) and Proposition 7,
(112)|H(E1i-E2i)+∫ℝ3‍qi(f1(t,p¯)-f2(t,p¯))(detg)1/2q0dq¯|≤|H||E1i-E2i|+C(detg)1/2gii∥f1-f2∥Ld2(ℝ3).
It follows by (6) that
(113)∥I1(t,E1¯,p1¯,f1,Φ1,U1)-I1(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ3≤C(|H|+(detg)1/2)×(∥E¯1-E¯2∥ℝ3+∥f1-f2∥Hd3(ℝ3)).
(b) We still have, by (102),
(114)I2(t,E1¯,p1¯,f1,Φ1,U1)-I2(t,E2¯,p2¯,f2,Φ2,U2)=(2Γ0ji(p2j-p1j))+((detg)1/2gijφkj[p2kp20∫ℝ3‍f2dq¯-p1kp10∫ℝ3‍f1dq¯]+(detg)1/2[E2i∫ℝ3‍f2dq¯-E1i∫ℝ3‍f1dq¯][p2kp20∫ℝ3‍f2dq¯-p1kp10∫ℝ3‍f1dq¯]).
But (5) gives
(115)|2Γ0ji(p2j-p1j)|≤C(∑i,j‍gij)2∥p1¯-p2¯∥ℝ3.
Invoking Proposition12, we have
(116)|(detg)1/2gijφkj[p2kp20∫ℝ3‍f2dq¯-p1kp10∫ℝ3‍f1dq¯]|≤C(detg)1/2(∥f1∥+∥E2¯∥)(∥p1¯-p2¯∥ℝ3+∥f1-f2∥),|(detg)1/2[E2i∫ℝ3‍f2dq¯-E1i∫ℝ3‍f1dq¯]|≤(detg)1/2|gijφkj(p2kp20-p1kp10)∫ℝ3‍f2dq¯+p1kp10∫ℝ3‍(f2-f1)dq¯(p2kp20-p1kp10)∫ℝ3‍f2dq¯|≤(detg)1/2∑i,j‍gij∑k,j‍|φkj|×(∑k‍CgkkCB1∥f1∥∥p1¯-p2¯∥ℝ3+∑k‍Cgkk∥f1-f2∥).
So by addition, we conclude that (106) holds.
(c) We also have, using (102),
(117)I3(t,E1¯,p1¯,f1,Φ1,U1)-I3(t,E2¯,p2¯,f2,Φ2,U2)=1p10Q(f1,f1,p1¯)-1p20Q(f2,f2,p2¯)=1p10(Q(f1,f1,p1¯)-Q(f2,f2,p1¯))+1p10(Q(f2,f2,p1¯)-Q(f2,f2,p2¯))+(1p10-1p20)Q(f2,f2,p2¯).
By Proposition6, we have
(118)∥1p10(Q(f1,f1,p1¯)-Q(f2,f2,p1¯))∥Hd3(0,T,ℝp¯3)≤C(T,gij0)(∥f1∥+∥f2∥)∥f1-f2∥.
Using also Proposition6, we find
(119)∥1p10(Q(f2,f2,p1¯)-Q(f2,f2,p2¯))∥Hd3(0,T,ℝp¯3)≤B1∥f2∥2∥p1¯-p2¯∥ℝ3.
Still using Proposition6 and invoking Proposition 12, we find
(120)∥(1p10-1p20)Q(f2,f2,p2¯)∥Hd3(0,T,ℝp¯3)≤CB2∥f2∥2∥p1¯-p2¯∥ℝ3.
Adding the last three inequalities, we obtain (107).
(d) Similarly, by (102),
(121)I4(t,E1¯,p1¯,f1,Φ1,U1)-I4(t,E2¯,p2¯,f2,Φ2,U2)=2U1-2U2=2U1+U2(U1-U2).
So(122)|I4(t,E1¯,p1¯,f1,Φ1,U1)-I4(t,E2¯,p2¯,f2,Φ2,U2)|≤2U1+U2|U1-U2|.
(e) Finally by (102) we have
(123)I5(t,E1¯,p1¯,f1,Φ1,U1)-I5(t,E2¯,p2¯,f2,Φ2,U2)=2H(U1-U2)+m02(Φ12U1-Φ22U2)+(gij(detg)1/2(E1j∫ℝ3‍qif1(t,p¯)q0dq¯-E2j∫ℝ3‍qif2(t,p¯)q0dq¯))=2H(U1-U2)+m02Φ1(2U1-2U2)+m022U2(Φ1-Φ2)+gij(detg)1/2∫ℝ3‍qif1(t,p¯)q0dq¯(E1j-E2j)+gijEj(∫ℝ3qi(f1(t,p¯)-f2(t,p¯))q0dq¯‍).
So by Propositions6 and 7, using (2) and the inequalities obtained for I1,I2,I4,
(124)|I5(t,E1¯,p1¯,f1,Φ1,U1)-I5(t,E2¯,p2¯,f2,Φ2,U2)|≤C7(∥E¯1-E¯2∥ℝ3+∥f1-f2∥+∥Φ1-Φ2∥ℝ+∥U1-U2∥ℝ),
where
(125)C7=C|H|+m02|Φ2|2U1+U2+m022U1+C(detg)1/2∑i,j‍gij(∥E¯2∥+∥f1∥).
This completes the proof of Proposition13.One requires in what follows that, for any real number
T
>
0,
(126)U(t)>U02,∀t∈[0,T].Theorem 14.
Lett0≥0, (Et0¯,pt0¯,ft0,Φt0,Ut0)∈ℝ3×ℝ3×Hd3(0,T,ℝp¯3)×ℝ×ℝ be given. Then the following holds.
There exists a real numberδ>0 such that the Cauchy problem (S) has a unique solution:
(127)(E¯,p¯,f,Φ,U)∈(𝒞([[t0,t0+δ];ℝ3]))2×Hd3(t0,t0+δ,ℝp¯3)×(𝒞([[t0,t0+δ];ℝ]))2
satisfying (E¯,p¯,f,Φ,U)(t0)=(Et0¯,pt0¯,ft0,Φt0,Ut0). Moreover, f satisfies the relation:
(128)|∥f∥|=sup{∥f(t)∥,t∈[t0,t0+δ]}≤|∥ft0∥|.Proof.
We apply the standard theory on the first-order differential systems to(E1)-(E2)-(E3)-(E4)-(E5).
Sincegij, ∂0gij, 1/gij, σ are continuous functions of  t, so is the function
(129)I(t,E¯,p¯,f,Φ,U)=(I1,I2,I3,I4,I5).
By continuity of the functionsz=gij,1/gij at t=t0, there exists a real number δ0>0 such that
(130)t∈]t0-δ0,t0+δ0[⟹|z(t)|≤|z(t0)|+1.
The previous relation implies, using (5) and (6) to bound z=gij,∂0gij,1/gij, that
(131)t∈]t0-δ0,t0+δ0[⟹|z(t)|≤(∑i,j‍gij0+∑i,j‍1gij0)eCt0+1.
Next, set(132)B(Et0¯,1)={E¯∈ℝ3,∥E¯-Et0¯∥<1},B(ft0,1)={f∈Hd3(0,T,ℝp¯3),∥f-ft0∥<1}.
Then(133)f∈B(ft0,1)⟹∥f∥≤∥ft0∥+1,E¯∈B(Et0¯,1)⟹∥E¯∥<∥Et0¯∥+1.
Now consider the neighborhoodVt0=]t0-δ0,t0+δ0[ × B(Et0¯,1)×ℝ3×B(ft0,1) × ]-Φt0+1,Φt0+1[×]-Ut0+1,Ut0+1[ of (t0,Et0¯,pt0¯,ft0,Φt0,Ut0) in the Banach space ℝ×ℝ3×ℝ3×Hd3(0,T,ℝp¯3)×ℝ×ℝ and take
(134)(t,E1¯,p1¯,f1,Φ1,U1),(t,E2¯,p2¯,f2,Φ2,U2)∈Vt0.
We deduce from the inequalities (105), (106), (107), (108), and (109) the definitions of  C3, C4, C5, C6, C7, the implications
(135)t∈]t0-δ0,t0+δ0[⟹|z(t)|≤(∑i,j‍gij0+∑i,j‍1gij0)eCt0+1,E¯∈B(Et0¯,1)⟹∥E¯∥<∥Et0¯∥+1,f∈B(ft0,1)⟹∥f∥≤∥ft0∥+1,Φ∈]-Φt0+1,Φt0+1[⟹|Φ|<|Φt0|+1,U∈]-Ut0+1,Ut0+1[⟹U<Ut0+1,
and the relation  (126) that there exists a constant
(136)C8=C8(t0,gij0,ft0,|Et0i|,|φij|,|Φt0|,Ut0)
such that(137)∥I(t,E1¯,p1¯,f1,Φ1,U1),-I(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ3×ℝ3×Hd3(0,T,ℝp¯3)×ℝ×ℝ≤C8(∥E¯1-E¯2∥ℝ3+∥p¯1-p¯2∥ℝ3+∥f1-f2∥Hd3(0,T,ℝp¯3)+∥Φ1-Φ2∥ℝ+∥U1-U2∥ℝ∥f1-f2∥Hd3(0,T,ℝp¯3))
which shows that I=(I1,I2,I3,I4,I5) is locally Lipschitzian in (E¯,p¯,f,Φ,U) with respect to the norm of the Banach space ℝ3×ℝ3×Hd3(0,T,ℝp¯3)×ℝ×ℝ. The existence of a unique solution (E¯,p¯,f,Φ,U) of the differential system (E1)-(E2)-(E3)-(E4)-(E5) on an interval [t0,t0+δ], δ>0, such that (E¯,p¯,f,Φ,U)(t0)=(Et0¯,pt0¯,ft0,Φt0,Ut0), is guaranteed by the standard theory on the first-order differential systems.
The relation|∥f∥|=sup{∥f(t)∥,t∈[0,δ]}≤|∥ft0∥| is established in [8].
As a direct consequence, we can deduce that there exists a real numberδ>0 such that the Maxwell-Boltzmann-Euler system (20)-(21)-(40)-(52) in all Bianchi types 1 to 8 spacetimes has a unique solution (F,f,Φ) on [0,δ] satisfying
(138)F0i(0)=E0i;Fij(0)=φij;f(0)=f0,Φ(0)=Φ0.


## 6. The Global Existence

### 6.1. The Method

Let[0,T[ be the maximal existence domain of solution, denoted here by (E¯~,p¯~,f~,Φ~,U~) and given by Theorem 14, of the system (E1)-(E2)-(E3)-(E4)-(E5), with the initial data (E0¯,p0¯,f0,Φ0,U0)∈ℝ3×ℝ3×Hd3(0,T,ℝp¯3)×ℝ×ℝ. We intend to prove that T=+∞.(a)
If we already haveT=+∞, then the problem of global existence is solved.(b)
We are going to show that if we supposeT<+∞, then the solution (E¯~,p¯~,f~,Φ~,U~) can be extended beyond T, which contradicts the maximality of T.(c)
The method is as follows: suppose0<T<+∞ and let t0∈[0,T[. We want to show that there exists a strictly positive number δ>0 independent of t0 such that the system (E1)-(E2)-(E3)-(E4)-(E5) on [t0,t0+δ], with the initial data (E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)) at t=t0, admits a unique solution (E¯,p¯,f,Φ,U) on [t0,t0+δ]. Then, by taking t0 sufficiently close to T, for example, t0 such that 0<T-t0<(δ/2), and hence T<t0+(δ/2), we can extend the solution (E¯~,p¯~,f~,Φ~,U~) to [0,t0+(δ/2)], which strictly contains [0,T[, and this contradicts the maximality of T. For the need to simplify the notations, it will be enough if we could look for a number δ such that 0<δ<1.(d)
In what follows we fix a numberr>0 and we take f0 such that ∥f0∥≤r.By (128) we can write
(139)∥ft0~∥≤∥f0∥.We also have from (139) using (128) that any solution f of the Boltzmann equation on [t0,t0+δ] such that f(t0)=f~(t0), satisfies the inequality:
(140)∥f(t)∥≤r,t∈[t0,t0+δ].Also notice that (140) shows that a solution (E¯,p¯,f,Φ,U) of the system (E1)-(E2)-(E3)-(E4)-(E5) on [t0,t0+δ], δ>0, such that (E¯,p¯,f,Φ,U)(t0)=(E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)), satisfies
(141)(E¯,p¯,f,Φ,U)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)×(𝒞([t0,t0+δ];ℝ))2.In what follows,[0,T[,T>0 is the maximal existence domain of solution (E¯~,p¯~,f~,Φ~,U~) of (E1)-(E2)-(E3)-(E4)-(E5) such that
(142)(E¯~,p¯~,f~,Φ~,U~)(0)=(E¯0~,p0¯~,f0~,Φ0~,U0~)∈ℝ3×ℝ3×Hd,r3(0,T,ℝp¯3)×ℝ×ℝ.The following result shows helpful in what is to follow.Lemma 15.
t
↦
E
¯
~
(
t
), t↦p¯~(t), t↦U~(t), and t↦Φ~(t) are uniformly bounded over [0,T[.Proof.
See [1].


### 6.2. Global Existence of Solutions

First of all, we consider, fort0∈[0,T[ and δ>0,
(143)(E¯¯,p¯¯,f¯)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3).Then, we built from system(E1)-(E2)-(E3)-(E4)-(E5) by setting in its r.h.s.I=(I1,I2,I3,I4,I5) which is given by (102) f=f¯ in I1, E¯=E¯¯, f=f¯ in I2, p¯=p¯¯ in I3, and f=f¯, E¯=E¯¯, in I5, the following useful differential system:
(144)dEidt=I1¯(t,E¯,p¯,f¯,Φ,U),(145)dpidt=I2¯(t,E¯¯,p¯,f¯,Φ,U),(146)dfdt=I3¯(t,E¯,p¯¯,f,Φ,U),(147)dΦdt=I4¯(t,E¯,p¯,f,Φ,U),(148)dUdt=I5¯(t,E¯¯,p¯,f¯,Φ,U),
where
(149)I1¯(t,E¯,p¯,f¯,Φ,U)=(HEi+∫ℝ3‍qif¯(detg)1/2q0dq¯)I2¯(t,E¯¯,p¯,f¯,Φ,U)=(-2Γ0jipj+[-Ei¯+gijpkφkjp0]∫ℝ3‍f¯(detg)1/2dq¯)I3¯(t,E¯,p¯¯,f,Φ,U)=1p0(p¯¯)Q(f,f,p¯¯)I4¯(t,E¯,p¯,f,Φ,U)=-2UI5¯(t,E¯¯,p¯,f¯,Φ,U)=2HU+m02Φ2U+gijEj¯∫ℝ3‍qif¯(detg)1/2q0dq¯,i=1,2,3.We prove the following.Proposition 16.
Lett0∈[0,T[, δ∈]0,1[, and (E¯¯,p¯¯,f¯)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3) be given. Then, the differential system (144)-(145)-(146)-(147)-(148) has a unique solution (E¯,p¯,f,Φ,U)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)×(𝒞([t0,t0+δ];ℝ))2 such that (E¯,p¯,f,Φ,U)(t0)=(E¯t0~,pt0¯~,ft0~,Φt0~,Ut0~).Proof.
(a) We consider (144) in E¯, with I1¯ defined by (149) in which f¯ is fixed. Since gij, 1/gij, ∂0gij, f¯ are continuous functions of  t, so is I1¯. Next, we deduce from (105) in which we set f1=f2=f¯ that
(150)∥I1(t,E1¯,p1¯,f¯,Φ1,U1)-I1(t,E2¯,p2¯,f¯,Φ2,U2)∥ℝ3≤C3∥E¯1-E¯2∥,
where C3=C(|H|+(detg)1/2).
Now we can use (5) and (6) to bound z=gij, 1/gij, ∂0gij and we obtain, for t∈[t0,t0+δ], then t≤t0+δ≤T+1,
(151)|z(t)|≤(C+∑i,j‍gij0)eC(T+1)+1,t∈[t0,t0+δ],z=gij,∂0gij.
We then deduce from (151) that
(152)C3≤C3′,whereC3′=C3′(gij0,T).
By (150) and (152), I1¯ is (globally) Lipschitzian with respect to the ℝ3-norm and the local existence of a solution E¯ of (144) such that E¯(t0)=E¯~(t0) is guaranteed by the standard theory of first-order differential systems.
Now, sinceE¯ satisfies (144) in which I1¯ is given by (149), following the same way as in the proof of Lemma 15, substituting E¯ to E¯~, f¯ to f~, and integrating this time over [t0,t0+t],t∈[0,δ[, lead to
(153)|Ei(t0+t)|≤(|Ei~(t0)|+C9iT)+C∫t0t0+t‍|Ei|(s)ds,|Ei(t0+t)|≤Ei(t0+t)Eikkit∈[0,δ[,i=1,2,3,
where C9i=C9i(gij0,T,r,|E0i|). However, by Lemma 15, we have, since t0[0,T[:|Ei~(t0)|≤(|E0i|+C9iT)eCT. Then, by Gronwall inequality,
(154)|Ei(t0+t)|≤((|E0i|+C9iT)eCT+C11iT)eC(T+1),|Ei(t0+t)|≤|Ei(t0+t)|it∈[0,δ[,i=1,2,3,
where (see [1]) C10i, C11i are two constants appearing in Lemma 15 when we bound t↦p¯~(t), which shows that every solution E¯ of (144) is uniformly bounded. By the standard theory of first-order differential systems, the solution E¯ is defined all over [t0,t0+δ[ and E¯∈𝒞([t0,t0+δ[;ℝ3).
(b) We also consider (145) in p¯, with I2¯ defined by (149) in which E¯¯, f¯ are fixed. Since gij, 1/gij, ∂0gij, E¯¯, f¯ are continuous functions of t, so is I2¯. Next, we deduce from (106) in which we set E1¯=E2¯=E¯¯, f1=f2=f¯ that
(155)∥I2(t,E¯¯,p1¯,f¯,Φ1,U1)-I2(t,E¯¯,p2¯,f¯,Φ2,U2)∥ℝ3≤C4∥p¯1-p¯2∥,
where
(156)C4=C(∑i,j‍gij)2+C(detg)1/2(∥f1∥+∥E2¯∥)+C(detg)1/2∑i,j‍gij∑k,j‍|φkj|(B1+∑k‍1gkk).
Now we can use (5) and (6) to bound z=gij,1/gij and we obtain, for t∈[t0,t0+δ], then t≤t0+δ≤T+1,
(157)|z(t)|≤(∑i,j‍gij0+∑i,j‍1gij0)eC(T+1)+1,t∈[t0,t0+δ],z=gij,1gij.
We then deduce from (157), using
(158)|Ei¯(t0+t)|≤((|E0i|+C9iT)eCT+C11iT)eC(T+1),t∈[0,δ[,i=1,2,3,
sinceE¯¯=E1¯=E2¯ is uniformly bounded, and ∥f¯∥≤∥|f¯|∥≤r, since f¯∈Hd,r3(t0,t0+δ,ℝ3) that
(159)C4≤C4′,whereC4′=C4′(gij0,|E0i|,|φij|,T,r).
By (155) and (159), I2¯ is (globally) Lipschitzian with respect to the ℝ3-norm and the local existence of a solution p¯ of (145) such that p¯(t0)=p¯~(t0) is guaranteed by the standard theory of first-order differential systems.
Now, sincep¯ satisfies (152) in which I2¯ is given by (149), following the same way as in the proof of Lemma 15, substituting E¯¯ to E¯~, p¯ to p¯~, and f¯ to f~, and integrating this time over [t0,t0+t], t∈[0,δ[, lead to
(160)|pi(t0+t)|≤(|pi~(t0)|+C12iT)+C∫t0t0+t‍|pi|(s)ds,t∈[0,δ[,i=1,2,3,
where C12i=C12i(gij0,|E0i|,|φij|,T,r). However, by Lemma 15, we have, since t0[0,T[:|pi~(t0)|≤(|p0i|+C8iT)eCT. Then, by Gronwall inequality,
(161)|pi(t0+t)|≤((|p0i|+C10iT)eCT+C12iT)eC(T+1),t∈[0,δ[,i=1,2,3,
which shows that every solution p¯ of (152) is uniformly bounded. By the standard theory of first-order differential systems, the solution p¯ is defined all over [t0,t0+δ[ and p¯∈𝒞([t0,t0+δ[;ℝ3).
(c) Next, we have proved in Theorem10 that the single equation (146) in f has a unique solution f∈Hd,r3(t0,t0+δ,ℝ3), substituting t0 to 0, t0+δ to T, such that f(t0)=f~(t0).
(d) For (147) in Φ, we have by (108)
(162)∥I4¯(t,E1¯,p1¯,f1,Φ1,U1)-I4¯(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ≤C6′∥U1-U2∥ℝ,
where
(163)C6′=2U0.
Equations (162) and (163) show that I4¯ is (globally) Lipschitzian with respect to the ℝ-norm and the local existence of a solution Φ of (147) such that Φ(t0)=Φ~(t0) is guaranteed by the standard theory on first-order differential systems.
SinceΦ satisfies (147) in which I4¯ is given by (149), substituting U to U~ in Lemma 15, we find(164)|Φ˙(t)|≤2((U0+C(|E¯0~|+∑i=13‍C9iT)eCTrT)+m02Tmax{Φ02,Φ2(T)})eCT,t∈[0,δ].
Applying the Gronwall inequality this time over[t0,t0+t], t∈[0,δ[, we find(165)|Φ(t)|≤|Φ0|+(1+T)2(((U0+C(|E¯0~|+∑i=13‍C9iT)eCTrT)+m02Tmax{Φ02,Φ2(T)})eCT)
and conclude that every solutionΦ of (147) is uniformly bounded. By the standard theory of first-order differential systems, the solution Φ is defined all over [t0,t0+δ[ and Φ∈𝒞([t0,t0+δ[;ℝ).
(e) In the end, for (148) in U, with I5¯ still defined by (149) in which E¯¯, f¯ are fixed. Since gij, 1/gij, ∂0gij, E¯¯, f¯ are continuous functions of t, so is I5¯. We deduce from (109) in which we set E¯1=E¯2=E¯¯, f1=f2=f¯ that(166)∥I5(t,E¯¯,p1¯,f¯,Φ1,U1)-I5(t,E¯¯,p2¯,f¯,Φ2,U2)∥ℝ≤C7(∥Φ1-Φ2∥ℝ+∥U1-U2∥ℝ),
where
(167)C7=C7=C|H|+m02|Φ2|2U1+U2+m022U1+C(detg)1/2∑i,j‍gij(∥E¯2∥+∥f1∥).
Invoking (157) and using |Ei¯(t0+t)|≤((|E0i|+C9iT)eCT+C11iT)eC(T+1),  t∈[0,δ[, i=1,2,3, since E¯¯=E1¯=E2¯ is uniformly bounded, ∥f¯∥≤∥|f¯|∥≤r, since f¯∈Hd,r3(t0,t0+δ,ℝ3), using(168)|Φ(t)|≤|Φ0|+(1+T)2(((U0+C(|E¯0~|+∑i=13‍C9iT)eCTrT)+m02Tmax{Φ02,Φ2(T)})eCT)
and Lemma15, we find
(169)C7≤C7′,whereC7′=C7′(gij0,T,m0,|E0¯|,|Φ0|,U0,r).
We conclude by (166), (169) that I5¯ is (globally) Lipschitzian with respect to the ℝ-norm and the local existence of a solution U of (148) such that U(t0)=U~(t0) is then guaranteed by the standard theory on first-order differential systems.
We similarly show that every solutionU of (148) is uniformly bounded and by the standard theory of first-order differential systems, U is defined all over [t0,t0+δ[ and U∈𝒞([t0,t0+δ[;ℝ).
This ends the proof of Proposition16.We now set(170)Xδt0=(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)(171)Yδt0=(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)×(𝒞([t0,t0+δ];ℝ))2.X
δ
t
0 is a complete metric subspace of the Banach space (𝒞([t0,t0+δ];ℝ3))2×Hd3(t0,t0+δ,ℝ3) and Yδt0 is a complete metric subspace of the Banach space (𝒞([t0,t0+δ];ℝ3))2×Hd3(t0,t0+δ,ℝ3) × (𝒞([t0,t0+δ];ℝ))2.With Proposition16 we define the map:
(172)g:Xδt0⟶Yδt0,(E¯¯,p¯¯,f¯)↦(E¯,p¯,f,Φ,U).We claim the following.Proposition 17.
Lett0[0,T[. There exists a number δ∈]0,1[,   independent oft0, such that the system (E1)-(E2)-(E3)-(E4)-(E5) has a unique solution (E¯,p¯,f,Φ,U)∈Yδt0 such that (E¯,p¯,f,Φ,U)(t0) = (E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)).Proof.
We will prove that there exists a numberδ∈]0,1[, independent oft0, such that the map g, defined by (172), induces a contraction of the complete metric space Xδt0 defined by (170), which will then have a fixed point (E¯,p¯,f) solution of the system (E1)-(E2)-(E3).
With the initial data(E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)) at t=t0, the differential system (144)-(145)-(146)-(147)-(148) is equivalent for i=1,2,3, and using notations (149) to the integral system
(173)Ei(t0+t)=E¯~(t0)+∫t0t0+t‍I1¯(t,E¯,p¯,f¯,Φ,U)(τ)dτ,(174)pi(t0+t)=pi~(t0)+∫t0t0+t‍I2¯(t,E¯¯,p¯,f¯,Φ,U)(τ)dτ,(175)f(t0+t)=f~(t0)+∫t0t0+t‍I3¯(t,E¯,p¯¯,f,Φ,U)(τ)dτ,(176)Φ(t0+t)=Φ~(t0)+∫t0t0+t‍I4¯(t,E¯,p¯,f,Φ,U)(τ)dτ,(177)U(t0+t)=U~(t0)+∫t0t0+t‍I5(t,E¯¯,p1¯,f¯,Φ1,U1)(τ)dτ,t∈[0,δ[.
To(Ej¯¯,pj¯¯,fj¯)∈Xδt0, j=1,2, correspond the solutions (Ej¯,pj¯,fj,Φj,Uj)∈Yδt0, j=1,2, whose existence is proved in Proposition 16. We now write the integral system (173)-(174)-(175)-(176)-(177) for j=1 and j=2, and taking the differences, we obtain the following, with i=1,2,3:
(178)(E1i-E2i)(t0+t)=∫t0t0+t‍[I1(t,E1¯,p1¯,f1¯,Φ1,U1)-I1(t,E2¯,p2¯,f2¯,Φ2,U2)](τ)dτ,(p1i-p2i)(t0+t)=∫t0t0+t‍[I2(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I2(t,E2¯¯,p2¯,f2¯,Φ2,U2)](τ)dτ(f1-f2)(t0+t)=∫t0t0+t‍[I3¯(t,E1¯¯,p1¯¯,f1,Φ1,U1)-I3¯(t,E2¯¯,p2¯¯,f2,Φ2,U2)](τ)dτ,(Φ1-Φ2)(t0+t)=∫t0t0+t‍[I4¯(t,E1¯,p1¯,f1,Φ1,U1)-I4¯(t,E2¯,p2¯,f2,Φ2,U2)](τ)dτ,(U1-U2)(t0+t)=∫t0t0+t‍[I5(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I5(t,E2¯¯,p2¯,f2¯,Φ2,U2)](τ)dτ.(a)
Now we can deduce, from (106) in which we set f1=f1¯, f2=f2¯,
(179)∥I1(t,E1¯,p1¯,f1¯,Φ1,U1)-I1(t,E2¯,p2¯,f2¯,Φ2,U2)∥ℝ3≤C3′(∥E¯1-E¯2∥ℝ3+∥f1¯-f2¯∥Hd3(ℝ3))(τ),
where C3′=C3′(gij0,T) is still given by (152).(b)
Next, since(Ej¯,pj¯,fj)∈Xδt0, we deduce from (107) in which we set E1¯=E1¯¯, E2¯=E2¯¯, f1=f1¯, f2=f2¯(180)∥I2(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I2(t,E2¯¯,p2¯,f2¯,Φ2,U2)∥ℝ3≤C4′(∥E1¯¯-E2¯¯∥ℝ3+∥p1¯-p2¯∥ℝ3+∥f1¯-f2¯∥Hd3(ℝ3))(τ),
where C4′=C4′(gij0,|E0i|,|φij|,T,r) is given by (159).(c)
We have by (108) in which we set p1¯=p1¯¯, p2¯=p2¯¯, since (Ej¯,pj¯,fj)∈Xδt0, and using C5=B3+C(T,gij0)(detg)1/2(∥f1∥+∥f2∥)+CB2 is still given by (108):
(181)∥I3¯(t,E1¯¯,p1¯¯,f1,Φ1,U1)-I3¯(t,E2¯¯,p2¯¯,f2,Φ2,U2)∥Hd3(ℝ3)≤C5′(∥p1¯¯-p2¯¯∥ℝ3+∥f1-f2∥Hd3(ℝ3))(τ),
where C5′=C5′(gij0,T,r).(d)
We also have, using (109),
(182)∥I4¯(t,E1¯,p1¯,f1,Φ1,U1)-I4¯(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ≤C6′(∥U1-U2∥ℝ)(τ),
where C6′ is given by (163).(e)
Finally, from (126) in which we set E1¯=E1¯¯, E2¯=E2¯¯, f1=f1¯, f2=f2¯, using the fact that Ej¯¯, Φj, Uj, j=1,2 are uniformly bounded and ∥fj¯(t)∥≤|∥fj¯∥|≤r, j=1,2, we deduce that
(183)∥I5(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I5(t,E2¯¯,p2¯,f2¯,Φ2,U2)∥ℝ≤C7′(∥E¯¯1-E¯¯2∥+∥f1¯-f2¯∥Hd3(ℝ3)+∥Φ1-Φ2∥ℝ+∥U1-U2∥ℝ∥E¯¯1-E¯¯2∥)(τ),
where C7′=C7′(gij0,T,m0,|E0¯|,|Φ0|,U0,r) is given by (169).
Already notice that the constantsC3′, C4′, C5′, C6′, and C7′ are absolute constants independent of t0.
Now using the inequalities (179), (180), (181), (182), and (183), we deduce from (178), using the norm |∥·∥| and since t∈[0,δ](184)|∥E¯1-E¯2∥|≤C3′δ(|∥E¯1-E¯2∥|+|∥f1¯-f2¯∥|)|∥p1¯-p2¯∥|≤C4′δ(|∥E1¯¯-E2¯¯∥|+|∥p1¯-p2¯∥|+|∥f1¯-f2¯∥|)|∥f1-f2∥|≤C5′δ(|∥p1¯¯-p2¯¯∥|+|∥f1-f2∥|)|∥Φ1-Φ2∥|≤C6′δ(|∥U1-U2∥|)|∥U1-U2∥|≤C7′δ(|∥E1¯¯-E2¯¯∥|+|∥f1¯-f2¯∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|∥E1¯¯-E2¯¯∥).
Now add (184) to obtain
(185)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|≤(C3′+C4′+C5′+C6′+C7′)δ×(|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|)+(C3′+C4′+C5′+C6′+C7′)δ×(|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|)+(C3′+C5′+C7′)δ×(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|).
First of all we takeδ such that
(186)(C3′+C4′+C5′+C6′+C7′)δ<12.
Then, reporting (186) in (185) and simplifying, we obtain
(187)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|≤2(C3′+C5′+C7′)δ×(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|).
Secondly choosingδ such that
(188)2(C3′+C5′+C7′)δ<12,
(187) becomes
(189)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|≤12(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|)
from which we deduce that(190)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|≤12(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|).
Consequently, if we take(191)0<δ<inf{1,14(C3′+C4′+C5′+C6′+C7′)},
(190) shows that g:Xδt0→Yδt0, (E¯¯,p¯¯,f¯)↦(E¯,p¯,f,Φ,U) defined by (172) induces a contraction (E¯¯,p¯¯,f¯)↦(E¯,p¯,f) in the complete metric space Xδt0 which then has a unique fixed point (E¯,p¯,f) and solution of the integral system (173)-(174)-(175) and, hence, of the differential system (E1)-(E2)-(E3) such that (E¯,p¯,f)(t0)=(E¯~(t0),p¯~(t0),f~(t0)). But using the fact that (E¯,p¯,f)∈(𝒞([t0,t0+δ];ℝ3))2 × Hd,r3(t0,t0+δ,ℝ3) and that t↦E¯(t), t↦p¯(t) are uniformly bounded, we conclude by Proposition 16 that the subsystem (E4)-(E5) in (Φ,U) is globally Lipschitzian in (Φ,U). So the unique solution (Φ,U), such that (Φ,U)(t0)=(Φ~(t0),U~(t0)), is global on [t0,t0+δ] since Φ and U remain uniformly bounded.
Consequently, for
t
0
∈
[
0
,
T
[
, there exists a number
δ
∈
]
0,1
[
, independent of
t
0, such that the system (E1)-(E2)-(E3)-(E4)-(E5) has a unique solution (E¯,p¯,f,Φ,U)∈Yδt0 such that
(192)(E¯,p¯,f,Φ,U)(t0)=(E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)).
This completes the proof of Proposition17.Based on the method detailed in Section6.1, we have proved the following result.Theorem 18.
LetE0¯=(E0i)∈ℝ3, φij∈ℝ, f0∈Hd3(0,T,ℝp¯3), Φ0∈ℝ, U0∈ℝ+*, be given, such that ∥f0∥≤r, where r>0 is a given real number. Then,(1)
the differential system (59)-(60)-(61)-(62) has a unique global solution (E¯,f,Φ,U) defined all over the interval [0,+∞[ and such that
(193)(E¯,f,Φ,U)(0)=(E0¯,f0,Φ0,U0),(2)
the Maxwell-Boltzmann-Euler system (20)-(21))-(40)-(41) in all Bianchi types I to VIII spacetimes has a unique global solution (F,f,Φ) defined all over the interval [0,+∞[ and satisfying E¯(0)=E0¯, Fij(0)=φij, f(0)=f0, Φ(0)=Φ0.


## 6.1. The Method

Let[0,T[ be the maximal existence domain of solution, denoted here by (E¯~,p¯~,f~,Φ~,U~) and given by Theorem 14, of the system (E1)-(E2)-(E3)-(E4)-(E5), with the initial data (E0¯,p0¯,f0,Φ0,U0)∈ℝ3×ℝ3×Hd3(0,T,ℝp¯3)×ℝ×ℝ. We intend to prove that T=+∞.(a)
If we already haveT=+∞, then the problem of global existence is solved.(b)
We are going to show that if we supposeT<+∞, then the solution (E¯~,p¯~,f~,Φ~,U~) can be extended beyond T, which contradicts the maximality of T.(c)
The method is as follows: suppose0<T<+∞ and let t0∈[0,T[. We want to show that there exists a strictly positive number δ>0 independent of t0 such that the system (E1)-(E2)-(E3)-(E4)-(E5) on [t0,t0+δ], with the initial data (E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)) at t=t0, admits a unique solution (E¯,p¯,f,Φ,U) on [t0,t0+δ]. Then, by taking t0 sufficiently close to T, for example, t0 such that 0<T-t0<(δ/2), and hence T<t0+(δ/2), we can extend the solution (E¯~,p¯~,f~,Φ~,U~) to [0,t0+(δ/2)], which strictly contains [0,T[, and this contradicts the maximality of T. For the need to simplify the notations, it will be enough if we could look for a number δ such that 0<δ<1.(d)
In what follows we fix a numberr>0 and we take f0 such that ∥f0∥≤r.By (128) we can write
(139)∥ft0~∥≤∥f0∥.We also have from (139) using (128) that any solution f of the Boltzmann equation on [t0,t0+δ] such that f(t0)=f~(t0), satisfies the inequality:
(140)∥f(t)∥≤r,t∈[t0,t0+δ].Also notice that (140) shows that a solution (E¯,p¯,f,Φ,U) of the system (E1)-(E2)-(E3)-(E4)-(E5) on [t0,t0+δ], δ>0, such that (E¯,p¯,f,Φ,U)(t0)=(E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)), satisfies
(141)(E¯,p¯,f,Φ,U)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)×(𝒞([t0,t0+δ];ℝ))2.In what follows,[0,T[,T>0 is the maximal existence domain of solution (E¯~,p¯~,f~,Φ~,U~) of (E1)-(E2)-(E3)-(E4)-(E5) such that
(142)(E¯~,p¯~,f~,Φ~,U~)(0)=(E¯0~,p0¯~,f0~,Φ0~,U0~)∈ℝ3×ℝ3×Hd,r3(0,T,ℝp¯3)×ℝ×ℝ.The following result shows helpful in what is to follow.Lemma 15.
t
↦
E
¯
~
(
t
), t↦p¯~(t), t↦U~(t), and t↦Φ~(t) are uniformly bounded over [0,T[.Proof.
See [1].


## 6.2. Global Existence of Solutions

First of all, we consider, fort0∈[0,T[ and δ>0,
(143)(E¯¯,p¯¯,f¯)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3).Then, we built from system(E1)-(E2)-(E3)-(E4)-(E5) by setting in its r.h.s.I=(I1,I2,I3,I4,I5) which is given by (102) f=f¯ in I1, E¯=E¯¯, f=f¯ in I2, p¯=p¯¯ in I3, and f=f¯, E¯=E¯¯, in I5, the following useful differential system:
(144)dEidt=I1¯(t,E¯,p¯,f¯,Φ,U),(145)dpidt=I2¯(t,E¯¯,p¯,f¯,Φ,U),(146)dfdt=I3¯(t,E¯,p¯¯,f,Φ,U),(147)dΦdt=I4¯(t,E¯,p¯,f,Φ,U),(148)dUdt=I5¯(t,E¯¯,p¯,f¯,Φ,U),
where
(149)I1¯(t,E¯,p¯,f¯,Φ,U)=(HEi+∫ℝ3‍qif¯(detg)1/2q0dq¯)I2¯(t,E¯¯,p¯,f¯,Φ,U)=(-2Γ0jipj+[-Ei¯+gijpkφkjp0]∫ℝ3‍f¯(detg)1/2dq¯)I3¯(t,E¯,p¯¯,f,Φ,U)=1p0(p¯¯)Q(f,f,p¯¯)I4¯(t,E¯,p¯,f,Φ,U)=-2UI5¯(t,E¯¯,p¯,f¯,Φ,U)=2HU+m02Φ2U+gijEj¯∫ℝ3‍qif¯(detg)1/2q0dq¯,i=1,2,3.We prove the following.Proposition 16.
Lett0∈[0,T[, δ∈]0,1[, and (E¯¯,p¯¯,f¯)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3) be given. Then, the differential system (144)-(145)-(146)-(147)-(148) has a unique solution (E¯,p¯,f,Φ,U)∈(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)×(𝒞([t0,t0+δ];ℝ))2 such that (E¯,p¯,f,Φ,U)(t0)=(E¯t0~,pt0¯~,ft0~,Φt0~,Ut0~).Proof.
(a) We consider (144) in E¯, with I1¯ defined by (149) in which f¯ is fixed. Since gij, 1/gij, ∂0gij, f¯ are continuous functions of  t, so is I1¯. Next, we deduce from (105) in which we set f1=f2=f¯ that
(150)∥I1(t,E1¯,p1¯,f¯,Φ1,U1)-I1(t,E2¯,p2¯,f¯,Φ2,U2)∥ℝ3≤C3∥E¯1-E¯2∥,
where C3=C(|H|+(detg)1/2).
Now we can use (5) and (6) to bound z=gij, 1/gij, ∂0gij and we obtain, for t∈[t0,t0+δ], then t≤t0+δ≤T+1,
(151)|z(t)|≤(C+∑i,j‍gij0)eC(T+1)+1,t∈[t0,t0+δ],z=gij,∂0gij.
We then deduce from (151) that
(152)C3≤C3′,whereC3′=C3′(gij0,T).
By (150) and (152), I1¯ is (globally) Lipschitzian with respect to the ℝ3-norm and the local existence of a solution E¯ of (144) such that E¯(t0)=E¯~(t0) is guaranteed by the standard theory of first-order differential systems.
Now, sinceE¯ satisfies (144) in which I1¯ is given by (149), following the same way as in the proof of Lemma 15, substituting E¯ to E¯~, f¯ to f~, and integrating this time over [t0,t0+t],t∈[0,δ[, lead to
(153)|Ei(t0+t)|≤(|Ei~(t0)|+C9iT)+C∫t0t0+t‍|Ei|(s)ds,|Ei(t0+t)|≤Ei(t0+t)Eikkit∈[0,δ[,i=1,2,3,
where C9i=C9i(gij0,T,r,|E0i|). However, by Lemma 15, we have, since t0[0,T[:|Ei~(t0)|≤(|E0i|+C9iT)eCT. Then, by Gronwall inequality,
(154)|Ei(t0+t)|≤((|E0i|+C9iT)eCT+C11iT)eC(T+1),|Ei(t0+t)|≤|Ei(t0+t)|it∈[0,δ[,i=1,2,3,
where (see [1]) C10i, C11i are two constants appearing in Lemma 15 when we bound t↦p¯~(t), which shows that every solution E¯ of (144) is uniformly bounded. By the standard theory of first-order differential systems, the solution E¯ is defined all over [t0,t0+δ[ and E¯∈𝒞([t0,t0+δ[;ℝ3).
(b) We also consider (145) in p¯, with I2¯ defined by (149) in which E¯¯, f¯ are fixed. Since gij, 1/gij, ∂0gij, E¯¯, f¯ are continuous functions of t, so is I2¯. Next, we deduce from (106) in which we set E1¯=E2¯=E¯¯, f1=f2=f¯ that
(155)∥I2(t,E¯¯,p1¯,f¯,Φ1,U1)-I2(t,E¯¯,p2¯,f¯,Φ2,U2)∥ℝ3≤C4∥p¯1-p¯2∥,
where
(156)C4=C(∑i,j‍gij)2+C(detg)1/2(∥f1∥+∥E2¯∥)+C(detg)1/2∑i,j‍gij∑k,j‍|φkj|(B1+∑k‍1gkk).
Now we can use (5) and (6) to bound z=gij,1/gij and we obtain, for t∈[t0,t0+δ], then t≤t0+δ≤T+1,
(157)|z(t)|≤(∑i,j‍gij0+∑i,j‍1gij0)eC(T+1)+1,t∈[t0,t0+δ],z=gij,1gij.
We then deduce from (157), using
(158)|Ei¯(t0+t)|≤((|E0i|+C9iT)eCT+C11iT)eC(T+1),t∈[0,δ[,i=1,2,3,
sinceE¯¯=E1¯=E2¯ is uniformly bounded, and ∥f¯∥≤∥|f¯|∥≤r, since f¯∈Hd,r3(t0,t0+δ,ℝ3) that
(159)C4≤C4′,whereC4′=C4′(gij0,|E0i|,|φij|,T,r).
By (155) and (159), I2¯ is (globally) Lipschitzian with respect to the ℝ3-norm and the local existence of a solution p¯ of (145) such that p¯(t0)=p¯~(t0) is guaranteed by the standard theory of first-order differential systems.
Now, sincep¯ satisfies (152) in which I2¯ is given by (149), following the same way as in the proof of Lemma 15, substituting E¯¯ to E¯~, p¯ to p¯~, and f¯ to f~, and integrating this time over [t0,t0+t], t∈[0,δ[, lead to
(160)|pi(t0+t)|≤(|pi~(t0)|+C12iT)+C∫t0t0+t‍|pi|(s)ds,t∈[0,δ[,i=1,2,3,
where C12i=C12i(gij0,|E0i|,|φij|,T,r). However, by Lemma 15, we have, since t0[0,T[:|pi~(t0)|≤(|p0i|+C8iT)eCT. Then, by Gronwall inequality,
(161)|pi(t0+t)|≤((|p0i|+C10iT)eCT+C12iT)eC(T+1),t∈[0,δ[,i=1,2,3,
which shows that every solution p¯ of (152) is uniformly bounded. By the standard theory of first-order differential systems, the solution p¯ is defined all over [t0,t0+δ[ and p¯∈𝒞([t0,t0+δ[;ℝ3).
(c) Next, we have proved in Theorem10 that the single equation (146) in f has a unique solution f∈Hd,r3(t0,t0+δ,ℝ3), substituting t0 to 0, t0+δ to T, such that f(t0)=f~(t0).
(d) For (147) in Φ, we have by (108)
(162)∥I4¯(t,E1¯,p1¯,f1,Φ1,U1)-I4¯(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ≤C6′∥U1-U2∥ℝ,
where
(163)C6′=2U0.
Equations (162) and (163) show that I4¯ is (globally) Lipschitzian with respect to the ℝ-norm and the local existence of a solution Φ of (147) such that Φ(t0)=Φ~(t0) is guaranteed by the standard theory on first-order differential systems.
SinceΦ satisfies (147) in which I4¯ is given by (149), substituting U to U~ in Lemma 15, we find(164)|Φ˙(t)|≤2((U0+C(|E¯0~|+∑i=13‍C9iT)eCTrT)+m02Tmax{Φ02,Φ2(T)})eCT,t∈[0,δ].
Applying the Gronwall inequality this time over[t0,t0+t], t∈[0,δ[, we find(165)|Φ(t)|≤|Φ0|+(1+T)2(((U0+C(|E¯0~|+∑i=13‍C9iT)eCTrT)+m02Tmax{Φ02,Φ2(T)})eCT)
and conclude that every solutionΦ of (147) is uniformly bounded. By the standard theory of first-order differential systems, the solution Φ is defined all over [t0,t0+δ[ and Φ∈𝒞([t0,t0+δ[;ℝ).
(e) In the end, for (148) in U, with I5¯ still defined by (149) in which E¯¯, f¯ are fixed. Since gij, 1/gij, ∂0gij, E¯¯, f¯ are continuous functions of t, so is I5¯. We deduce from (109) in which we set E¯1=E¯2=E¯¯, f1=f2=f¯ that(166)∥I5(t,E¯¯,p1¯,f¯,Φ1,U1)-I5(t,E¯¯,p2¯,f¯,Φ2,U2)∥ℝ≤C7(∥Φ1-Φ2∥ℝ+∥U1-U2∥ℝ),
where
(167)C7=C7=C|H|+m02|Φ2|2U1+U2+m022U1+C(detg)1/2∑i,j‍gij(∥E¯2∥+∥f1∥).
Invoking (157) and using |Ei¯(t0+t)|≤((|E0i|+C9iT)eCT+C11iT)eC(T+1),  t∈[0,δ[, i=1,2,3, since E¯¯=E1¯=E2¯ is uniformly bounded, ∥f¯∥≤∥|f¯|∥≤r, since f¯∈Hd,r3(t0,t0+δ,ℝ3), using(168)|Φ(t)|≤|Φ0|+(1+T)2(((U0+C(|E¯0~|+∑i=13‍C9iT)eCTrT)+m02Tmax{Φ02,Φ2(T)})eCT)
and Lemma15, we find
(169)C7≤C7′,whereC7′=C7′(gij0,T,m0,|E0¯|,|Φ0|,U0,r).
We conclude by (166), (169) that I5¯ is (globally) Lipschitzian with respect to the ℝ-norm and the local existence of a solution U of (148) such that U(t0)=U~(t0) is then guaranteed by the standard theory on first-order differential systems.
We similarly show that every solutionU of (148) is uniformly bounded and by the standard theory of first-order differential systems, U is defined all over [t0,t0+δ[ and U∈𝒞([t0,t0+δ[;ℝ).
This ends the proof of Proposition16.We now set(170)Xδt0=(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)(171)Yδt0=(𝒞([t0,t0+δ];ℝ3))2×Hd,r3(t0,t0+δ,ℝ3)×(𝒞([t0,t0+δ];ℝ))2.X
δ
t
0 is a complete metric subspace of the Banach space (𝒞([t0,t0+δ];ℝ3))2×Hd3(t0,t0+δ,ℝ3) and Yδt0 is a complete metric subspace of the Banach space (𝒞([t0,t0+δ];ℝ3))2×Hd3(t0,t0+δ,ℝ3) × (𝒞([t0,t0+δ];ℝ))2.With Proposition16 we define the map:
(172)g:Xδt0⟶Yδt0,(E¯¯,p¯¯,f¯)↦(E¯,p¯,f,Φ,U).We claim the following.Proposition 17.
Lett0[0,T[. There exists a number δ∈]0,1[,   independent oft0, such that the system (E1)-(E2)-(E3)-(E4)-(E5) has a unique solution (E¯,p¯,f,Φ,U)∈Yδt0 such that (E¯,p¯,f,Φ,U)(t0) = (E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)).Proof.
We will prove that there exists a numberδ∈]0,1[, independent oft0, such that the map g, defined by (172), induces a contraction of the complete metric space Xδt0 defined by (170), which will then have a fixed point (E¯,p¯,f) solution of the system (E1)-(E2)-(E3).
With the initial data(E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)) at t=t0, the differential system (144)-(145)-(146)-(147)-(148) is equivalent for i=1,2,3, and using notations (149) to the integral system
(173)Ei(t0+t)=E¯~(t0)+∫t0t0+t‍I1¯(t,E¯,p¯,f¯,Φ,U)(τ)dτ,(174)pi(t0+t)=pi~(t0)+∫t0t0+t‍I2¯(t,E¯¯,p¯,f¯,Φ,U)(τ)dτ,(175)f(t0+t)=f~(t0)+∫t0t0+t‍I3¯(t,E¯,p¯¯,f,Φ,U)(τ)dτ,(176)Φ(t0+t)=Φ~(t0)+∫t0t0+t‍I4¯(t,E¯,p¯,f,Φ,U)(τ)dτ,(177)U(t0+t)=U~(t0)+∫t0t0+t‍I5(t,E¯¯,p1¯,f¯,Φ1,U1)(τ)dτ,t∈[0,δ[.
To(Ej¯¯,pj¯¯,fj¯)∈Xδt0, j=1,2, correspond the solutions (Ej¯,pj¯,fj,Φj,Uj)∈Yδt0, j=1,2, whose existence is proved in Proposition 16. We now write the integral system (173)-(174)-(175)-(176)-(177) for j=1 and j=2, and taking the differences, we obtain the following, with i=1,2,3:
(178)(E1i-E2i)(t0+t)=∫t0t0+t‍[I1(t,E1¯,p1¯,f1¯,Φ1,U1)-I1(t,E2¯,p2¯,f2¯,Φ2,U2)](τ)dτ,(p1i-p2i)(t0+t)=∫t0t0+t‍[I2(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I2(t,E2¯¯,p2¯,f2¯,Φ2,U2)](τ)dτ(f1-f2)(t0+t)=∫t0t0+t‍[I3¯(t,E1¯¯,p1¯¯,f1,Φ1,U1)-I3¯(t,E2¯¯,p2¯¯,f2,Φ2,U2)](τ)dτ,(Φ1-Φ2)(t0+t)=∫t0t0+t‍[I4¯(t,E1¯,p1¯,f1,Φ1,U1)-I4¯(t,E2¯,p2¯,f2,Φ2,U2)](τ)dτ,(U1-U2)(t0+t)=∫t0t0+t‍[I5(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I5(t,E2¯¯,p2¯,f2¯,Φ2,U2)](τ)dτ.(a)
Now we can deduce, from (106) in which we set f1=f1¯, f2=f2¯,
(179)∥I1(t,E1¯,p1¯,f1¯,Φ1,U1)-I1(t,E2¯,p2¯,f2¯,Φ2,U2)∥ℝ3≤C3′(∥E¯1-E¯2∥ℝ3+∥f1¯-f2¯∥Hd3(ℝ3))(τ),
where C3′=C3′(gij0,T) is still given by (152).(b)
Next, since(Ej¯,pj¯,fj)∈Xδt0, we deduce from (107) in which we set E1¯=E1¯¯, E2¯=E2¯¯, f1=f1¯, f2=f2¯(180)∥I2(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I2(t,E2¯¯,p2¯,f2¯,Φ2,U2)∥ℝ3≤C4′(∥E1¯¯-E2¯¯∥ℝ3+∥p1¯-p2¯∥ℝ3+∥f1¯-f2¯∥Hd3(ℝ3))(τ),
where C4′=C4′(gij0,|E0i|,|φij|,T,r) is given by (159).(c)
We have by (108) in which we set p1¯=p1¯¯, p2¯=p2¯¯, since (Ej¯,pj¯,fj)∈Xδt0, and using C5=B3+C(T,gij0)(detg)1/2(∥f1∥+∥f2∥)+CB2 is still given by (108):
(181)∥I3¯(t,E1¯¯,p1¯¯,f1,Φ1,U1)-I3¯(t,E2¯¯,p2¯¯,f2,Φ2,U2)∥Hd3(ℝ3)≤C5′(∥p1¯¯-p2¯¯∥ℝ3+∥f1-f2∥Hd3(ℝ3))(τ),
where C5′=C5′(gij0,T,r).(d)
We also have, using (109),
(182)∥I4¯(t,E1¯,p1¯,f1,Φ1,U1)-I4¯(t,E2¯,p2¯,f2,Φ2,U2)∥ℝ≤C6′(∥U1-U2∥ℝ)(τ),
where C6′ is given by (163).(e)
Finally, from (126) in which we set E1¯=E1¯¯, E2¯=E2¯¯, f1=f1¯, f2=f2¯, using the fact that Ej¯¯, Φj, Uj, j=1,2 are uniformly bounded and ∥fj¯(t)∥≤|∥fj¯∥|≤r, j=1,2, we deduce that
(183)∥I5(t,E1¯¯,p1¯,f1¯,Φ1,U1)-I5(t,E2¯¯,p2¯,f2¯,Φ2,U2)∥ℝ≤C7′(∥E¯¯1-E¯¯2∥+∥f1¯-f2¯∥Hd3(ℝ3)+∥Φ1-Φ2∥ℝ+∥U1-U2∥ℝ∥E¯¯1-E¯¯2∥)(τ),
where C7′=C7′(gij0,T,m0,|E0¯|,|Φ0|,U0,r) is given by (169).
Already notice that the constantsC3′, C4′, C5′, C6′, and C7′ are absolute constants independent of t0.
Now using the inequalities (179), (180), (181), (182), and (183), we deduce from (178), using the norm |∥·∥| and since t∈[0,δ](184)|∥E¯1-E¯2∥|≤C3′δ(|∥E¯1-E¯2∥|+|∥f1¯-f2¯∥|)|∥p1¯-p2¯∥|≤C4′δ(|∥E1¯¯-E2¯¯∥|+|∥p1¯-p2¯∥|+|∥f1¯-f2¯∥|)|∥f1-f2∥|≤C5′δ(|∥p1¯¯-p2¯¯∥|+|∥f1-f2∥|)|∥Φ1-Φ2∥|≤C6′δ(|∥U1-U2∥|)|∥U1-U2∥|≤C7′δ(|∥E1¯¯-E2¯¯∥|+|∥f1¯-f2¯∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|∥E1¯¯-E2¯¯∥).
Now add (184) to obtain
(185)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|≤(C3′+C4′+C5′+C6′+C7′)δ×(|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|)+(C3′+C4′+C5′+C6′+C7′)δ×(|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|)+(C3′+C5′+C7′)δ×(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|).
First of all we takeδ such that
(186)(C3′+C4′+C5′+C6′+C7′)δ<12.
Then, reporting (186) in (185) and simplifying, we obtain
(187)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|≤2(C3′+C5′+C7′)δ×(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|).
Secondly choosingδ such that
(188)2(C3′+C5′+C7′)δ<12,
(187) becomes
(189)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|+|∥Φ1-Φ2∥|+|∥U1-U2∥|≤12(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|)
from which we deduce that(190)|∥E¯1-E¯2∥|+|∥p¯1-p¯2∥|+|∥f1-f2∥|≤12(|∥E¯¯1-E¯¯2∥|+|∥p1¯¯-p2¯¯∥|+|∥f1¯-f2¯∥|).
Consequently, if we take(191)0<δ<inf{1,14(C3′+C4′+C5′+C6′+C7′)},
(190) shows that g:Xδt0→Yδt0, (E¯¯,p¯¯,f¯)↦(E¯,p¯,f,Φ,U) defined by (172) induces a contraction (E¯¯,p¯¯,f¯)↦(E¯,p¯,f) in the complete metric space Xδt0 which then has a unique fixed point (E¯,p¯,f) and solution of the integral system (173)-(174)-(175) and, hence, of the differential system (E1)-(E2)-(E3) such that (E¯,p¯,f)(t0)=(E¯~(t0),p¯~(t0),f~(t0)). But using the fact that (E¯,p¯,f)∈(𝒞([t0,t0+δ];ℝ3))2 × Hd,r3(t0,t0+δ,ℝ3) and that t↦E¯(t), t↦p¯(t) are uniformly bounded, we conclude by Proposition 16 that the subsystem (E4)-(E5) in (Φ,U) is globally Lipschitzian in (Φ,U). So the unique solution (Φ,U), such that (Φ,U)(t0)=(Φ~(t0),U~(t0)), is global on [t0,t0+δ] since Φ and U remain uniformly bounded.
Consequently, for
t
0
∈
[
0
,
T
[
, there exists a number
δ
∈
]
0,1
[
, independent of
t
0, such that the system (E1)-(E2)-(E3)-(E4)-(E5) has a unique solution (E¯,p¯,f,Φ,U)∈Yδt0 such that
(192)(E¯,p¯,f,Φ,U)(t0)=(E¯~(t0),p¯~(t0),f~(t0),Φ~(t0),U~(t0)).
This completes the proof of Proposition17.Based on the method detailed in Section6.1, we have proved the following result.Theorem 18.
LetE0¯=(E0i)∈ℝ3, φij∈ℝ, f0∈Hd3(0,T,ℝp¯3), Φ0∈ℝ, U0∈ℝ+*, be given, such that ∥f0∥≤r, where r>0 is a given real number. Then,(1)
the differential system (59)-(60)-(61)-(62) has a unique global solution (E¯,f,Φ,U) defined all over the interval [0,+∞[ and such that
(193)(E¯,f,Φ,U)(0)=(E0¯,f0,Φ0,U0),(2)
the Maxwell-Boltzmann-Euler system (20)-(21))-(40)-(41) in all Bianchi types I to VIII spacetimes has a unique global solution (F,f,Φ) defined all over the interval [0,+∞[ and satisfying E¯(0)=E0¯, Fij(0)=φij, f(0)=f0, Φ(0)=Φ0.


## 7. Conclusion

The physical significance of the work we did in the present paper is the study of the global dynamics of a kind of fast moving, massive, and charged particles, in the case where the gravitational forces are neglected in front of the electromagnetic forces. We have coupled the Maxwell-Boltzmann system with the Euler equations which simply express the conservation of the Stress-matter tensor for the unknownΦ representing a massive scalar field. Notice that this present work follows our paper titled “Global regular solutions to the Maxwell-Boltzmann-Euler system in a Bianchi type 1 spacetime in presence of a massive scalar field,” where the unknowns were like now, the electromagnetic field F, subject to the Maxwell equations, the distribution function f, subject to the Boltzmann equation, and the massive scalar field Φ, subject to the Euler equations. We have generalized in this present work the result obtained for the Bianchi types 1 spacetime to all Bianchi types 1 to 8 spacetimes. We have only excluded the case of Bianchi type 9 spacetime for the main reason we gave in Section 1. We have also improved the process to establish energy inequalities and the definition of function spaces. In our future investigations we intend to study the same system coupled this time with the Einstein equations with the cosmological constant. The investigation of the global existence of solutions to this Einstein-Maxwell-Boltzmann-Euler system with cosmological constant Λ is with a great interest, in the sense that some recent observations show that the whole universe is in an accelerated expansion, and it is the presence of the cosmological constant in the Einstein equations which mathematically shapes this phenomenon, which is also very important for the evolution of our universe and consequently for the humanity.



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*Source: 679054-2013-09-01.xml*