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

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.


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 their distribution 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 field  created by the fast moving charged particles.We consider the case where the electromagnetic field  is generated, through the Maxwell equations by the Maxwell current defined by the distribution function  of the colliding particles, a charge density , and a future pointing unit vector , 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 function , the electromagnetic field , and a massive scalar field Φ, which depends only on the time .
The Euler equations simply express the conservation of the energy-momentum tensor.
The system is coupled in the sense that , 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 , which is subject to the Maxwell equations, is in the Lie derivative of  with respect to the vectors field tangent to the trajectories of the particles. 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][2][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 Section 2, we introduce the spacetime and we give the unknowns.
In Section 3, we describe the Maxwell-Boltzmann-Euler system.
In Section 4, we define the function spaces and we establish the energy inequalities.
In Section 5, we study the local existence of the solution.
In Section 6, we prove the global existence of the solution.

The Spacetime and the Unknowns
Greek indexes , , , . . .range from 0 to 3, and Latin indexes , , , . . .from 1 to 3. We adopt the Einstein summation convention: 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 (R 4 , g) and denote by   = ( 0 ,   ) = (,   ) the usual coordinates in R 4 , where  0 =  represents the time and (  ) the space; g stands for the given metric tensor of Lorentzian signature (−, +, +, +) which writes where   are continuously differentiable functions on R, components of a 3-symmetric metric tensor  = (  ), whose variable is denoted by .
The expression of the Levi-Civita connection ∇ associated with g, which is gives directly Recall that Γ   = Γ   .We require the assumption that  0   /  are bounded.This implies that there exists a constant  > 0 such that            0               ≤ .
As a direct consequence, we have, for  ∈ R + , where  0  =   (0).The massive particles have a rest mass  > 0, normalized to the unity, that is,  = 1.We denote by (R 4 ) the tangent bundle of R 4 with coordinates (  ,   ), where  = (  ) = ( 0 , ) stands for the momentum of each particle and  = (  ),  = 1, 2, 3. Really the charged particles move on the future sheet of the mass shell or the mass hyperboloid (R 4 ) ⊂ (R 4 ), whose equation is   () : g (, ) − 1 or, equivalently, using expression (2) of g: where the choice  0 > 0 symbolizes the fact that, naturally, the particles eject towards the future.Setting if  > 1, the relations ( 6) and ( 7) also show that in any interval [0, ],  > 0: where  = () > 0,  = () > 0 are constants.The invariant volume element in   () reads where We denote by  the distribution function which measures the probability of the presence of particles in the plasma. is a nonnegative unknown real-valued function of both the position (  ) and the 4-momentum of the particles  = (  ), so: We define a scalar product on R 3 by setting for  = ( 0 , ) = ( 0 ,   ) and  = ( 0 , ) = ( 0 ,   ): In this paper we consider the homogeneous case for which  depends only on the time  0 =  and .According to the Laplace law, the fast moving and charged particles Advances in Mathematical Physics 3 create an unknown electromagnetic field  which is a 2closed antisymmetric form and locally writes So in the homogeneous case we consider In the presence of the electromagnetic field , the trajectories   → (  (),   ()) of the charged particles are no longer the geodesics of spacetime (R 4 , ) but the solutions of the differential system: where where  =  () denotes the charge density of particles.Notice that the differential system (16) shows that the vectors field  () defined locally by where   is given by ( 17), is tangent to the trajectories.The charged particles also create a current  = (  ),  = 0, 1, 2, 3, called the Maxwell current which we take in the form in which  = (  ) is a unit future pointing timelike vector, tangent to the time axis at any point, which means that  0 = 1,   =   = 0, and  = 1, 2, 3.The particles are then supposed to be spatially at rest.The electromagnetic field  = ( 0 ,   ), where  0 and   stand for the electric and magnetic parts, respectively, is subject to the Maxwell equations.

The Maxwell-Boltzmann-Euler
System in , , and Φ 3.1.The Maxwell Equations in .The Maxwell system in  can be written, using the covariant notation: Equations ( 20) and (21) are, respectively, the first and second groups of the Maxwell equations, and ∇  stands for the convariant derivative in .In (20),   represents the Maxwell current we take in the form (19). Now the well-known identity ∇  ∇    = 0 imposes, given (20), that the current   is always subject to the conservation law: However using  = 0 in (20), we obtain since  = (),   = −  and by (4) that  0 = 0.
(23) By ( 23), the expression (19) of   in which we set  = 0 then allows to compute  and gives, since which shows that  determines .
The second set (21) of the Maxwell equations is identically satisfied since  = (), and the first set reduces to   = 0. Then   is constant and This physically shows that the magnetic part of  does not evolve and stays in its primitive state.It remains to determine the electric part  0 .

The Relativistic Boltzmann Equation in 𝑓.
The relativistic Boltzmann equation in , for charged particles in the Bianchi types 1 to 8 spacetimes, can be written: where   is the Lie derivative of  with respect to the vectors field () defined by (18) and (, ), the collision operator we now introduce.
According to Lichnerowicz and Chernikov, we consider a scheme, in which, at a given position (,   ), 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 ,  stand for the two momenta before the shock and   ,   for the two momenta after the shock, then we have The collision operator  is then defined, using functions  and  on R 3 , and the previous notations by where whose elements we now introduce step by step, specifying properties and hypotheses we adopt: (i)  2 is the unit sphere of R 3 , whose area element is denoted by Ω; (ii)  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  1 > 0 is a constant: where ‖‖ = (∑ 3 =1 (  ) 2 ) 1/2 =  is the norm in R 3 .
(iii) The conservation law  +  =   +   splits into Equation (34) expresses, using (7), the conservation of the quantity: in which p = / 0 , ẽ 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 , , Ω and that these integrals with respect to  and Ω give functions  + (, ) and  − (, ) of the single variable .
Using now the usual properties of the determinants, we compute the Jacobian of the change of variables (, )  → (  ,   ) defined by (37) and find But  = (, ), so using (7), the Boltzmann equation (29) leads to the following form: 3.3.The Euler Equations.The Euler equations only express the conservation of the energy-momentum tensor   and write In (41), where (i)  1  is the energy-momentum tensor associated with ; (ii)   is the Maxwell tensor associated with ; is the energy-momentum tensor associated with the scalar field Φ whose mass is denoted by  0 , with  0 > 0.
Equation (42) shows that (41) writes But it is proved in [5] that if  verifies the Boltzmann equation (40), then  1, defined by (43) verifies ∇   1, = 0; (46) reduces then to Now, using (21), we have and using (45), where ◻ g = ∇  ∇  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: For  = , (50) leads to the constraints system: between the unknown functions  and , constraints which we have to solve in what is to follow.For  = 0, (50) leads to a nonlinear differential equation of second order: where  is defined in (27).
Setting in (52) it comes that One supposes in what follows that Φ is continuously differentiable, is not a constant, and is decreasing.This implies that (55) Equation ( 52) is then equivalent to the nonlinear firstorder differential system given as follows: where   =  0 .

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.

Advances in Mathematical Physics
Definition 2 (   (0, , R 3 )).Let  > 0,  ∈ N,  ∈ R be given.We define    (R 3 ) as (R 3 ) will be endowed with the norm We also define Endowed with the norm (0, , R 3 ) is a Banach space.   (0, , R 3  ) will be the completion of    (0, , R 3  ) for the norm |‖ ⋅ ‖|    (0,,R 3  ) .For  > 0 to be given, we define Endowed with the induced distance by the norm Remark 4. The reasons for the choice of the function space    (0, , R 3  ) for  = 3 and  > 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  = (, ) which is continuously differentiable; in particular we can search  = (, ⋅) belonging to the space C 1  (R 3  ).We want to use the Faedo-Galerkin method which is applied for separable Hilbert spaces.That is the case for the Sobolev spaces    (0, , R 3  ),  ∈ N.
We need then to find an integer  such that But we know by the Sobolev theorems that Since in our case we have  = 3,  = 2, and  = 1 (  2 =   ), we must choose  such that The smallest integer  satisfying  > 5/2 is naturally  = 3.
Consequently we have then where  1 2 (R 3  ) is defined in [1].It then results that We can now state the following results which will be fundamental.
Lemma 5.There exists a real number  > 0 such that Furthermore, one has and the function (, , Ω)  → , and one has where  =  () > 0. Moreover Proof.We simply use Lemma 5.For the details, see [2].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 for  is  3  (0, , R 3  ).The framework we will refer to for  is R 3 , whose norm is denoted by ) is a Banach space for the norm: The framework we will refer to for Φ and  is R, whose norm is denoted by ) is a Banach space for the norm: (iii) We will consider the Cauchy problem ( 59)-( 60)-( 61)-(62) for the initial data: where  0 is given in  3 , (0, , R 3  ),   0 ,   , Φ 0 ∈ R, ,  = 1, 2, 3, and  0 ∈ R + .
Next, let us introduce the subgroup  of O 3 defined by A function  on R 3 is said to be invariant under  if Using the observation that  0 is invariant under , it is proved in [6] that if  0 is invariant under , then so will be the solution  of the Boltzmann equation satisfying  0 () = (0,).It is also proved in [7] that One requires in all what follows that the initial datum  0 = (0; ⋅) of the distribution function  is not invariant under .The immediate consequence is that Now, computing the determinant of the system (64), we conclude that, under our requirement, the problem of constraints (64) admits on [0, ] a nontrivial solution: where  is the unique solution to the Boltzmann equation ( 60) on [0, ] in which Ẽ is given.
Advances in Mathematical Physics 9 Let us now state the following result which shows helpful in what is to follow.
The framework we will refer to for  is R 3 , whose norm is denoted by ‖ ⋅ ‖ or ‖ ⋅ ‖ R 3 .

One requires in what follows that, for any real number
Then the following holds.There exists a real number  > 0 such that the Cauchy problem () has a unique solution: Proof.We apply the standard theory on the first-order differential systems to ( 1 )-( 2 )-( 3 )-( 4 )-( 5 ).

The Global Existence
6.1.The Method.Let [0, [ be the maximal existence domain of solution, denoted here by ( Ẽ, p, f, Φ, Ũ) and given by Theorem 14, of the system We intend to prove that  = +∞.
(a) If we already have  = +∞, then the problem of global existence is solved.
(b) We are going to show that if we suppose  < +∞, then the solution ( Ẽ, p, f, Φ, Ũ) can be extended beyond , which contradicts the maximality of .

(d)
In what follows we fix a number  > 0 and we take  0 such that ‖ 0 ‖ ≤ .
Proof.(a) We consider (144) in , with  1 defined by (149) in which  is fixed.Since   , 1/  ,  0   ,  are continuous functions of , so is  1 .Next, we deduce from (105) in which we set  1 =  2 =  that       1 (, We conclude by (166), (169) that  5 is (globally) Lipschitzian with respect to the R-norm and the local existence of a solution  of (148) such that ( 0 ) = Ũ( 0 ) is then guaranteed by the standard theory on first-order differential systems.
We similarly show that every solution  of ( 148) is uniformly bounded and by the standard theory of first-order differential systems,  is defined all over [ 0 ,  0 + [ and  ∈ C([ 0 ,  0 + [; R).
This ends the proof of Proposition 16.
Proof.We will prove that there exists a number  ∈]0, 1[, independent of  0 , such that the map , defined by (172), induces a contraction of the complete metric space   0  defined by (170), which will then have a fixed point (, , ) solution of the system ( 1 )-( 2 )-( 3 ).

Remark 8 .
The hypothesis of Proposition 6 concerning the collision kernel  is a supplementary hypothesis for the investigation of the solution to the Boltzmann equation.