Advances in Mathematical Physics

Volume 2013 (2013), Article ID 679054, 18 pages

http://dx.doi.org/10.1155/2013/679054

Research Article

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

Raoul Domingo Ayissi, Norbert Noutchegueme, and Hugues Paulin Mbeutcha Tchagna

Department of Mathematics, Faculty of Science, University of Yaoundé I, P.O. Box 812,Yaoundé, Cameroon

Received 6 April 2013; Accepted 17 June 2013

Academic Editor: B. G. Konopelchenko

Copyright © 2013 Raoul Domingo Ayissi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 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 , 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 [1–3] 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.

2. The Spacetime and the Unknowns

Greek indexes range from to , and Latin indexes from to . 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 and denote by the usual coordinates in , where represents the time and the space; stands for the given metric tensor of  Lorentzian signature which writes

where are continuously differentiable functions on , components of a 3-symmetric metric tensor , whose variable is denoted by .

The expression of the Levi-Civita connection associated with , which is

gives directly

Recall that .

We require the assumption that are bounded. This implies that there exists a constant such that

As a direct consequence, we have, for ,

where .

The massive particles have a rest mass , normalized to the unity, that is, . We denote by the tangent bundle of with coordinates , where stands for the momentum of each particle and . Really the charged particles move on the future sheet of the mass shell or the mass hyperboloid , whose equation is or, equivalently, using expression (2) of :

where the choice symbolizes the fact that, naturally, the particles eject towards the future.

Setting

if , the relations (6) and (7) also show that in any interval , :

where , 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 by setting for and :

In this paper we consider the homogeneous case for which depends only on the time and . According to the Laplace law, the fast moving and charged particles create an unknown electromagnetic field which is a 2-closed 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 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 , , 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 , , and . The particles are then supposed to be spatially at rest.

The electromagnetic field , where and stand for the electric and magnetic parts, respectively, is subject to the Maxwell equations.

3. The Maxwell-Boltzmann-Euler System in , , and

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 imposes, given (20), that the current is always subject to the conservation law:

However using in (20), we obtain since , and by (4) that

By (23), the expression (19) of in which we set 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 . 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 .

Writing (19) for , using (4), , and , implies that

By (20), we obtain the linear in which writes

Remark 1. In (27), the expression

represents the second fundamental form in .  Really is the trace of the 2-symmetric tensor where . is called the middle curvature of . Since is given, so is .

3.2. The Relativistic Boltzmann Equation in

The relativistic Boltzmann equation in , for charged particles in the Bianchi types 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 , and the previous notations by

where

whose elements we now introduce step by step, specifying properties and hypotheses we adopt:

(i) is the unit sphere of , whose area element is denoted by ;(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 is a constant:

where is the norm in .(iii)The conservation law splits into

Equation (34) expresses, using (7), the conservation of the quantity:

called the elementary energy of the unit rest mass particles; we can interpret (35) by setting, following Glassey and Strauss in [4, equation ],

in which is a real-valued function. Using (7) to express , in terms of   , and next (37) to express , in terms of  , we prove that (34) leads to a quadratic equation in , which solves to give the only nontrivial solution:

in which , 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 , , and that these integrals with respect to and 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) is the energy-momentum tensor associated with ;(ii) is the Maxwell tensor associated with ;(iii) is the energy-momentum tensor associated with the scalar field whose mass is denoted by , with .

Equation (42) shows that (41) writes

But it is proved in [5] that if verifies the Boltzmann equation (40), then defined by (43) verifies ; (46) reduces then to

Now, using (21), we have

and using (45),

where is the D'Alembertian.

We deduce from (20), (48), and (49) that the Euler equations (41) are satisfied if 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 , (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 is continuously differentiable, is not a constant, and is decreasing. This implies that

Equation (52) is then equivalent to the nonlinear first-order differential system given as follows:

where .

3.4. The Coupled System

From (17), using (4), we obtain

Using (24), (27), (40), (56), and (57), the Maxwell-Boltzmann-Euler system in reduces to the following form:

which is an integrodifferential system to solve in what is to follow.

We are searching a solution of the Cauchy problem (59)-(60)-(61)-(62) globally in time on for the initial data:

3.5. The Problem of Constraints

We must find a nontrivial solution of the Cauchy problem (59)-(60)-(61)-(62) satisfying the system (51) of constraints which writes after computation

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 (). Let , , be given.

We define as

will be endowed with the norm

will be the completion of in the norm .

We also define

Endowed with the norm

is a Banach space.

will be the completion of for the norm .

For to be given, we define

Endowed with the induced distance by the norm , is a complete metric subspace of .

Remark 3. If , then , so will be denoted by .

Remark 4. The reasons for the choice of the function space for and .

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 .

We want to use the Faedo-Galerkin method which is applied for separable Hilbert spaces. That is the case for the Sobolev spaces , .

We need then to find an integer such that

But we know by the Sobolev theorems that

Since in our case we have , , and , we must choose such that

The smallest integer satisfying is naturally .

Consequently we have

Furthermore if

then

where 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 such that

Furthermore, one has

and the function , is bounded.

Proof. See [2].

Proposition 6. Let , and , be given.

If , then , and one has

where .

Moreover

Proof. We simply use Lemma 5. For the details, see [2].

Proposition 7. Let be given. Then , .

Proof. See [2].

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.

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 .

The framework we will refer to for is , whose norm is denoted by or :

is a Banach space for the norm:

The framework we will refer to for and is , whose norm is denoted by :

is a Banach space for the norm:

(i)We consider on the norm

(ii)We consider on the norm:

(iii)We will consider the Cauchy problem (59)-(60)-(61)-(62) for the initial data:

where is given in , , , , and .

5. The Local Existence of Solution

Theorem 9. Let be given, and let be fixed. Then the linearized partial differential equation

whose unknown is , with , has in a local unique and bounded -weak solution.

Proof. We use the Faedo-Galerkin method in the function space . For the other details, see [1].

Theorem 10. Let , ,  be fixed. Then the Boltzmann equation,

has in a local unique -weak solution such that .

Proof. We use the Banach fixed point theorem in for the map:

where satisfies (88).

(i)We firstly prove, using a sequence of approximations of , the Banach-Alaoglu theorem and the fact that is a reflexive space (see [1]) that we can choose and such that

(ii)Let now be given, and let be two solutions of (88). Then

Let and .

Then we get

Conveniently using energy inequalities established in [1], the system (92), and remembering that , we obtain

where is a positive constant.

Then taking the in (94), for and , we get

which implies

The relations (91), (96) show clearly that , is a contracting map, so by the Banach theorem has a unique fixed point and the proof of Theorem 10 is complete.

Next, let us introduce the subgroup of defined by

A function on is said to be invariant under if

Using the observation that   is invariant under , it is proved in [6] that if is invariant under , then so will be the solution of the Boltzmann equation satisfying . It is also proved in [7] that , if and only if is invariant under .

One requires in all what follows that the initial datum 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 a nontrivial solution:

where is the unique solution to the Boltzmann equation (60) on in which 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 :

Proof. See [1].

The framework we will refer to for is , whose norm is denoted by or .

Let denote the . of ----, that is,

It then appears that, on the contrary to the uncharged case studied in [6, 8], the momentum also becomes an unknown in the charged case. Note that and are now independent variables for the system ----. In this context, the collision operator defined by (31) will depend on only through the collision kernel , and we show it by writing now instead of  . One must from now be careful in order to avoid any confusion between the unknown of the system ---- and the variable in (26), (27), (57), and (62), for example. For this reason, we denoted the variable in the integrals in , , and by instead of   .

Proposition 12. Let , , , be given. Then

where

Proof. See [7].

We prove the following.

Proposition 13. Let ,  , be given. Then

where

Proof. (a) We have, using (102),

So by (5) and Proposition 7,

It follows by (6) that

(b) We still have, by (102),

But (5) gives

Invoking Proposition 12, we have

So by addition, we conclude that (106) holds.

(c) We also have, using (102),

By Proposition 6, we have

Using also Proposition 6, we find

Still using Proposition 6 and invoking Proposition 12, we find

Adding the last three inequalities, we obtain (107).

(d) Similarly, by (102),

So

(e) Finally by (102) we have

So by Propositions 6 and 7, using (2) and the inequalities obtained for ,

where

This completes the proof of Proposition 13.

One requires in what follows that, for any real number ,

Theorem 14. Let , be given. Then the following holds.

There exists a real number such that the Cauchy problem has a unique solution:

satisfying . Moreover, satisfies the relation:

Proof. We apply the standard theory on the first-order differential systems to ----.

Since , , , are continuous functions of  , so is the function

By continuity of the functions at , there exists a real number such that

The previous relation implies, using (5) and (6) to bound , that

Next, set

Then

Now consider the neighborhood × × of in the Banach space and take

We deduce from the inequalities (105), (106), (107), (108), and (109) the definitions of  , , , , , the implications

and the relation  (126) that there exists a constant

such that

which shows that is locally Lipschitzian in with respect to the norm of the Banach space . The existence of a unique solution of the differential system ---- on an interval , , such that , is guaranteed by the standard theory on the first-order differential systems.

The relation is established in [8].

As a direct consequence, we can deduce that there exists a real number such that the Maxwell-Boltzmann-Euler system (20)-(21)-(40)-(52) in all Bianchi types to spacetimes has a unique solution on satisfying

6. The Global Existence

6.1. The Method

Let be the maximal existence domain of solution, denoted here by and given by Theorem 14, of the system ----, with the initial data . 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 can be extended beyond , which contradicts the maximality of .(c)The method is as follows: suppose and let . We want to show that there exists a strictly positive number independent of such that the system ---- on , with the initial data at , admits a unique solution on . Then, by taking sufficiently close to , for example, such that , and hence , we can extend the solution to , which strictly contains , and this contradicts the maximality of . For the need to simplify the notations, it will be enough if we could look for a number such that .(d)In what follows we fix a number and we take such that .

By (128) we can write

We also have from (139) using (128) that any solution of the Boltzmann equation on such that , satisfies the inequality:

Also notice that (140) shows that a solution of the system ---- on , , such that , satisfies

In what follows, is the maximal existence domain of solution of ---- such that

The following result shows helpful in what is to follow.

Lemma 15. , , , and are uniformly bounded over .

Proof. See [1].

6.2. Global Existence of Solutions

First of all, we consider, for and ,

Then, we built from system ---- by setting in its which is given by (102) in , , in , in , and , , in , the following useful differential system:

where

We prove the following.

Proposition 16. Let , , and be given. Then, the differential system (144)-(145)-(146)-(147)-(148) has a unique solution such that .

Proof. (a) We consider (144) in , with defined by (149) in which is fixed. Since , , , are continuous functions of  , so is . Next, we deduce from (105) in which we set that

where .

Now we can use (5) and (6) to bound , , and we obtain, for , then ,

We then deduce from (151) that

By (150) and (152), is Lipschitzian with respect to the -norm and the local existence of a solution of (144) such that is guaranteed by the standard theory of first-order differential systems.

Now, since satisfies (144) in which is given by (149), following the same way as in the proof of Lemma 15, substituting to , to , and integrating this time over , lead to

where . However, by Lemma 15, we have, since . Then, by Gronwall inequality,

where (see [1]) , are two constants appearing in Lemma 15 when we bound , which shows that every solution of (144) is uniformly bounded. By the standard theory of first-order differential systems, the solution is defined all over and .

(b) We also consider (145) in , with defined by (149) in which , are fixed. Since , , , , are continuous functions of , so is . Next, we deduce from (106) in which we set , that

where

Now we can use (5) and (6) to bound and we obtain, for , then ,

We then deduce from (157), using

since is uniformly bounded, and , since that

By (155) and (159), is Lipschitzian with respect to the -norm and the local existence of a solution of (145) such that is guaranteed by the standard theory of first-order differential systems.

Now, since satisfies (152) in which is given by (149), following the same way as in the proof of Lemma 15, substituting to , to , and to , and integrating this time over , , lead to

where . However, by Lemma 15, we have, since . Then, by Gronwall inequality,

which shows that every solution of (152) is uniformly bounded. By the standard theory of first-order differential systems, the solution is defined all over and .

(c) Next, we have proved in Theorem 10 that the single equation (146) in has a unique solution , substituting to , to , such that .

(d) For (147) in , we have by (108)

where

Equations (162) and (163) show that is Lipschitzian with respect to the -norm and the local existence of a solution of (147) such that is guaranteed by the standard theory on first-order differential systems.

Since satisfies (147) in which is given by (149), substituting to in Lemma 15, we find

Applying the Gronwall inequality this time over , , we find

and conclude that every solution of (147) is uniformly bounded. By the standard theory of first-order differential systems, the solution is defined all over and .

(e) In the end, for (148) in , with still defined by (149) in which , are fixed. Since , , , , are continuous functions of , so is . We deduce from (109) in which we set , that

where

Invoking (157) and using ,  , , since is uniformly bounded, , since , using

and Lemma 15, we find

We conclude by (166), (169) that is Lipschitzian with respect to the -norm and the local existence of a solution of (148) such that 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 and .

This ends the proof of Proposition 16.

We now set

is a complete metric subspace of the Banach space and is a complete metric subspace of the Banach space × .

With Proposition 16 we define the map:

We claim the following.

Proposition 17. Let . There exists a number ,   independent of , such that the system ---- has a unique solution such that = .

Proof. We will prove that there exists a number , independent of , such that the map , defined by (172), induces a contraction of the complete metric space defined by (170), which will then have a fixed point solution of the system --.

With the initial data at , the differential system (144)-(145)-(146)-(147)-(148) is equivalent for , and using notations (149) to the integral system

To , , correspond the solutions , , whose existence is proved in Proposition 16. We now write the integral system (173)-(174)-(175)-(176)-(177) for and , and taking the differences, we obtain the following, with :

(a)Now we can deduce, from (106) in which we set , ,

where is still given by (152).(b)Next, since , we deduce from (107) in which we set , , ,

where is given by (159).(c)We have by (108) in which we set , , since , and using is still given by (108):

where .(d)We also have, using (109),

where is given by (163).(e)Finally, from (126) in which we set , , , , using the fact that , , , are uniformly bounded and , , we deduce that

where is given by (169).

Already notice that the constants , , , , and are absolute constants independent of .

Now using the inequalities (179), (180), (181), (182), and (183), we deduce from (178), using the norm and since

Now add (184) to obtain

First of all we take such that

Then, reporting (186) in (185) and simplifying, we obtain

Secondly choosing such that

(187) becomes

from which we deduce that

Consequently, if we take

(190) shows that , defined by (172) induces a contraction in the complete metric space which then has a unique fixed point and solution of the integral system (173)-(174)-(175) and, hence, of the differential system -- such that . But using the fact that × and that , are uniformly bounded, we conclude by Proposition 16 that the subsystem - in is globally Lipschitzian in . So the unique solution , such that , is global on since and remain uniformly bounded.

Consequently, for , there exists a number , independent of , such that the system ---- has a unique solution such that

This completes the proof of Proposition 17.

Based on the method detailed in Section 6.1, we have proved the following result.

Theorem 18. Let , , , , , be given, such that , where is a given real number. Then,

(1)the differential system (59)-(60)-(61)-(62) has a unique global solution defined all over the interval and such that

(2)the Maxwell-Boltzmann-Euler system (20)-(21))-(40)-(41) in all Bianchi types to spacetimes has a unique global solution defined all over the interval and satisfying , , , .

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 unknown 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 spacetime in presence of a massive scalar field,” where the unknowns were like now, the electromagnetic field , subject to the Maxwell equations, the distribution function , subject to the Boltzmann equation, and the massive scalar field , subject to the Euler equations. We have generalized in this present work the result obtained for the Bianchi types spacetime to all Bianchi types to spacetimes. We have only excluded the case of Bianchi type 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 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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