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