%0 Journal Article
%T The Maxwell-Boltzmann-Euler System with a Massive Scalar Field in All Bianchi Spacetimes
%A Raoul Domingo Ayissi
%A Norbert Noutchegueme
%A Hugues Paulin Mbeutcha Tchagna
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/679054
%X 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
%U http://www.hindawi.com/journals/amp/2013/679054/