Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson–Walker space-time

Abstract

In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson-Walker space-time. Unique global (in time) mild solution is obtained in a suitable weighted space. written as:(1.1) in (1.1), R(t) is given and is called the cosmological expansion factor. In fact, the RW metric is an exact solution of Einstein's field equations of General Relativity; it describes a homogeneous, isotropic expanding or contracting universe. The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time.We consider in this work particles with the same rest-mass m that can be rescaled to m = 1. The particles are then required to move on the future sheet of the mass-shell whose equationThe main difficulty while studying the Boltzmann equation lays in the collision kernel. Glassey derived the collision kernel in [8] for the Newtonian case, but for the relativistic case for a more detail description of the scattering kernel, we refer to [5].Choquet-Bruhat, Y.[4] defined the µ − N regularity of the collision operator, which appears in the r.h.s of the Boltzmann equation. Under the condition of µ − N regularity, several authors studied and proved some existence theorems for the relativistic Boltzmann equation. Takou, E., Noutchegueme, N. and Dongo, D.[18,19,20] proved some global results in the homogeneous cases under assumptions close to µ − N regularity. Bancel, D.[1] also proved one local result under the µ − N regularity condition. Glassey, R.[9] proved a global existence theorem with near vacuum initial data in the Minkowski space-time.The use of µ − N regularity doesn't allow a very good physical description of the collision operator. In fact, this operator depends on several terms including the collision kernel, the relative momentum and the energy in the center of momentum. Furthermore, one of the main terms in the collision kernel is the scattering kernel which measures interactions between particles.In the Newtonian Boltzmann equation, scattering kernels are usually classified into soft and hard potentials. This classification was originally adapted in the relativistic case by Dudyński, M. and Ekiel-Jerżewska, M.[6] and recently reformulated by Strain, R. in [21]. This reformulation increases the importance and the interest of the relativistic Boltzmann equation. With this reformulation, Strain, R. op.cit. proved one global result in the Minkowski space-time. Lee, H. and Rendall, A., D. proved in [13] the positivity of a possible solution of the coupled Einstein-Boltzmann equation and also proved global solution in certain homogeneous cases in [14,15].In this paper, we consider the relativistic Boltzmann equation w...

The relativistic Boltzmann equation on a spherically symmetric gravitational field

“…The notable work of Glassey [8] provides global existence of mild solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data for an appropriate class of scattering cross sections; the work was carried out in a static Minkowski spacetime. With homogeneous Robertson-Walker spacetime, the authors proved in [22] an unique global (in time) mild solution in a suitable weighted space. To our knowledge, there is not yet such a result for the relativistic Boltzmann equation in an inhomogeneous spacetime as background.…”