The relativistic Boltzmann equation on a spherically symmetric gravitational field

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. We consider this equation on a spherically symmetric gravitational field spacetime. The collision kernel considered here is for the hard potentials case. We prove the existence of a unique global (in time) mild solution in a suitable weighted space.

Viscosity solutions for the relativistic inhomogeneous Vlasov equation in Schwarzschild outer space-time in the presence of the Yang-Mills field

“…Thus, for the transformation of the equation (2.1), we shall need expressions of the Christofell symbols Γ i αβ which are given by (see [20]):…”

“…In order to find a more simplified form of equation (2.1), we consider the new momenta variables as in ( [13], [20]). Before we define the geometric framework on R 3 characterized by vectors defined in the following way:…”

“…Note that, here we are working in the presence of Yang-Mills particles, see ( [2], [9], [11], [16]) and references therein for the notion on Yang-Mills charges. Many authors have already studied the relativistic Vlasov equation, taking it alone or in association with other field equations, see ( [2], [7], [9], [10], [11], [13], [16], [18], [20]). In the work carried out in ( [9]) by two of authors, the study of the Vlasov equation was made in a homogeneous background and they obtained a Cauchy problem for the Hamilton-Jacobi equation.…”

“…This work is therefore an extension of work [9] for two reasons: we work here in a non-homogeneous framework and the Hamilton-Jacobi problem that we obtain is a Cauchy-Dirichlet problem. The rest of the paper is organized as follows: In section 2, following Etienne Takou and Fidèle L. Ciake Ciake [20], we provide the correct formulation of the relativistic Vlasov equation in Cartesian coordinates, taking as the background space-time a Schwarzschild outer space-time in presence of a given Yang-Mills field. In section 3, we give the relativistic Vlasov equation in the new variables following [20], next we give the Hamilton-Jacobi equation equivalent of the relativistic Vlasov equation and we prove energy estimates.…”

“…The rest of the paper is organized as follows: In section 2, following Etienne Takou and Fidèle L. Ciake Ciake [20], we provide the correct formulation of the relativistic Vlasov equation in Cartesian coordinates, taking as the background space-time a Schwarzschild outer space-time in presence of a given Yang-Mills field. In section 3, we give the relativistic Vlasov equation in the new variables following [20], next we give the Hamilton-Jacobi equation equivalent of the relativistic Vlasov equation and we prove energy estimates. In section 4, we prove existence and uniqueness theorems for the relativistic Vlasov equation.…”