Viscosity Solutions for the One-Body Liouville Equation in Yang–Mills Charged Bianchi Models with Non-Zero Mass

Abstract

Recently in 2005, Briani and Rampazzo (Nonlinear Differ Equ Appl 12:71-91, 2005) gave, using results of Crandall and Lions (Ill J Math 31:665-688, 1987), Ishii (Indiana Univ Math J 33: 721-748, 1984, Bull Fac Sci Eng 28: 33-77, 1985 and Ley (Adv Diff Equ 6:547-576, 2001) a density approach to Hamilton-Jacobi equations with t-measurable Hamiltonians. In this paper we show, using an important result of Briani and Rampazzo (Nonlinear Differ Equ Appl 12:71-91, 2005) the existence and uniqueness of viscosity solutions to the one-body Liouville relativistic equation in Yang-Mills charged Bianchi space times with non-zero mass. To our knowledge, the method used here is original and thus, totally different from those used in Alves (C R Acad Sci Paris Sér

Regular Solution for the Generalized Relativistic Boltzmann Equation in Yang-Mills Field

“…Proof. For i), see [3]. 2i) is immediate using i) and the fact that the Yang-Mills potential A and the Yang-Mills field F are with compact support.…”

“…In the collisionless case, the Boltzmann equation is replaced by the Vlasov equation, many authors have already studied this kind of phenomenon: Y. Choquet-Bruhat and N. Noutchegueme in [7] studied the Yang-Mills-Vlasov system using the characteristic method and obtained a local in time existence result. They also studied in [8] the Yang-Mills-Vlasov system only for the zero mass particles case and obtained a global existence theorem in Minkowski space-time for small initial data; N. Noutchegueme and P. Noudjeu in [19] proved a local in time existence theorem of solutions of the Cauchy problem for the Yang-Mills system in temporal gauge with current generated by a distribution function that satisfies a Vlasov equation; R. D. Ayissi and al in [3] obtained the viscosity solutions for the one-Body Liouville equation in Yang-Mills charged Bianchi models with non-zero mass.…”

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

“…For more details about the notion of viscosity solutions, we refer interested readers to ( [3], [5], [6], [14], [15]) and references therein. The method has not yet been widely used in kinetic theory where one of the main equations is that of Vlasov, more specifically in the framework of general relativity, excluding works ( [2], [9]). The relativistic Vlasov equation describes the collision free evolution of massives particles where speeds are relatively high.…”

“…The characteristic feature of kinetic theory is that, its models are statistical and particles system are described by a distribution function f = f (t, x, p, q); which represents the density of particles at a given space-time position (t, x) ∈ R + × R 3 , with momentum p ∈ R 4 and non-Abelian charges of particles q ∈ R N . 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]).…”

Regular Solutions for the relativistic Boltzmann equation in Yang-Mills field

“…N. Noutchegueme and P. Noudjeu in [15] proved a local in time existence theorem of solutions of the Cauchy problem for the Yang-Mills system in temporal gauge with current generated by a distribution function that satisfies a Vlasov equation. R. D. Ayissi and al in [1] obtained the viscosity solutions for the one-Body Liouville equation in Yang-Mills charged Bianchi models with non-zero mass.…”