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

Main Article Content

Foko Kamseu Maturin
Dongo David
Abel Nguelemo Kenfack
Woukeng Jean Louis


This paper provide evidence of the existence and uniqueness result for the viscosity solutions of inhomogeneous Vlasov equation. we consider the Cauchy-Dirichlet problem for the relativistic Vlasov equation with near vacuum initial data where the distribution function depends on the time, the position, the momentum and the non-Abelian charge of particles. We consider this equation on a
Schwarzschild outer space-time with Yang-Mills field.

Article Details

How to Cite
Maturin, F. K. ., David, D., Nguelemo Kenfack, A., & Jean Louis, W. (2022). Viscosity solutions for the relativistic inhomogeneous Vlasov equation in Schwarzschild outer space-time in the presence of the Yang-Mills field. Tamkang Journal of Mathematics, 54(3), 237–256.


A. Alves, Equation de liouville pour les particules de masse nulle, C. R. Acad. Sci. paris,278, (1987) 1151-1154.

R. D. Ayissi, R. M. Etoua and H. P. M. Tchagna, Viscosity solutions for the one-body liouville equation in Yang-Mills charged bianchi models with non-zero mass, Lett. Math. Phys., 105,(2015) 1289-1299.

G. Barles, Solutions de viscosit´e des ´equations de Hamilton-Jacobi, Springer-Verlag Berlin Heidelberg, (1994).

J. Binney and S. Tremaine, Galactic dynamics, Princeton University Press, (1987).

A. Bressan, Viscosity solutions of hamilton-jacobi equations and optimal control problems, Adv. Pure Appl. Math., (2001).

A. Briani and F. Rampazzo, A densty approach to hamilton-jacobi equation with measurable hamiltonians, Nonlinear differ. equa. appl., 12, (2005).

Y. Choquet-Bruhat, Existence and uniqueness for the Einstein-Maxwell-Vlasov system, Volume dedicated to Petrov, (1971).

M. G. Crandall and H. Ishii and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equation, Amer. Math. Soc., 27(1), (1992), 1-67.

D. Dongo and M. K. Foko, Viscosity solutions for the vlasov equation in the presence of a Yang-Mills Field in temporal gauge, Afr. Diaspora J. Math., 23(1), (2020) 24-39.

Y. Choquet-Bruhat et N. Noutchegueme, Solution globale des équations de Yang-Mills-Vlasov (masse nulle), C. R. Acad. Sci. Paris, 311, (1991).

Y. Choquet-Bruhat et N. Noutchegueme, Syst`eme de Yang-Mills-Vlasov en jauge temporelle, Ann. Inst. Henri Poincaré, 55, (1991).

L. C. Evans, Partial Differential Equations, A. M. S., (1997).

G. Rein and A. D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data commun, Math. Phys., 150, 561-583 (1992).

Shigeaki Koike, A Beginner’s Guide to the Theory of viscosity solutions, Society of Japon, (2002).

P. L. Lions, Generalized solutions of Hamilton-Jacobi equation, Research Notes in Mathematics, 69, (1982).

N. Noutchegueme and P. Noundjeu, Système de Yang-Mills-Valsov pour des particules avec des densités de charge de jauge non-abélienne sur un espace-temps courbe, Ann. Inst. Henri Pointcaré, (2000).

A. D. Rendall, Global properties of locally spatially homogeneous cosmological models with matter, Math. Proc. Camb. Phil. Soc, 118, (1995), 511-526.

A. D. Rendall and C. Uggla, Dynamics of spatially homogeneous locally rotationally symmetric solutions of the Einsein-Vlasov equations, Class. Quantum Grav, 17, (2000), 4697- 4714.

G. Borner, The early universe, Springer, 1971.

E. Takou and F. L. C. Ciake, The relativistic boltzmann equation on a spherically, symmetric gravitational Field Class. Quantum Grav., 34(1), (2017), 33.

Server S, Dragomir and Young-Ho Kim On nonlinear integral inequalities of Gronwall type in two variables, RGMIA research report collection, 6, (2003).

Most read articles by the same author(s)