Abstract
The Einstein-Vlasov-Fokker-Planck system describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity. We study the Cauchy problem for spatially homogeneous and isotropic solutions and prove the existence of both global-in-time solutions and solutions that blow-up in finite time depending on the size of certain functions of the initial data. We also derive information on the large-time behavior of global solutions and toward the singularity for solutions which blow-up in finite time. Our results entail the existence of a phase of decelerated expansion followed by a phase of accelerated expansion, in accordance with the physical expectations in cosmology.
Highlights
The purpose of this paper is to study spatially homogeneous and isotropic solutions of the Einstein-Vlasov-Fokker-Planck system
It is assumed that diffusion takes place in a cosmological scalar field, which can be identified with the dark energy source responsible for the phase of accelerated expansion of the Universe
We show that for an open set of initial data which describe an initially decelerating expanding Universe, global solutions will eventually give rise to a phase of accelerated expansion, in agreement with the expectations in Cosmology [14]
Summary
The purpose of this paper is to study spatially homogeneous and isotropic solutions of the Einstein-Vlasov-Fokker-Planck system. The model describes the kinetic diffusion dynamics of self-gravitating particles within the Einstein theory of general relativity. It is assumed that diffusion takes place in a cosmological scalar field, which can be identified with the dark energy source responsible for the phase of accelerated expansion of the Universe. When relativistic effects are neglected, the motion of self-gravitating kinetic particles undergoing diffusion is described by the frictionless Vlasov-Poisson-FokkerPlanck system in the gravitational case, which is given by. Einstein-Vlasov-Fokker-Planck, blow-up, global existence, asymptotic behavior, cosmological expansion.
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