Abstract:
The kinetic motion of the stars of a galaxy is considered within the framework of a relativistic scalar theory of gravitation. This model, even though unphysical, may represent a good laboratory where to study in a rigorous, mathematical way those problems, like the influence of the gravitational radiation on the dynamics, which are still beyond our present understanding of the physical model represented by the Einstein--Vlasov system. The present paper is devoted to derive the equations of the model and to prove the existence of spherically symmetric equilibria with finite radius.

Abstract:
A new model to describe the dynamics of particles undergoing diffusion in general relativity is proposed. The evolution of the particle system is described by a Fokker-Planck equation without friction on the tangent bundle of spacetime. It is shown that the energy-momentum tensor for this matter model is not divergence-free, which makes it inconsistent to couple the Fokker-Planck equation to the Einstein equations. This problem can be solved by postulating the existence of additional matter fields in spacetime or by modifying the Einstein equations. The case of a cosmological scalar field term added to the left hand side of the Einstein equations is studied in some details. For the simplest cosmological model, namely the flat Robertson-Walker spacetime, it is shown that, depending on the initial value of the cosmological scalar field, which can be identified with the present observed value of the cosmological constant, either unlimited expansion or the formation of a singularity in finite time will occur in the future. Future collapse into a singularity also takes place for a suitable small but positive present value of the cosmological constant, in contrast to the standard diffusion-free scenario.

Abstract:
We consider a modified version of the Vlasov-Maxwell system in which the usual Maxwell fields are replaced by their retarded parts. We show that solutions of this modified system exist globally for a small number of particles and that they describe a system without incoming radiation

Abstract:
The relativistic Boltzmann equation for a constant differential cross section and with periodic boundary conditions is considered. The speed of light appears as a parameter $c>c_0$ for a properly large and positive $c_0$. A local existence and uniqueness theorem is proved in an interval of time independent of $c>c_0$ and conditions are given such that in the limit $c\to +\infty$ the solutions converge, in a suitable norm, to the solutions of the non-relativistic Boltzmann equation for hard spheres.

Abstract:
The asymptotic properties at future null infinity of the solutions of the relativistic Vlasov-Maxwell system whose global existence for small data has been established by the author in a previous work are investigated. These solutions describe a collisionless plasma isolated from incoming radiation. It is shown that a non-negative quantity associated to the plasma decreases as a consequence of the dissipation of energy in form of outgoing radiation. This quantity represents the analogue of the Bondi mass in general relativity.

Abstract:
A new type of fluid matter model in general relativity is introduced, in which the fluid particles are subject to velocity diffusion without friction. In order to compensate for the energy gained by the fluid particles due to diffusion, a cosmological scalar field term is added to the left hand side of the Einstein equations. This hypothesis promotes diffusion to a new mechanism for accelerated expansion in cosmology. It is shown that diffusion alters not only quantitatively, but also qualitatively the global dynamical properties of the standard cosmological models.

Abstract:
A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the velocity field, the energy function and the diffusion matrix, it is shown that global regular solutions converge in time to equilibrium with exponential rate. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by Villani. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by Bakry and Emery. The result applies to the relativistic Fokker-Planck equation in the low temperature regime, for which exponential trend to equilibrium was previously unknown.

Abstract:
The Nordstr\"om-Vlasov system describes the evolution of self-gravitating collisionless matter in the framework of a relativistic scalar theory of gravitation. We prove global existence and uniqueness of classical solutions for the corresponding initial value problem in three dimensions when the initial data for the scalar field are smooth and the initial particle density is smooth with compact support.

Abstract:
The relativistic Vlasov-Maxwell system of plasma physics is considered with initial data on a past light cone. This characteristic initial value problem arises in a natural way as a mathematical framework to study the existence of solutions isolated from incoming radiation. Various consequences of the mass-energy conservation and of the absence of incoming radiation condition are first derived assuming the existence of global smooth solutions. In the spherically symmetric case, the existence of a unique classical solution in the future of the initial cone follows by arguments similar to the case of initial data at time $t=0$. The total mass-energy of spherically symmetric solutions equals the (properly defined) mass-energy on backward and forward light cones.

Abstract:
The existence and the properties of isolated solutions to the relativistic Vlasov-Maxwell system with initial data on the backward hyperboloid $t=-\sqrt{1+|x|^2}$ are investigated. Isolated solutions of Vlasov-Maxwell can be defined by the condition that the particle density is compactly supported on the initial hyperboloid and by imposing the absence of incoming radiation on the electromagnetic field. Various consequences of the mass-energy conservation laws are derived by assuming the existence of smooth isolated solutions which match the inital data. In particular, it is shown that the mass-energy of isolated solutions on the backward hyperboloids and on the surfaces of constant proper time are preserved and equal, while the mass-energy on the forward hyperboloids is non-increasing and uniformly bounded by the mass-energy on the initial hyperboloid. Moreover the global existence and uniqueness of classical solutions in the future of the initial surface is established for the one dimensional version of the system.