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4.2 Kinetic theory

Collisionless matter is known to admit a global singularity-free evolution in many cases. For self-gravitating collisionless matter, which is described by the Vlasov-Poisson system, there is a general global existence theorem [266236]. There is also a version of this which applies to Newtonian cosmology [280Jump To The Next Citation Point]. A more difficult case is that of the Vlasov-Maxwell system, which describes charged collisionless matter. Global existence is not known for general data in three space dimensions, but has been shown in two space dimensions [154155] and in three dimensions with one symmetry [153] or with almost spherically symmetric data [270].

A model system which has attracted some interest (see [18]) is the Nordström-Vlasov system where the Vlasov equation is coupled to a scalar field as in Nordström’s theory of gravitation. This is not a physically correct model but may be useful for obtaining mathematical insights. A similar procedure was used to look for numerical insights in [321]. At the moment the state of knowledge concerning this system can be summed up by saying that it is roughly equal to that available for the Vlasov-Maxwell system.

The nonlinear stability of static solutions of the Vlasov-Poisson system describing Newtonian self-gravitating collisionless matter has been investigated using the energy-Casimir method. For information on this see [165] and its references. The energy-Casimir method has been applied to the Einstein equations in [350].

For the classical Boltzmann equation, global existence and uniqueness of smooth solutions has been proved for homogeneous initial data and for data that are small or close to equilibrium. For general data with finite energy and entropy, global existence of weak solutions (without uniqueness) was proved by DiPerna and Lions [132]. For information on these results and on the classical Boltzmann equation in general see [7980]. Despite the non-uniqueness it is possible to show that all solutions tend to equilibrium at late times. This was first proved by Arkeryd [27] by non-standard analysis and then by Lions [235] without those techniques. It should be noted that since the usual conservation laws for classical solutions are not known to hold for the DiPerna-Lions solutions, it is not possible to predict which equilibrium solution a given solution will converge to. In the meantime, analogues of several of these results for the classical Boltzmann equation have been proved in the relativistic case. Global existence of weak solutions was proved in [134]. Global existence and convergence to equilibrium for classical solutions starting close to equilibrium was proved in [156]. On the other hand, global existence of classical solutions for small initial data is not known. Convergence to equilibrium for weak solutions with general data was proved by Andréasson [15]. Until recently there was no existence and uniqueness theorem in the literature for general spatially homogeneous solutions of the relativistic Boltzmann equation. A paper claiming to prove existence and uniqueness for solutions of the Einstein-Boltzmann system which are homogeneous and isotropic [250] contains fundamental errors. These problems were corrected in [263] and a global existence theorem for the special relativistic Boltzmann equation was obtained. In [262] this was generalized to a global existence theorem for LRS Bianchi type I solutions of the Einstein-Boltzmann system.

Further information on kinetic theory and its relation to general relativity can be found in the Living Review of Andréasson [17].

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