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 [122] and its references. The energy-Casimir method has been applied to the Einstein equations in [242].

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 [94]. For information on these results and on the classical Boltzmann equation in general see [56, 55]. Despite the non-uniqueness it is possible to show that all solutions tend to equilibrium at late times. This was first proved by Arkeryd [16] by non-standard analysis and then by Lions [168] 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 [96]. Global existence and convergence to equilibrium for classical solutions starting close to equilibrium was proved in [113]. 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 [7]. There is still 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 [180] contains fundamental errors.)

Theorems on Existence and Global Dynamics for the
Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr-2002-6
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