Long - time behaviour through entropy methods

The Einstein-Vlasov System Cosmic Censorship Håkan Andreasson A central problem in general relativity is the cosmic censorship conjecture, i.e. the question about uniqueness in the large. This concerns the global properties of spacetime and a necessary prerequisite for attacking this problem is to obtain global existence results. In general this seems too difficult and symmetry assumptions on spacetime are imposed to reduce the complexity of the equations. It is also essential to carefully choose a matter model so that singularities do not form as a result of the matter model itself. A kinetic matter model has shown to be satisfying in this respect and the resulting system is the Einstein-Vlasov system. The global existence results for this system are reviewed in both the asymptotically flat case and in the cosmological case. In the latter case global existence results have been established for large initial data (for some symmetry classes), whereas in the asymptotically flat case only a small data result is known under the assumption of spherical symmetry. In order to investigate cosmic censorship for spherical symmetric gravitational collapse large data has to be considered and the difficulties (compared to the Vlasov-Poisson system) of this problem are discussed. A related model the NordströmVlasov model is also discussed. For this model global existence has recently been proved in the spherically symmetric case. Self-similar solutions and self-similar asymptotics for the Boltzmann equation with elastic and inelastic interactions. A. Bobylev (joint work with Carlo Cercignani) We present a review of recent results obtained jointly with Carlo Cercignani. The review is based on series of our publications in ”Journal of Statistical Physics” in 2002-2003. We consider the spatially homogeneous non-linear Boltzman equation (BE) for elastic and inelastic Maxwell models. Then, it is easy to show at the formal level that the equation admits a class of self-similar solutions of the form f(v, t) = A(t)F (v exp(−at)), a = const. Two main questions are: (a) does such non-negative solutions really exist? and (b) do they describe a large time asymptotics for some classes of initial data (self-similar asymptotics)? The answer to both questions is ”yes”. Moreover, we constructed two such solutions for the classical (elastic) BE in explicit form. For the same equation, we proved the existence of such solutions for any a > 0 and that they are asymptotic states for some (different for different a) classes of initial data having infinite second moment (energy). We note that the solutions are obviously eternal, they exist for both positive and negative values of time t, this is the first example of eternal solutions of BE. A by-product result was a clarification of the Krook-Wu conjecture (1977), we proved sufficient conditions under which the conjecture is true and large time asymptotics is described by the BKW mode. In case of inelastic Boltzmann equation, we proved recent Ernst-Brito conjecture (2002): any solution having at t = 0 the finite moment of an order p > 2 has the self-similar large time asymptotics (this result was obtained jointly also with G. Toscani). It was also shown that the limiting self-similar solution has a power-like high energy tail for almost all values of the restitution coefficient.