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Abstract. Solutions of the Boltzmann equation are proved to exist, globally in time, under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough. It is also proved, as might be expected in this case, that the density of the gas at each point in space goes to zero as time goes to infinity.

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We prove global existence and uniqueness of classical solutions of the Wigner-Poisson and SchrodingerPoisson systems of equations for both repulsive and attractive potentials. In the repulsive case, we prove decay estimates for the particle density, the potential and the solutions.

We consider a system of N hard disks in U 2 in the Boltzmann-Grad limit (i.e. N-» oo, d \ 0, ΛΓ d->Λ~ * >0, where d is the diameter of the disks). If λ is sufficiently small and if the joint distribution densities factorize at time zero, we prove that the time-evolved one-particle distribution converges for all times to the solution of the Boltzmann equation with the same initial datum.