  We study the non-equilibrium statistical mechanics of a system of freely
moving particles, in which binary encounters lead either to an elastic
collision or to the disappearance of the pair. Such a system of {\em ballistic
annihilation} therefore constantly looses particles. The dynamics of
perturbations around the free decay regime is investigated from the spectral
properties of the linearized Boltzmann operator, that characterize linear
excitations on all time scales. The linearized Boltzmann equation is solved in
the hydrodynamic limit by a projection technique, which yields the evolution
equations for the relevant coarse-grained fields and expressions for the
transport coefficients. We finally present the results of Molecular Dynamics
simulations that validate the theoretical predictions.
