  The hydrodynamic description of probabilistic ballistic annihilation, for
which no conservation laws hold, is an intricate problem with hard sphere-like
dynamics for which no exact solution exists. We consequently focus on
simplified approaches, the Maxwell and very hard particles (VHP) models, which
allows us to compute analytically upper and lower bounds for several
quantities. The purpose is to test the possibility of describing such a far
from equilibrium dynamics with simplified kinetic models. The motivation is
also in turn to assess the relevance of some singular features appearing within
the original model and the approximations invoked to study it. The scaling
exponents are first obtained from the (simplified) Boltzmann equation, and are
confronted against Monte Carlo simulation (DSMC technique). Then, the
Chapman-Enskog method is used to obtain constitutive relations and transport
coefficients. The corresponding Navier-Stokes equations for the hydrodynamic
fields are derived for both Maxwell and VHP models. We finally perform a linear
stability analysis around the homogeneous solution, which illustrates the
importance of dissipation in the possible development of spatial
inhomogeneities.
