  Following the seminal work of F. Bouchut on zero pressure gas dynamics which
has been extensively used for gas particle-flows, the present contribution
investigates quadrature-based velocity moments models for kinetic equations in
the framework of the infinite Knudsen number limit, that is, for dilute clouds
of small particles where the collision or coalescence probability
asymptotically approaches zero. Such models define a hierarchy based on the
number of moments and associated quadrature nodes, the first level of which
leads to pressureless gas dynamics. We focus in particular on the four moment
model where the flux closure is provided by a two-node quadrature in the
velocity phase space and provide the right framework for studying both smooth
and singular solutions. The link with both the kinetic underlying equation as
well as with zero pressure gas dynamics is provided and we define the notion of
measure solutions as well as the mathematical structure of the resulting system
of four PDEs. We exhibit a family of entropies and entropy fluxes and define
the notion of entropic solution. We study the Riemann problem and provide a
series of entropic solutions in particular cases. This leads to a rigorous link
with the possibility of the system of macroscopic PDEs to allow particle
trajectory crossing (PTC) in the framework of smooth solutions. Generalized
$\delta$-choc solutions resulting from Riemann problem are also investigated.
Finally, using a kinetic scheme proposed in the literature without mathematical
background in several areas, we validate such a numerical approach in the
framework of both smooth and singular solutions.
