Modeling multi-stream flow in collisionless matter: approximations for large-scale structure beyond shell-crossing

The generally held view that a model of large-scale structure, formed by collisionless matter in the Universe, can be based on the matter model ``dust'' fails in the presence of multi-stream flow, i.e., velocity dispersion. We argue that models for large-scale structure should rather be constructed for a flow which describes the average motion of a multi-stream system. We present a clearcut reasoning how to approach the problem and derive an evolution equation for the mean peculiar-velocity relative to background solutions of Friedmann-Lema\^\i tre type. We consider restrictions of the nonlinear problem and show that the effect of velocity dispersion gives rise to an effective viscosity of non-dissipative gravitational origin. We discuss subcases which arise naturally from this approach: the ``sticky particle model'' and the ``adhesion approximation''. We also construct a novel approximation that features adhesive action in the multi-stream regime while conserving momentum, which was considered a drawback of the standard approximation based on Burger's equation. We finally argue that the assumptions made to obtain these models should be relaxed and we discuss how this can be achieved.