Merging and fragmentation in the Burgers dynamics
  We explore the noiseless Burgers dynamics in the inviscid limit, the
so-called ``adhesion model'' in cosmology, in a regime where (almost) all the
fluid particles are embedded within point-like massive halos. Following
previous works, we focus our investigations on a ``geometrical'' model, where
the matter evolution within the shock manifold is defined from a geometrical
construction. This hypothesis is at variance with the assumption that the usual
continuity equation holds but, in the inviscid limit, both models agree in the
regular regions. Taking advantage of the formulation of the dynamics of this
``geometrical model'' in terms of Legendre transforms and convex hulls, we
study the evolution with time of the distribution of matter and the associated
partitions of the Lagrangian and Eulerian spaces. We describe how the halo mass
distribution derives from a triangulation in Lagrangian space, while the dual
Voronoi-like tessellation in Eulerian space gives the boundaries of empty
regions with shock nodes at their vertices. We then emphasize that this
dynamics actually leads to halo fragmentations for space dimensions greater or
equal to 2 (for the inviscid limit studied in this article). This is most
easily seen from the properties of the Lagrangian-space triangulation and we
illustrate this process in the two-dimensional (2D) case. In particular, we
explain how point-like halos only merge through three-body collisions while
two-body collisions always give rise to two new massive shock nodes (in 2D).
This generalizes to higher dimensions and we briefly illustrate the
three-dimensional (3D) case. This leads to a specific picture for the
continuous formation of massive halos through successive halo fragmentations
and mergings.
