Ballistic aggregation in symmetric and non-symmetric flows

We consider exact solutions to the problem of ballistic aggregation in a flow of adhesive particles, providing a model for large-scale structure formation in cosmology within the framework of the Zel'dovich approximation. Two different explicit variational constructions, suggested for this problem by (A. Shnirelman, 1986) and (E Weinan et al, 1996) in the case of flat symmetry, are shown to be equivalent. Explicit solutions are constructed for flows of cylindric and spherical symmetry. Two explicit counterexamples, showing that natural generalizations of both variational constructions fail to provide solutions for nonsymmetric multidimensional flows, are presented.