  Structure formation in the Universe has been well-studied within the Eulerian
and Lagrangian perturbation theories, where the latter performs substantially
better in comparison with N-body simulations. Standing out is the celebrated
Zel'dovich approximation for dust matter. In this work, we recall the
description of gravitational noncollisional systems and extend both the
Eulerian and Lagrangian approaches by including, possibly anisotropic, velocity
dispersion. A simple case with plane symmetry is then studied with an exact,
nonperturbative approach, and various approximations of the derived model are
then compared numerically. A striking result is that linearized Lagrangian
solutions outperform models based on Burgers' equation in the multistream
regime in comparison with the exact solution. These results are finally
extended to a 3D case without symmetries, and master equations for the
evolution of all parts of the perturbations are derived. The particular 3D case
studied corresponds to a maximally anisotropic collapse, which involves an
approximation based on the estimation of importance of the different levels of
spatial derivatives of the local deformation field.
