Correspondence between the adhesion model and the velocity dispersion   for the cosmological fluid
  Basing our discussion on the Lagrangian description of hydrodynamics, we studied the evolution of density fluctuation for nonlinear cosmological dynamics. Adhesion approximation (AA) is known as a phenomenological model that describes the nonlinear evolution of density fluctuation rather well and that does not form a caustic. In addition to this model, we have benefited from discussion of the relation between artificial viscosity in AA and velocity dispersion. Moreover, we found it useful to regard whether the velocity dispersion is isotropic produces effective `pressure' or viscosity terms. In this paper, we analyze plane- and spherical-symmetric cases and compare AA with Lagrangian models where pressure is given by a polytropic equation of state.   From our analyses, the pressure model undergoes evolution similar to that of AA until reaching a quasi-nonlinear regime. Compared with the results of a numerical calculation, the linear approximation of the pressure model seems rather good until a quasi-nonlinear regime develops. However, because of oscillation arising from the Jeans instability, we could not produce a stable nonlinear structure. 
Takayuki Tatekawa