The large-scale structure of the Universe in the frame of the model equation of non-linear diffusion

The evolution of density inhomogeneities and the velocity field in an expanding continuous medium is studied. The consideration is based on the model equation of non-linear diffusion (Burgers' equation), that, together with the equation of continuity incorporating mass density, gives an approximate description of density inhomogeneity growth at the advanced non-linear stage of gravitational instability. The model describes the formation of pancakes, filaments and compact clumps of mass as well as the disruption of the cellular and filamentary structure and the process of merging of the clumps that follows later. At this stage, the statistics of non-linear density peaks, as well as the velocity field, are determined by statistical properties of the fluctuations of the gravitational potential at linear stage. The model is in a sense an extrapolation of Zel'dovich's non-linear solution for gravitational instability of an expanding universe. It does not attempt to describe the internal structure of density enhancements but rather follows the general evolution of inhomogeneities. The model provides an analytic expression for the velocity field and makes it possible to calculate statistical characteristics of an ensemble of density clumps including their masses, velocities and spatial distribution.