Abstract:  Smoluchowski coagulation kinetic equation (1916) assumes that two colliding particles just merge together with some probability. However, the experiments demonstrate that the increase in the particle size is overlapped by splitting the particles during a shorter time interval. These observations contradict the Smoluchowski's approach and demonstrate the necessity to reconsider it on a new basis. We simulate such collisions from a new viewpoint with short-time splitting taken into account and derive a new balance kinetic model. We compare analytically and numerically the results for both coagulation models (Smoluchowski and the new one) and show a surprising coincidence in the results: both models have same or almost same time-dependent solutions, equilibria, critical phase transition times, etc. Such a coincidence explains why Smoluchowski equation fits well to the reality though this well known model is based on the assumptions that are far from the reality.

If the coagulation processes happen along with fragmentation then the analysis of the equilibrium solution leads us to the derivation of a hydrodynamic limit equation in the form of a scalar conservation law. This new hydrodynamic model substitutes the Navier-Stokes equations for the dynamics of coagulating-fragmenting flows.

In this talk I would like to describe the derivation of these new models and discuss their principal mathematical properties.