Fluid Dynamic Limit of the Boltzmann Kinetic Equation Arising in the Coagulation—Fragmentation Dynamics

Abstract

This paper is devoted to establishing a connection between the kinetic coagulation—fragmentation theory and macroscopic dynamics. We derive the fluid dynamic limit and discuss the corollaries. For celebrated Boltzmann equation from gas dynamics, which describes elastic interactions of particles, the fluid dynamics limit is, at some assumptions, just five Euler equations of hydrodynamics. This number of hydrodynamic equations arises due to five conservation laws for solutions of the Boltzmann equation of gas kinetics. In our case particles may merge and/or split and, thus, the interactions are nonelastic. So, the energy conservation law fails. The conservation law for the total number of particles fails, too. The conservation laws for impulse turns into the only conservation law — mass (or density) conservation law expressed by the first moment of the distribution function. So, the aim of this paper is derivation and justification a new equation of macroscopic dynamics for density. The importance of this research is pointed out, e.g., in book 1, p.175, where an attempt to derive the fluid dynamic limit for coagulation processes is done. However, the resulting equations in1 (they are not justified mathematically) include integrals of the distribution function and, thus, cannot be considered as a closed macrodynamic system. So, the aim of this paper is deriving and justifying a closed macroscopic equation for evolution of mass density of the substance.