On a discrete Boltzmann-Smoluchowski equation with rates bounded in the velocity variables
The Boltzmann-Smoluchowski equation (BSE) describes a system of particles that undergo both velocity scattering and coalescence phenomena. In the space-homogeneous case, considered by the authors, each particle is characterized by its mass \(m \in (0,\infty)\) and velocity \(v \in {\mathbb R}^3\) and the BSE is written for the time-dependent density \(f(t,m,v) \geq 0\). The equation is characterized by suitable scattering and coagulation rate functions. In the present paper the authors consider the simplified case in which the mass variable is discrete and the rate functions are assumed to be bounded in the velocity variables, deferring the study of the general case to a subsequent work. The BSE is studied in a ``indirect'' way, the paper being rather focused on the associated pure jump Markov process \((M_t,V_t)_{t \geq 0}\). This can be interpreted as the evolution of mass and velocity of a ``typical particle''. The relation to the solution of the BSE is given by \[ f(t,m,v) = \tfrac{1}{m}\,P \left[ M_t = m, V_t \in dv \right]. \] Using such stochastic interpretation of the BSE, the authors achieve two main results: 1. an exact simulation scheme of the process \((M_t,V_t)_{t \geq 0}\) is derived, which yields an immediate and constructive existence proof for the BSE; 2. it is shown that, as \(t\) tends to infinity, \(M_t\) tends almost surely to infinity, which in particular implies that the solution \(f(t,\,\cdot\,,\,\cdot\,)\) of the BSE tends to 0 in \(L^1\).
scientific article; zbMATH DE number 5004776