Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data

Abstract

The Cauchy problem for the inelastic Boltzmann equation is studied for small data. Existence and uniqueness of mild and weak solutions is obtained for sufficiently small data that lies in the space of functions bounded by Maxwellians. The technique used to derive the result is the well known iteration process of Kaniel & Shinbrot. The theory developed by DiPerna & Lions in the 90’s [13] on what is called Renormalized solutions has been a great success in finding existence theorems for the Boltzmann equation (BE): The Cauchy problem in [13] and the Boundary value problem [21]. The theory is strong and flexible and can be adapted to find solutions for different problems, for instance: The Vlasov-Poisson-Boltzmann system (VPB) [24], the treatment of the BE with infinite energy [26], the relativistic Boltzmann equation [14] and others. Indeed, a great deal of applications of this theory has been written in the last 18 years on BE related problems. The theory is based in the bounded entropy which is a feature of the elastic BE solutions. Unfortunately, it is not known how to obtain an a priori estimate that confirms such feature for the inelastic BE solutions, even in the case where a cut-off is imposed to the collision kernel. This simple fact creates a big upset for the theory in the inelastic case. It is important to say that in the one dimensional case, Benedetto & Pulvirenti overcame this problem in [4] provided that the initial datum is essentially bounded and has compact support in the velocity space. The proof uses an iterative process introduced by J. M. Bony that relies strongly on the dimension. The technique allows to find a uniform control on the entropy, and hence, to prove existence and uniqueness provided the initial datum has the afore mentioned properties. It is clear that more understanding in the collision operator is needed to solve the inelastic BE in its simplest form: The Cauchy problem. More complex problems, like the initial/boundary value problem or the VPB system, are still out of hand. This paper returns to the late 70’s and presents an application for the inelastic Boltzmann problem of the technique introduced by Kaniel & Shinbrot in that time. Known as Kaniel & Shinbrot iterates [23], this technique was created by these authors to find existence and uniqueness