On existence and uniqueness to homogeneous Boltzmann flows of monatomic gas mixtures

Abstract

We solve the Cauchy problem for the full non-linear homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions in three dimensions. More precisely, we show existence and uniqueness of the vector value solution by means of an existence theorem for ODE systems in Banach spaces under the transition probability rates assumption corresponding to hard potentials rates in the interval $(0,1]$, with an angular section modeled by an integrable function of the angular transition rates modeling binary scattering effects. The initial data for the vector valued solutions needs to be a vector of non-negative measures with finite total number density, momentum and strictly positive energy, as well as to have a finite $L^1_{k_*}(\mathbb{R}^3)$-integrability property corresponding to a sum across each species of $k_*$-polynomial weighted norms depending on the corresponding mass fraction parameter for each species as much as on the intermolecular potential rates, referred as to the scalar polynomial moment of order $k_*$. The existence and uniqueness rigorous results rely on a new angular averaging lemma adjusted to vector values solution that yield a Povzner estimate with constants that decay with the order of the corresponding dimensionless scalar polynomial moment. In addition, such initial data yields global generation of such scalar polynomial moments at any order as well as their summability of moments to obtain estimates for corresponding scalar exponentially decaying high energy tails, referred as to scalar exponential moments associated to the system solution. Such scalar polynomial and exponential moments propagate as well.