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 γ ∈ (0, 1], with a bounded angular section modeled by a bounded function of the scattering angle. The initial data for the vector valued solutions needs to be a vector of nonnegative positive measures that will have finite mass, momentum and strictly positive energy, as well as to have finite L 1 2+2γ (R 3 )-integrability corresponding to a sum across each species of 2 + 2γ-polynomial weighted norms depending in the corresponding mass fraction parameter for each species, referred as to the scalar polynomial moment of order 2 + 2γ. In addition there is no assumption on the finiteness of the initial system associated scalar entropy. This set of initial data assumptions allows for some of the species to be a singular mass initially.The existence and uniqueness proof relies on a new angular averaging lemma adjusted to vector values solution that yield a Povzner estimate with explicit constants that decay with the order of the corresponding 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.

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

“…Finally, we also want to mention the very recent work [18] on the homogeneous multi-species Boltzmann system, for which it seems to be rather hard to conduct the sensitivity analysis and study the long-time behavior in the UQ setting, since the logarithmic entropy functional cannot be evaluated for the z-derivatives of the distribution function, due to their lack of positivity.…”

BGK model for multi-component gases near a global Maxwellian

“…The Vlasov-Poisson-Boltzmann equation was considered in [30] about large time asymptotic profiles when the different-species gases tend to two distinct global Maxwellians. In [32], the existence and uniqueness are constructed in spatially homogeneous settings when an initial data has upper and lower bounds for some polynomial moments. The authors in [15] obtained some energy estimates.…”

A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases

“…Furthermore, rigorous connections between (1.1) and the compressible Navier-Stokes equations for mixtures of fluids have been established [12]. Explicit solutions to the space homogeneous variant of (1.1) have been studied [10] in addition to recent proofs of global well posedness and propagation of moments [18,15]. However, despite the mathematical progress on the subject of the system (1.1), no work has been done on rigorously deriving the system from a system of hard spheres.…”