An asymptotics leading from the reactive Boltzmann equation towards reaction--diffusion equations
has been introduced in  (cf. also , for an analogous scaling starting from
reactive BGK equations).
We propose here a justification of this asymptotics, at the formal level,
based on a non--dimensional form of the original equations.
In the frame of a kinetic Boltzmann-type approach to modeling market economies, a random conservative-in-the-mean scheme is proposed for binary transactions among agents. The scheme extends a very successful model recently introduced by Cordier, Pareschi and Toscani. Effects of the risky market on the overall output after the trade of each agent are accounted for by random variables affecting not only the wealth of that agent before the trade, but also the one of his partner. Variations induced by this generalization on steady distribution, existence of moments, and Pareto index are discussed. In particular, the continuous trading limit and the relevant limiting Fokker-Planck equation are commented on in detail.
A recently proposed consistent BGK-type approach for chemically reacting gas mixtures is discussed,
which accounts for the correct rates of transfer for mass, momentum and energy, and recovers the exact conservation equations and collision equilibria,
including mass action law. In particular, the hydrodynamic limit is derived by a Chapman-Enskog procedure,
and compared to existing results for the reactive and non-reactive cases.
Steady one-dimensional flame structure is investigated in a binary mixture made up by two components of the same chemical species
undergoing binary irreversible exothermic reactive encounters. A kinetic model at the Boltzmann level,
accounting for chemical transitions as well as for mechanical collisions, is proposed and its main features are analyzed.
In the case of slow chemical reactions and collision dominated regime, the model is the starting point for a consistent derivation,
via suitable asymptotic expansion of Chapman-Enskog type, of reactive Navier-Stokes equations at the fluid-dynamic scale.
The resulting set of ordinary differential equations is investigated in the frame of the qualitative theory of dynamical systems,
and numerical results are presented and briefly commented on for illustrative purposes.
The Generalized Burnett Equations, very recently introduced by Bobylev [3,4], are tested versus Fluid--Dynamic applications,
considering the classical steady evaporation/condensation problem. By means of the methods of the qualitative theory of dynamical systems,
comparison is made to other kinetic and hydrodynamic models, and indications on an appropriate choice of the disposable parameters are
The aim of this paper is to compare different kinetic approaches to a polyatomic rarefied gas: the kinetic approach via a continuous energy parameter $I$ and the mixture-like one, based on discrete internal energy. We prove that if we consider only $6$ moments for a non-polytropic gas the two approaches give the same symmetric hyperbolic differential system previously obtained by the phenomenological Extended Thermodynamics. Both meaning and role of dynamical pressure become more clear in the present analysis.