Some remarks about the scaling of systems of reactive Boltzmann equations
Marzia Bisi Laurent Desvillettes
Kinetic & Related Models 2008, 1(4): 515-520 doi: 10.3934/krm.2008.1.515
An asymptotics leading from the reactive Boltzmann equation towards reaction--diffusion equations has been introduced in [1] (cf. also [10], 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.
keywords: Reactive Boltzmann equations; Reaction--diffusion systems
A Boltzmann-type model for market economy and its continuous trading limit
Marzia Bisi Giampiero Spiga
Kinetic & Related Models 2010, 3(2): 223-239 doi: 10.3934/krm.2010.3.223
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.
keywords: Boltzmann equation; Wealth distribution; Econophysics.
On a kinetic BGK model for slow chemical reactions
Marzia Bisi Giampiero Spiga
Kinetic & Related Models 2011, 4(1): 153-167 doi: 10.3934/krm.2011.4.153
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.
keywords: Kinetic theory; Chemical reaction; BGK model.
On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases
Céline Baranger Marzia Bisi Stéphane Brull Laurent Desvillettes
Kinetic & Related Models 2018, 11(4): 821-858 doi: 10.3934/krm.2018033

In this paper, we propose a formal derivation of the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic gases. We use a direct extension of the model devised in [8,16] for treating the internal energy with only one continuous parameter. This model is based on the Borgnakke-Larsen procedure [6]. We detail the dissipative terms related to the interaction between the gradients of temperature and the gradients of concentrations (Dufour and Soret effects), and present a complete explicit computation in one case when such a computation is possible, that is when all cross sections in the Boltzmann equation are constants.

keywords: Boltzmann equations gas mixtures Chapman-Enskog procedure
A general consistent BGK model for gas mixtures
Alexander V. Bobylev Marzia Bisi Maria Groppi Giampiero Spiga Irina F. Potapenko
Kinetic & Related Models 2018, 11(6): 1377-1393 doi: 10.3934/krm.2018054

We propose a kinetic model of BGK type for a gas mixture of an arbitrary number of species with arbitrary collision law. The model features the same structure of the corresponding Boltzmann equations and fulfils all consistency requirements concerning conservation laws, equilibria, and H-theorem. Comparison is made to existing BGK models for mixtures, and the achieved improvements are commented on. Finally, possible application to the case of Coulomb interaction is briefly discussed.

keywords: Kinetic theory Boltzmann equation BGK models, gas mixtures gas mixtures exchange rates
Flame structure from a kinetic model for chemical reactions
Marzia Bisi Maria Groppi Giampiero Spiga
Kinetic & Related Models 2010, 3(1): 17-34 doi: 10.3934/krm.2010.3.17
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.
keywords: Kinetic theory; Irreversible chemical reactions; Deflagration waves.
Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics
Marzia Bisi Tommaso Ruggeri Giampiero Spiga
Kinetic & Related Models 2018, 11(1): 71-95 doi: 10.3934/krm.2018004

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.

keywords: Extended thermodynamics kinetic theory moment equations polyatomic gas dynamical pressure
Qualitative analysis of the generalized Burnett equations and applications to half--space problems
Marzia Bisi Maria Paola Cassinari Maria Groppi
Kinetic & Related Models 2008, 1(2): 295-312 doi: 10.3934/krm.2008.1.295
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 obtained.
keywords: Burnett equations; Hydrodynamics; Half-space Problems.

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