A gaseous mixture of four constituents undergoing a reversible bimolecular reaction is modeled by means of a Bhatnagar, Gross, and Krook ͑BGK͒-type equation in a flow regime close to chemical equilibrium. In the proposed relaxation method, elastic and chemistry collision terms are approximated separately, introducing different reference distribution functions which assure the correct balance laws. A Chapman-Enskog procedure is applied in order to provide explicitly the transport coefficients of diffusion, shear viscosity and thermal conductivity in dependence on elastic and reactive collision frequencies, mass concentrations of each species and temperature of the whole mixture. The closure of the balance equations is performed at the Navier-Stokes level and plane wave solutions are characterized. For the ͑H 2 , Cl, HCl, H͒ system, transport coefficients, as well as the Prandtl number of the mixture, are represented as functions of the temperature and compared with the inert case in order to discuss the influence of chemical reaction. Moreover, the thermal conductivity for nondiffusive and homogeneous mixtures are compared. For the problem of longitudinal wave propagation the phase velocity, attenuation coefficient and affinity are analyzed as functions of the wave frequency.

We consider the modified simple reacting spheres (MSRS) kinetic model that, in addition to the conservation of energy and momentum, also preserves the angular momentum in the collisional processes. In contrast to the line-of-center models or chemical reactive models considered in [23], in the MSRS (SRS) kinetic models, the microscopic reversibility (detailed balance) can be easily shown to be satisfied, and thus all mathematical aspects of the model can be fully justified. In the MSRS model, the molecules behave as if they were single mass points with two internal states. Collisions may alter the internal states of the molecules, and this occurs when the kinetic energy associated with the reactive motion exceeds the activation energy. Reactive and non-reactive collision events are considered to be hard spheres-like. We consider a four component mixture A, B, A * , B * , in which the chemical reactions are of the type A + B A * + B * , with A * and B * being distinct species from A and B. We provide fundamental physical and mathematical properties of the MSRS model, concerning the consistency of the model, the entropy inequality for the reactive system, the characterization of the equilibrium solutions, the macroscopic setting of the model and the spatially homogeneous evolution. Moreover, we show that the MSRS kinetic model reduces to the previously considered SRS model (e.g., [21], [27]) if the reduced masses of the reacting pairs are the same before and after collisions, and state in the Appendix the more important properties of the SRS system.

We construct a mathematical model of kinetic type in order to describe the immune system interactions in the context of autoimmune disease. The interacting populations are self-antigen presenting cells, self reactive T cells and the set of immunosuppressive cells consisting of regulatory T cells and Natural Killer cells. The main aim of our work is to develop a qualitative analysis of the model equations and investigate the existence of biologically realistic solutions. Having this goal in mind we describe the interactions between cells during an autoimmune reaction based on biological considerations that are given in the literature and we show that the corresponding system of integro-differential equations has finite positive solutions. The asymptotic behaviour of the solution of the system is also studied. We complement our mathematical analysis with numerical simulations that study the sensitivity of the model to parameters related to proliferation of immunosuppressive cells, destruction of self-antigen presenting cells and self reactive T cells and tolerance of SRTCs to self-antigens.

We consider physical and mathematical aspects of the model of simple reacting spheres (SRS) in the kinetic theory of chemically reacting fluids. The SRS, being a natural extension of the hard-sphere collisional model, reduces itself to the revised Enskog theory when the chemical reactions are turned off. In the dilute-gas limit, it provides an interesting kinetic model of chemical reactions that has not been considered before. In contrast to other reactive kinetic theories (e.g., line-of-centers models), the SRS has built-in detailed balance and microscopic reversibility conditions. The mathematical analysis of the work consists of global existence result for the system of partial differential equations for the model of SRS.

In this work we present some results on the kinetic theory of chemically reacting gases, concerning the model of simple reacting spheres (SRS) for a gaseous mixture undergoing a chemical reaction of type A 1 + A 2 A 3 + A 4. Starting from the approach developed in paper [11], we provide properties of the SRS system needed in the mathematical and physical analysis of the model. Our main result in this proceedings provides basic properties of the SRS system linearized around the equilibrium, including the explicit representations of the kernels of the linearized SRS operators.