Entropy dissipation estimates for the linear Boltzmann operator

We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignani's conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, both with and without a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to Matthes and Toscani (2012) and Villani (1998). For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities

A BGK relaxation model for polyatomic gas mixtures

We present a BGK approximation of a kinetic Boltzmann model for a mixture of polyatomic gases, in which non-translational degrees of freedom of each gas are represented by means of a set of discrete internal energy levels. We also deal with situations, in which even chemical reactions implying transfer of mass may occur. The consistency of the proposed BGK model is proved in both inert and reactive frames, and numerical simulations in space homogeneous settings are presented

EQUILIBRIUM SOLUTION TO THE INELASTIC BOLTZMANN EQUATION DRIVEN BY A PARTICLES THERMAL BATH

International audienceWe show the existence of smooth stationary solutions for the inelastic Boltzmann equation under the thermalization induced by a host-medium with a fixed distribution. This is achieved by controlling the Lp-norms, the moments and the regularity of the solutions for the Cauchy problem together with arguments related to a dynamical proof for the existence of stationary states

A general framework for the kinetic modeling of polyatomic gases

A general framework for the kinetic modelling of non-relativistic polyatomic gases is proposed,where each particle is characterized both by its velocity and by its internal state, and the Boltzmann collisionoperator involves suitably weighted integrals over the space of internal energies. The description of the internalstructure of a molecule is kept highly general, and this allows classical and semi-classical models, such asthe monoatomic gas description, the continuous internal energy structure, and the description with discreteinternal energy levels, to fit our framework. We prove the H-Theorem for the proposed kinetic equation ofBoltzmann type in this general setting, and characterize the equilibrium Maxwellian distribution and thethermodynamic number of degrees of freedom. Euler equations are derived, as zero-order approximation in asuitable asymptotic expansion. In addition, within this general framework it is possible to build up new models,highly desirable for physical applications, where rotation and vibration are precisely described. Examples ofmodels for the Hydrogen Fluoride gas are presented.Comment: 37 pages, 2 figure

Fluid-dynamic equations for reacting gas mixtures

summary:Starting from the Grad 13-moment equations for a bimolecular chemical reaction, Navier-Stokes-type equations are derived by asymptotic procedure in the limit of small mean paths. Two physical situations of slow and fast reactions, with their different hydrodynamic variables and conservation equations, are considered separately, yielding different limiting results

Optimal control of leachate recirculation for anaerobic processes in landfills

A mathematical model for the degradation of the organic fraction of solid waste in landfills, by means of an anaerobic bacterial population, is proposed. Additional phenomena, like hydrolysis of insoluble substrate and biomass decay, are taken into account. The evolution of the system is monitored by controlling the effects of leachate recirculation on the hydrolytic process. We investigate the optimal strategies to minimize substrate concentration and recirculation operation costs. Analytical and numerical results are presented and discussed for linear and quadratic cost functionals

From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell-Stefan type

In this paper we perform a formal asymptotic analysis on a kinetic model for reactive mixtures in order to derive a reaction-diffusion system of Maxwell-Stefan type. More specifically, we start from the kinetic model of simple reacting spheres for a quaternary mixture of monatomic ideal gases that undergoes a reversible chemical reaction of bimolecular type. Then, we consider a scaling describing a physical situation in which mechanical collisions play a dominant role in the evolution process, while chemical reactions are slow, and compute explicitly the production terms associated to the concentration and momentum balance equations for each species in the reactive mixture. Finally, we prove that, under the isothermal assumption, the limit equations for the scaled kinetic model is the reaction diffusion system of Maxwell-Stefan type.B.A. and A.J.S. thank Centro de Matematica da Universidade do Minho, Portugal, and the FCT/Portugal Project UID/MAT/00013/2013. B.A. thanks the FCT/Portugal for the support through the PhD grant PD/BD/128188/2016. P.G. thanks FCT/Portugal for the support through the project UID/MAT/04459/2013 and the French Ministry of Education through the grant ANR (EDNHS). The authors thank the Program Pessoa of Cooperation between Portugal and France with reference 406/4/4/2017/S.info:eu-repo/semantics/acceptedVersio