62 research outputs found
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
Exponential convergence to equilibrium via Lyapounov functionals for reaction-diffusion equations arising from non reversible chemical kinetics
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
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