Kinetic and Related Models
February 2020 , Volume 13 , Issue 1
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In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in
We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [
In this article we deduce a mathematical model of Maxwell-Stefan type for a reactive mixture of polyatomic gases with a continuous structure of internal energy. The equations of the model are derived in the diffusive limit of a kinetic system of Boltzmann equations for the considered mixture, in the general non-isothermal setting. The asymptotic analysis of the kinetic system is performed under a reactive-diffusive scaling for which mechanical collisions are dominant with respect to chemical reactions. The resulting system couples the Maxwell-Stefan equations for the diffusive fluxes with the evolution equations for the number densities of the chemical species and the evolution equation for the temperature of the mixture. The production terms due to the chemical reaction and the Maxwell-Stefan diffusion coefficients are moreover obtained in terms of general collision kernels and parameters of the kinetic model.
We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus
The asymptotic behavior of the solutions of the second order linearized Vlasov-Poisson system around homogeneous equilibria is derived. It provides a fine description of some nonlinear and multidimensional phenomena such as the existence of Best frequencies. Numerical results for the
Kinetic exchange models of markets utilize Boltzmann-like kinetic equations to describe the macroscopic evolution of a community wealth distribution corresponding to microscopic binary interaction rules. We develop such models to study a form of welfare called need-based transfer (NBT). In contrast to conventional centrally organized wealth redistribution, NBTs feature a welfare threshold and binary donations in which above-threshold individuals give from their surplus wealth to directly meet the needs of below-threshold individuals. This structure is motivated by examples such as the gifting of cattle practiced by East African Maasai herders or food sharing among vampire bats, and has been studied using agent-based simulation. From the regressive to progressive kinetic NBT models developed here, moment evolution equations and simulation are used to describe the evolution of the community wealth distribution in terms of efficiency, shape, and inequality.
The semiconductor Boltzmann-Dirac-Benney equation
is a model for ultracold atoms trapped in an optical lattice. The global existence of a solution is shown for small
The key technique is based of the ideas of Mouhot and Villani by using Gevrey-type norms which vary over time. The global existence result for small initial data is also generalized to
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