American Institute of Mathematical Sciences

In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in \begin{document}$ \ell^\infty $\end{document} for transport equations. We show in particular that the Neumann numerical boundary condition is a stable, local, and absorbing numerical boundary condition for discretized transport equations. Our main result is proved for explicit two time level numerical approximations of transport operators with arbitrarily wide stencils. The proof is based on the energy method and bypasses any normal mode analysis.

We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [4]. We consider a conservative underlying kinetic model, the Vlasov-BGK-Poisson system. We perform a scaling on this system in order to obtain its hydrodynamic limit. We present a deterministic numerical method to approximate this kinetic system. The method is shown to be Asymptotic-Preserving in the hydrodynamic limit, which means that any stability condition of the method is independant of any parameter \begin{document}$ \varepsilon $\end{document}, with \begin{document}$ \varepsilon \rightarrow 0 $\end{document}. We prove that the method is, under appropriate choices, consistant with the solution for bi-temperature Euler. Finally, our method is compared to methods for the fluid model (HLL, Suliciu).

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 \begin{document}$ (x,v) \in \mathbb{T}^d \times \mathbb{R}^d $\end{document} or on the whole space \begin{document}$ (x,v) \in \mathbb{R}^d \times \mathbb{R}^d $\end{document} with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively \begin{document}$ L^1 $\end{document} or weighted \begin{document}$ L^1 $\end{document} norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.

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 \begin{document}$ 1D\times1D $\end{document} and \begin{document}$ 2D\times2D $\end{document} Vlasov-Poisson system illustrate the effectiveness of this approach.

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

\begin{document}$ \partial_t f + \nabla\epsilon(p)\cdot\nabla_x f - \nabla \rho_f(x,t)\cdot\nabla_p f = \frac{\mathcal F_\lambda(p)-f}\tau, \quad x\in\mathbb{R}^d,\ p\in B, \ t>0 $\end{document}

is a model for ultracold atoms trapped in an optical lattice. The global existence of a solution is shown for small \begin{document}$ \tau>0 $\end{document} assuming that the initial data are analytic and sufficiently close to the Fermi-Dirac distribution \begin{document}$ \mathcal F_\lambda $\end{document}. This system contains an interaction potential \begin{document}$ \rho_f: = \int_B fdp $\end{document} being significantly more singular than the Coulomb potential, which causes major structural difficulties in the analysis.

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

\begin{document}$ \partial_t f + Lf = Q(f), $\end{document}

where \begin{document}$ L $\end{document} is a generator of an \begin{document}$ C^0 $\end{document}-group with \begin{document}$ \|e^{tL}\|\leq Ce^{\omega t} $\end{document} for all \begin{document}$ t\in\mathbb R $\end{document} and \begin{document}$ \omega>0 $\end{document} and, where further additional analytic properties of \begin{document}$ L $\end{document} and \begin{document}$ Q $\end{document} are assumed.