Incompressible Hydrodynamic Limits of the Boltzmann Equation : a Survey of Mathematical Results
Moyennisation des champs de vecteurs et ÉDP
The mean-field limit for the dynamics of large particle systems
This short course explains how the usual mean-field evolution PDEs in Statistical Physics - such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations - are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems.
From the Boltzmann Equation to Hydrodynamic Equations in thin Layers
The present paper discusses an asymptotic theory for the Boltzmann equation leading to either the Prandtl incompressible boundary layer equations, or the incompressible hydrostatic equations. These results are formal, and based on the same moment method used in [C. Bardos, F. Golse, D. Levermore, J. Stat. Phys 63 (1991), pp. 323-344] to derive the incompressible Euler and Navier-Stokes equations from the Boltzmann equation.
On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
We prove that the Boltzmann-Grad limit of the Lorentz gas with periodic distribution of scatterers cannot be described with a linear Boltzmann equation. This is at variance with the case of a Poisson distribution of scatterers, for which the convergence to the linear Boltzmann equation was proved by Gallavotti [ , 308 (1969)]. The arguments presented here complete the analysis in [Golse-Wennberg, , 1151 (2000)], where the impossibility of a kinetic description was...
New Results in Velocity Averaging
This paper discusses two new directions in velocity averaging. One is an improvement of the known velocity averaging results for functions. The other shows how to adapt some of the ideas of velocity averaging to a situation that is essentially a new formulation of the Vlasov-Maxwell system.
On the distribution of free path lengths for the periodic Lorentz gas II
On the distribution of free path lengths for the periodic Lorentz gas II
Consider the domain and let the free path length be defined as In the Boltzmann-Grad scaling corresponding to , it is shown that the limiting distribution of is bounded from below by an expression of the form , for some . A numerical study seems to indicate that asymptotically for large , . This is an extension of a previous work [J. Bourgain , (1998) 491-508]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describe the...
Derivation of a homogenized two-temperature model from the heat equation
This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat,...
Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic riemannian manifolds
Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system.
A domain decomposition analysis for a two-scale linear transport problem
We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating...
La méthode de l’entropie relative pour les limites hydrodynamiques de modèles cinétiques
A Domain Decomposition Analysis for a Two-Scale Linear Transport Problem
We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating...
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