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HILBERT’S 6TH PROBLEM: EXACT AND APPROXIMATE HYDRODYNAMIC MANIFOLDS FOR KINETIC EQUATIONS
"... Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the ..."
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Abstract. The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically both as the result of summation of the Chapman–Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grad’s moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the shortwave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilbert’s 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newton’s iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the
Derivation of the Fick’s law for the Lorentz model in a low density regime arXiv
"... Abstract. We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics which converges to the stationary solution of the heat equation, namely to the linear profile of ..."
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Abstract. We consider the Lorentz model in a slab with two mass reservoirs at the boundaries. We show that, in a low density regime, there exists a unique stationary solution for the microscopic dynamics which converges to the stationary solution of the heat equation, namely to the linear profile of the density. In the same regime the macroscopic current in the stationary state is given by the Fick’s law, with the diffusion coefficient determined by the GreenKubo formula.
Approaches to derivation of the Boltzmann equation with hard sphere collisions
, 2013
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Geometric analysis of the linear Boltzmann equation II. Localization properties of the spectrum
, 2014
"... Abstract. This work is devoted to the analysis of the linear Boltzmann equation in a bounded domain, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset ..."
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Abstract. This work is devoted to the analysis of the linear Boltzmann equation in a bounded domain, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity. We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We handle several cases of phase spaces, including those associated to specular reflection
The Boltzmann–Grad Limit of a Hard Sphere System: Analysis of the Correlation Error
"... In memory of Oscar Erasmus Lanford III Abstract. We present a quantitative analysis of the Boltzmann–Grad (low–density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propa ..."
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In memory of Oscar Erasmus Lanford III Abstract. We present a quantitative analysis of the Boltzmann–Grad (low–density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order k measures the event where k tagged particles are connected by a chain of interactions preventing the factorization. We prove that, provided k is not too large, such an error flows to zero with the hard spheres diameter ε, for short times, as εγk, for some γ> 0. This requires a new analysis of many–recollision events, and improves previous estimates of high order
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"... From microscopic dynamics to macroscopic equations: scaling limits for the Lorentz gas ..."
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From microscopic dynamics to macroscopic equations: scaling limits for the Lorentz gas
DISTRIBUTION OF SCATTERERS
"... Abstract. We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weakcoupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat ..."
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Abstract. We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weakcoupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the GreenKubo formula associated to the generator of the diffusion process dictated by the linear Landau equation. 1.