Homogenization of linear Boltzmann equations in the context of algebras with mean value
- PDF / 503,970 Bytes
- 22 Pages / 547.087 x 737.008 pts Page_size
- 16 Downloads / 173 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Homogenization of linear Boltzmann equations in the context of algebras with mean value P. Fouegap, R. Kenne Bogning, G. Nguetseng, D. Dongo and J. L. Woukeng
Abstract. The paper deals with the homogenization of linear Boltzmann equations by the means of the sigma-convergence method. Replacing the classical periodicity hypothesis on the coefficients of the collision operator by an abstract assumption covering a great variety of physical behaviours, we prove that the density of the particles converges to the solution of a drift-diffusion equation. We then illustrate this abstract setting by working out a few concrete homogenization problems such as the periodic one, the almost periodic one and others. To achieve our goal, we use the Krein–Rutman theorem for locally convex spaces together with the Fredholm alternative to solve the so-called corrector problem. Mathematics Subject Classification. 35B40, 45M05, 82C70, 85A25. Keywords. Deterministic homogenization, Boltzmann equations, Algebras with mean value, Sigma-convergence.
1. Introduction and the main result An important topic in multiscale analysis is the derivation of macroscopic model equations from the microscopic ones arising from kinetic theory. One of the most important kinetic equations is the Boltzmann equation, which roughly reads as ∂f + v · ∇x f = Qf . ∂t
(1.1)
In (1.1), the left-hand side accounts for the total derivative that takes into account the free streaming of particles, while the right-hand side is the collision operator describing interactions between particles. The unknown function f refers to the distribution function and is physically interpreted as the probability of the density of particles in a given volume. In this work, we deal with a specific type of collision operator (see (1.4)) leading to the linear Boltzmann equation. The linear Boltzmann equation is a kinetic model used in many different contexts. It appeared (for the first time; see [20]) on the motion of electrons in metals and has been since then used in various branches of mathematics and physics such as radiative transfer [22,25], neutron transfer theory [30]. The macroscopic effects come to light when the time elapsing between collisions is much smaller than the observation time scale. This amounts to saying that when the average distance between two successive collisions is smaller than the given specimen length scale. In that case, it therefore becomes interesting to seek the solution f of (1.1) under the form f (t, x, v) = fε (εt, x, v) where 0 < ε 0 such that ⎪ ⎩ μ({v ∈ V : |a(v) · ξ| ≤ h}) ≤ Chγ for all ξ ∈ S d−1 , h > 0 where S d−1 stands for the d-dimensional sphere in Rd . Next, we also need the following assumption on σ: 2,∞ (Rdy )). (1.9) σ ∈ B(Rdx × V × V ; BA In (1.9), A is a given algebra with mean value on Rd (that is, a closed subalgebra of the C ∗ -Banach algebra of bounded uniformly continuous real-valued functions on Rd that contains the constants, is close under complex conjugation, is translation inv
Data Loading...
