Homogenization of the linear Boltzmann equation in a domain with a periodic distribution of holes

Consider a linear Boltzmann equation posed on the Euclidian plane with a periodic system of circular holes and for particles moving at speed 1. Assuming that the holes are absorbing -- i.e. that particles falling in a hole remain trapped there forever, we discuss the homogenization limit of that equation in the case where the reciprocal number of holes per unit surface and the length of the circumference of each hole are asymptotically equivalent small quantities. We show that the mass loss rate due to particles falling into the holes is governed by a renewal equation that involves the distribution of free-path lengths for the periodic Lorentz gas. In particular, it is proved that the total mass of the particle system at time t decays exponentially fast as t tends to infinity. This is at variance with the collisionless case discussed in [Caglioti, E., Golse, F., Commun. Math. Phys. 236 (2003), pp. 199--221], where the total mass decays as Const./t as the time variable t tends to infinity.Comment: 29 pages, 1 figure, submitted; figure 1 corrected in new versio

Recent developments in the numerical simulation of shallow water equations I: boundary conditions

Shallow water equations (briefly, SWE) provide a model to describe fluid dynamical processes of various nature, and find therefore widespread application in science and engineering. A rigorous mathematical analysis is not available, unless for few specific cases under strict assumptions on the problem's data. In particular, the issue of which kind of boundary conditions are allowed is not completely understood yet. Here we investigate several sets of boundary conditions of physical interest that are admissible from the mathematical viewpoint. By that we mean that, when plugged into the integral form of SWE, these boundary conditions allow the proof of a priori estimates for the unknowns of physical interest: the velocity field and the elevation on the fluid (or its pressure). In our investigation we consider the most general case in which the physical boundary is partitioned into two sets: one closed (this is typically a coast or a shore), the other open (this is a virtual boundary delimiting the domain of investigation). In the latter we further distinguish among inflow and outflow boundary. Several kinds of conditions are investigated on each boundary component. The paper is concluded showing how to achieve a priori estimates corresponding to three different choices of boundary conditions. The correct treatment of boundary terms is crucial for both mathematical and numerical analysis of SWE. The characterization of the set of boundary conditions of physical interest that are mathematically admissible is important in view of the numerical simulation of this kind of phenomena. This paper is the first part of an investigation that the authors have carried out in this field. A second one shows how to implement these boundary conditions in the framework of discrete methods based on a finite element approximation in space, and several kind of time-marching techniques [11]. In particular, the a priori estimates obtained throughout this paper are extended in order to show stability properties for the approximate solution. Numerical experiments based on test cases corresponding to the various sets of boundary conditions considered here are presented in [10,12]. © 1994

A mathematical approach in the design of arterial bypass using unsteady Stokes equations

In this paper we present an approach for the study of Aorto-Coronaric bypass anastomoses configurations using unsteady Stokes equations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary according to several optimality criteria. \ua9 2006 Springer Science+Business Media, Inc

Fluid Dynamic Limits of the Kinetic Theory of Gases

Abstract These three lectures introduce the reader to recent progress on the hydrodynamic limits of the kinetic theory of gases. Lecture 1 outlines the main mathematical results in this direction, and explains in particular how the Euler or Navier-Stokes equations for compressible as well as incompressible fluids, can be derived from the Boltzmann equation. It also presents the notion of renormalized solution of the Boltzmann equation, due to P.-L. Lions and R. DiPerna, together with the mathematical methods used in the proofs of the fluid dynamic limits. Lecture 2 gives a detailed account of the derivation by L. Saint-Raymond of the incompressible Euler equations from the BGK model with constant collision frequency [L. Saint-Raymond, Bull. Sci. Math. 126 (2002), 493–506]. Finally, lecture 3 sketches the main steps in the proof of the incompressible Navier-Stokes limit of the Boltzmann equation, connecting the DiPerna-Lions theory of renormalized solutions of the Boltzmann equation with Leray’s theory of weak solutions of the Navier-Stokes system, following [F. Golse, L. Saint-Raymond, J. Math. Pures Appl. 91 (2009), 508–552]. As is the case of all mathematical results in continuum mechanics, the fluid dynamic limits of the Boltzmann equation involve some basic properties of isotropic tensor fields that are recalled in Appendices 1-2