Asymptotic Analysis - Volume 31, issue 2

Authors: Boatto, Stefanella | Golse, François

Article Type: Research Article

Abstract: In this article we study a model for the dynamics of particles moving freely between two horizontal plates. Our goal is to characterize the diffusive behavior for such systems in the long‐time and large (horizontal) scale limits. Using homogenization techniques for PDEs, we obtain a diffusion equation as a limit of the original kinetic equation, appropriately scaled. This limiting diffusion can be understood intuitively by observing that in the limit of long times and vanishing vertical distance between the plates, for a given particle, the mean free path between two successive reflections at the plates also vanishes, and therefore the …number of reflections grows unboundedly with the length of the time interval. The nature of a particular reflection law then efficiently randomizes the particle motion. More specifically, we study the dependence of the diffusion constant Dα on mixed boundary conditions: the case of specular reflection (based on the “Arnold cat map”) with an isotropic component (i.e., with small “accommodation” coefficient α). We find that the diffusion tensor Dα is positive definite for every α∈(0,1]. Furthermore, in the limit of vanishing isotropic component (α→0), we recover the result of Bardos, Golse and Colonna [Physica D, 104 (1997), 32–60]. Show more

Citation: Asymptotic Analysis, vol. 31, no. 2, pp. 93-111, 2002

Authors: Collard, Christophe | Miara, Bernadette

Article Type: Research Article

Abstract: In this paper, we apply the asymptotic analysis to thin piezoelectric shells in the framework of geometrically exact formulation. The formal mathematical approach used here is based neither on geometrical nor on mechanical assumptions and rigorously justifies the limiting constitutive nonlinear two‐dimensional equations. More precisely, we formally obtain two‐dimensional membrane and flexural models written on the middle surface of the shell. We show that the coupling between the limit displacement field and the limit electric potential inherent to piezoelectricity appears in the membrane model but not in the flexural model. Finally, we suggest a “full” new model for piezoelectric shells …using membrane and flexural effects. Show more

Citation: Asymptotic Analysis, vol. 31, no. 2, pp. 113-151, 2002

Authors: Berthelin, F. | Bouchut, F.

Article Type: Research Article

Abstract: We obtain rigorously the correspondence between kinetic and macroscopic invariant domains for a vector kinetic BGK system considered in a previous paper of the authors. We establish the convergence, as the relaxation parameter tends to 0, to weak entropy solutions of isentropic gas dynamics equations.

Keywords: isentropic gas dynamics, kinetic BGK model, kinetic entropy, kinetic invariant domain, relaxation limit

Citation: Asymptotic Analysis, vol. 31, no. 2, pp. 153-176, 2002

Authors: Braides, Andrea | Buttazzo, Giuseppe | Fragalà, Ilaria

Article Type: Research Article

Abstract: We show that the class of smooth and isotropic Riemannian metrics is dense in the class of all lower semicontinuous Finsler metrics, with respect to the Γ‐convergence of energy integrals.

Keywords: Riemannian and Finsler metrics, gamma convergence

Citation: Asymptotic Analysis, vol. 31, no. 2, pp. 177-187, 2002