Asymptotic Analysis - Volume 33, issue 1

Authors: Grébert, B. | Kappeler, T.

Article Type: Research Article

Abstract: For various spaces of potentials we prove that the set of finite gap potentials of the Zakharov–Shabat system is dense. In particular our result holds for Sobolev spaces and for spaces of analytic potentials of a given type.

Citation: Asymptotic Analysis, vol. 33, no. 1, pp. 1-8, 2003

Authors: Bellettini, Giovanni | Fusco, Giorgio

Article Type: Research Article

Citation: Asymptotic Analysis, vol. 33, no. 1, pp. 9-50, 2003

Authors: Marušić‐Paloka, Eduard

Article Type: Research Article

Abstract: We study the junction of m pipes that are either thin or long (i.e., they have small ratio between the cross‐section and the length, denoted by ε). Pipes are filled with incompressible Newtonian fluid and the values of the pressure pi at end of each pipe are prescribed. By rigorous asymptotic analysis, as ε→0, we justify the analog of the Kirchhoff law for computing the junction pressure. In interior of each pipe the effective flow is the Poiseuille flow governed by the pressure drop between the end of the pipe and the junction point. The pressure at the junction …point is equal to a weighted mean value of the prescribed pi ‐s (Kirchhoff law). In the vicinity of the junction there is an interior layer, with thickness εlog (1/ε). To get a better approximation and to control the velocity gradient in vicinity of the junction, first order asymptotic approximation has to be corrected by solving an appropriate Leray problem. We prove the asymptotic error estimate for the approximation. Show more

Keywords: thin pipes, Navier–Stokes equations, Poiseuille flow, junction, Leray problem

Citation: Asymptotic Analysis, vol. 33, no. 1, pp. 51-66, 2003

Authors: Lods, V. | Piétrus, A. | Rakotoson, J.M.

Article Type: Research Article

Abstract: We consider a crystal constituted by an elastic substrate and a film with a small thickness. This crystal being constrained, it appears morphological instabilities. We are interested in the evolution of the free boundary of the film, which is parametrized by a function denoted by f. The three‐dimensional model here considered is detailed in [8]. This model consists in solving a coupled system of partial derivative equations. The first equations are the linearized elasticity equations posed in the solid, the boundary of which depends on the evolution surface. The second equation is the evolution equation, depending on the elastic …displacement. This model is first classically simplified in order to obtain a two‐dimensional model by assuming that the crystal is infinite in one dimension. Besides, under some hypotheses, we derive a wide class of models which the unknown is the map of the film–vapor surface and solves a nonlinear partial derivatives equation, which is independent of the displacement of the solid. Some of those models might blow up in finite time as the physicists expect. In this paper, we study the existence and uniqueness of solution of this model, by constructing approximated problems under very weak assumptions. To this end, we assume that the initial map of the free boundary is small enough in an appropriate space. Show more

Keywords: linear elasticity, free boundary problem, fourth order curvature operator, interpolations inequalities

Citation: Asymptotic Analysis, vol. 33, no. 1, pp. 67-91, 2003