Asymptotic Analysis

Dans cet article, on étudie les modèles discrets de l'équation de Boltzmann avec termes linéaires et quadratiques dans l'espace unidimensionnel. On montre d'abord l'existence d'une solution avec estimation explicite pour les modèles droite‐gauche, une généralisation des modèles de Broadwell. Après avoir montré l'existence globale d'une solution (resp. avec estimation explicite) pour les modèles discrets à données (localement) bornées (resp. et sommable), on étudie le comportement asymptotique des solutions et montre deux phémonénas distinctes correspondant aux conditions imposées pour les coefficients de termes linéaires.

The problem of small vibrations of a smooth inhomogeneous string damped at an interior point and fixed at the endpoints is reduced to a three‐point Sturm–Liouville boundary problem. This problem is considered as an eigenvalue problem for a nonmonic quadratic operator pencil of a special type with the spectrum located in the upper half‐plane of the spectral parameter. Concerning the corresponding inverse problem it is shown that the spectrum does not determine the potential of the Sturm–Liouville problem (and consequently the density of the string) uniquely. The conditions are given sufficient for a sequence of complex numbers to be the spectrum of the considered Sturm–Liouville problem with real‐valued potential which belongs to L₂(0,a). In order to recover the potential uniquely the spectrum of the corresponding so to say truncated Sturm–Liouville problem is chosen as an additional information. Then the problem of recovering of the potential turns out to be overdetermined. The self‐consistency of the two spectra is discussed.

In this paper we give a definition of asymptotic development of an analytic function of several complex variables. So far, there existed two different definitions which are joined by the definition given in this paper. This definition generalizes, in a natural sense, the one given for one variable. The generalization is made by changing the object polysector by a conoidal domain which is more general because all the polysectors are conoidal domains but not all the conoidal domains are polysectors.

It is given a characterization of the analytic functions that admit asymptotic developments in a cone as those which restricted to a closed subcone admit an extension of class 𝒞^∞. It is proven that, through this definition, the asymptotic developments of analytic functions of several variables verify all the good properties of the asymptotic developments of analytic functions of one variable.

The present work investigates boundary conditions that modelize the penetration of a time‐harmonic wave into a dissipative obstacle. The exact model is given through a transmission problem that couples the propagations into an absorbing domain and an exterior domain. The conditions arise from an asymptotic analysis of the interior solution whose propagation is studied by using pseudodifferential technics classically involved for the construction of radiation boundary conditions. Herein, first and second‐order conditions are analyzed and some numerical experiments illustrate their validity domain.

We study the set of residual boundary conditions for a one‐dimensional hyperbolic system of conservation laws set in x>0 which appears when the diffusion coefficient of Dirichlet's problem for a parabolic perturbation tends to zero. We show that this set is a submanifold in a vicinity of a point where the Evans function of the associated profile of boundary layer is such that D(0)≠0.

Next we linearize a multidimensional hyperbolic problem about a constant state in the set of residual conditions and a viscous approximation about the associated profile of boundary layer. We show that the Evans function for the viscous problem reduces in the long‐wave limit to the Lopatinsky determinant. We deduce that inviscid well‐posedness is necessary for stability of the boundary layer.