Asymptotic Analysis

In a preceding paper, an asymptotic analysis of the one dimensional Vlasov–Poisson was presented. That analysis accounted for the “Child–Langmuir regime”, when the injected particle energy is small compared with the applied external bias. In the present paper, we propose a numerical algorithm for solving the Vlasov-Poisson equations in the “Child–Langmuir regime”. The crucial step of the algorithm is a penalization of the “Child–Langmuir emission condition”, which is proved to converge in the one-dimensional case.

We consider −h²Δ+V on Rⁿ, when V is smooth, real-valued and has a unique minimum at 0 which is nondegenerate: V(0)=0,V″(0)>0. We also assume that lim inf _|x|→∞V(x)>0. For any fixed δ>0 we get uniform asymptotic formulas for the eigenvalues up to h^δ, when h→0. The proofs use Birkhoff normal forms and pseudodifferential functional calculus.

In this paper we are extending the classical preparation theorem and division theorem to an asymptotic case.

We consider a structure consisting of two parts: a three-dimensional linearly elastic body and a linearly elastic plate. The plate is inserted into the three-dimensional body. We perform an asymptotic analysis of the time-dependent behavior of the structure during the time-interval [0,T] when the thickness of the plate goes to zero. Under specific assumptions on the data (Lame constants, mass densities, loads), we establish the existence of a limit problem. This limit problem is a system of coupled partial differential equations posed over a three-dimensional body with a slit and the middle surface of the plate. Strong convergence in L²(0,T;H¹) of the time-dependent displacements (with appropriate scaling) is proved.

On considére une structure constituée de deux parties: un corps tridimensionnel élastique et une plaque élastique. La plaque est insérée dans le bloc tridimensionnel. On étudie le comportement asymptotique de la structure pendant un intervalle de temps [0,T] lorsque l'epaisseur de la plaque tend vers zéro. Sous des hypothèses précises sur les données (constantes de Lamé, masses volumiques, forces appliquées), on établit l'existence d'un modèle limite. Ce modèle est un système d'équations aux dérivées partielles couplées écrites sur un domaine tridimensionnel présentant une fente et sur la surface moyenne de la plaque. On démontre la convergence forte des déplacements (convenablement mis à l'échelle) dans l'espace L²(0,T;H¹).