Approximation des ensembles ω-limites des difféomorphismes par des orbites périodiques

Let M be a compact manifold, D the set of its C¹-diffeomorphisms (possibly symplectic or volume preserving). We prove that there exists a dense G_δ of D such that if , every ω-limit set of f is the limit (for the Hausdorff topology) of a sequence of periodic orbits. This has certain interesting consequences concerning the structure of the ω-limit sets. Moreover, we define a new notion of attractors and describe them precisely in different cases.     

Nouvelle approche à la stabilisation forte des systèmes distribués

We propose a new method to prove the strong asymptotic stability of certain distributed evolutionary systems. It can be applied in some cases where LaSalle's invariance principle cannot be applied due to the lack of compactness.     

Approximation grossière dʼun problème elliptique à coefficients hautement oscillants

Our wish is to approximate an elliptic problem with highly oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillating coefficients in the equation can be incomplete or entirely missing. We investigate the links between this particular question and the classical theory of homogenization. On some illustrating examples we show the potential practical interest of the approach.     

Approximation du système quasigéostrophique à l'aide des systèmes primitifs

In this paper we show that the quasigeostrophic system is well approximated by the primitive systems. More precisely, we prove that if the initial data are well-prepared, then the maximal time existence of the regular solution of the primitive system goes to infinity and the regular solution goes to the solution of the quasigeostrophic system, strongly on an arbitrary time interval. By “well-prepared initial data” we mean that the initial data of the primitive systems is converging to an initial data with zero oscillating part.     

Un modèle du calcul de Dirac dans la théorie relative des ensembles

The aim of this Note is a new justification of the pertinence of the Dirac calculus (see [1]). We give a model of it in a relatively consistent set theory,the Relative Set Theory (see [2]). We have obtained this result by introducing a derivation of step functions.     

Heat Semigroup Bounds on BV(Ω) and Related Estimates

Let \(\overline{\Omega}\) be a compact Riemannian manifold with nonempty boundary. We note that if \(f\in C^\infty(\overline{\Omega})\) does not vanish identically on the boundary, then the heat semigroup \(e^{t\Delta_D}\) (with the Dirichlet boundary condition) acting on f produces a family bounded in \(H^{1,p}(\overline{\Omega})\) if and only if p?=?1. This observation motivates the main result of this paper, which is that the heat semigroup is uniformly bounded on BV(Ω), the space of functions on Ω with bounded variation.     

Synchronisation exacte dʼun système couplé dʼéquations des ondes par des contrôles frontières de Dirichlet

In this Note, the exact synchronization for a coupled system of wave equations with Dirichlet boundary controls and some related concepts are introduced. By means of the exact null controllability of a reduced coupled system, under certain conditions of compatibility, the exact synchronization, the exact synchronization by groups, and the exact null controllability and synchronization by groups are all realized by suitable boundary controls.     

La série entière 1+zΓ(1+i)+z2Γ(1+2i)+z3Γ(1+3i)+⋯ possède une frontière naturelle !

The lacunary series are the most classic examples among all the power series whose circle of convergence constitutes a natural boundary (Dienes, 1931 [4, 493–94, pp. 372–383], Titchmarsh, 1939 [8, 47.43, p. 223], …). In this Note, we study a family of non-lacunary power series whose coefficients are given by means of values of the Gamma function over vertical line. We explain how to transform these series into lacunary Dirichlet series, which allows us to conclude the existence of their natural boundary. Our results, which illustrate in what manner the Gamma function may have an unpredictable behaviour on any vertical line, may also be partially understood in the framework of our forthcoming work on a class of differential q-difference equations, namely, on pantograph type equations (meanwhile see Kato and McLeod (1971) [6]).     

Problèmes elliptiques à frontière libre axi-symétriques: Estimation du diamètre de la section au moyen de la capacité

Résumé Le but de cet article est d'estimer la capacité d'un pétit segment [a(1–), a(1+)]×{z=0} du démi-plan := {x = (r, z)|r > 0, z } de ² par rapport à la capacité définie par la norme . Ce résultat est ensuite utilisé pour estimer le diamètre de la surface libre de certains problèmes elliptiques non linéaires.     

Weak-Type (1-1) Inequality for the Monge-Ampère SIO's

The purpose of this article is to prove a weak-type (1-1) estimate for a class of singular integral operators related to the real analysis of the Monge-Ampère equation. The method uses the theory of homogeneous spaces. It should be noted that the class of operators has been widened since [3] by replacing the Lipschitz type condition with a weaker Hölder-aalpha type condition ($0 < alpha leq 1$0 < alpha leq 1).     

Les SurfacesW (Weingarten) à Lignes de Courbure planes dans les deux Systèmes

Sans résumé     

Bornes sur la densité pour un problème de Navier–Stokes compressible à frontière variable avec conditions aux limites de Dirichlet

We extend a compactness result shown by P.-L. Lions in 1998 to an isentropic compressible Navier–Stokes problem ($\gamma ?1$) defined on a time dependent domain with Dirichlet boundary conditions. This result can be useful for the study of some fluid–structure interaction problems, for the analysis of some pollution water problems (shallow water equations with free boundary: $\gamma =1$) or for the modelling of a river level. To cite this article: F. Flori, B. Giudicelli, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble

Let \(K\) be a global field and \(G\) a finite solvable \(K\) -group. Under certain hypotheses concerning the extension splitting \(G\) , we show that the homogeneous space \(V=G'/G\) with \(G'\) a semi-simple simply connected \(K\) -group has the weak approximation property. We use a more precise version of this result to prove the Hasse principle for homogeneous spaces \(X\) under a semi-simple simply connected \(K\) -group \(G'\) with finite solvable geometric stabilizer \({\bar{G}}\) , under certain hypotheses concerning the \(K\) -kernel (or \(K\) -lien) \(({\bar{G}},\kappa )\) defined by \(X\) .     

Classification des Objets Galoisiens de U q (𝔤) à Homotopie PrèS

Pour toute algèbre enveloppante quantique U_q(𝔤) de Drinfeld–Jimbo et toute famille λ = (λ_ij)1≤i ∈ k^? d'éléments inversibles du corps de base, nous construisons explicitement par générateurs et relations un objet galoisien A_λ de U_q(𝔤) et nous montrons que tout objet galoisien de U_q(𝔤) est homotope à un unique objet de la forme A_λ.

For any Drinfeld–Jimbo quantum enveloping algebra U_q(𝔤) and for any family λ = (λ_ij)_{1≤i ∈ k^? of invertible elements of the base field, we explicitly construct a Galois object Aλ of Uq(𝔤) by generators and relations and we prove that any Galois object of Uq(𝔤) is homotopic to a unique object of type Aλ.}