Comportement des noyaux de la chaleur des opérateurs de Schrödinger et applications à certaines équations paraboliques semi-linéaires

We consider general Schrödinger operators on domains of Riemannian manifolds with possibly exponential volume growth. We prove sharp large time Gaussian upper bounds. These bounds are then used to prove new Lp-Lp estimates for the corresponding semigroups. Applications to semi-linear parabolic equations are given.     

Étude de l'algèbre de Lie double des arbres enracinés décorés

The double Lie algebra LD of rooted trees decorated by a set D is introduced, generalising the construction of Connes and Kreimer. It is shown that it is a simple Lie algebra. Its derivations and its automorphisms are described, as well as some central extensions. Finally, the category of lowest weight modules is introduced and studied.     

Solutions sous-analytiques globales de certaines équations d'Hamilton–Jacobi

In the 1980 Crandall and Lions introduced the concept of viscosity solution in order to get existence and/or unicity results for Hamilton–Jacobi equations. In this Note we focus on the Dirichlet problem for Hamilton–Jacobi equations stemming from calculus of variations, and assert that if the data are analytic then the viscosity solution is moreover subanalytic. We extend this result to generalized eikonal equations, stemming from sub-Riemannian geometry problems. To cite this article: E. Trélat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Solutions périodiques de certaines équations aux derivées partielles

Résumé On donne des critères assurant l'existence et l'unicité des solutions périodiques des problèmes(1.6)–(1.7) et(2.1)–(2.2). Entrata in Redazione il 16 settembre 1968.     

Unicité des solutions des équations de Navier–Stokes dans les espaces de Morrey–Campanato

In a recent work [P.G. Lemarié-Rieusset, Uniqueness for the Navier–Stokes problem: Remarks on a theorem of Jean-Yves Chemin, Nonlinearity 20 (2007) 1475–1490], P.G. Lemarié-Rieusset proved the uniqueness of solution to the Navier–Stokes equations in the space provided that p>2 and q>d. In this paper, we prove a local version of this result which covers the limit case q=d. Precisely, we prove the uniqueness of solution to the Navier–Stokes equations in the space for every p>2 and r>2 where is the closure of the test functions in the Morrey–Campanato space Mr,d(Rd). The prove of our result relies on an extension of the Comparison Theorem of P.G. Lemarié-Rieusset (Theorem 21.1 in [P.G. Lemarié-Rieusset, Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC, 2002]). Moreover, this extension allows us to prove the uniqueness of solution to the Navier–Stokes equations in a functional space closed to the critical space C([0,T],M2,d(Rd)).     

Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles

Dissipative solutions [12] of the Euler equations of incompressible fluids are obtained as the hydrodynamic limit of a properly scaled BGK equation. This stability result comes from refined entropy and entropy dissipation bounds. It uses in a crucial way the local conservation laws which are known to hold for weak solutions of this simplified model of the Boltzmann equation.     

A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles

We prove existence and regularity results for a general inequality including fixed point problems, variational and quasi variational inequalities. Examples of such problems are considered.     

Étude du cas rationnel de la théorie des formes linéaires de logarithmes

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called rational case. More precisely, let k be a number field and v₀ be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u∈Lie(G(Cv₀)) a logarithm of a point p∈G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)k_⊗Cv₀. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height loga of p, removing a polynomial term in logloga. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy.     

Méthodes d'ordre dans l'interprétation de certaines inéquations variationnelles et applications

Using the Riesz space structure of Sobolev spaces of real positive order less than one we prove a new trace theorem and a dual estimate for the solution of variational inequalities of Neumann type.     

Sur l'analyticité des solutions de certaines équations aux opérateurs pseudodifférentiels

Sommaire Dans ce travail on obtient un résultat concernant l'analyticité des solutions de certaines équations où interviennent certains opérateurs pseudodifférentiels du type elliptique, tels qu'ils ont été definis dans [1].

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Sunto In questo lavoro si ottiene un risultato concernente l'analiticità delle soluzioni di certe equazioni nelle quali intervengono certi operatori pseudo differenziali di tipo ellittico.

     

Sur l'existence, l'unicité, la stabilité et les propriétés de grandes déviations des solutions d'équations différentielles stochastiques rétrogrades à horizon aléatoire. Application à des problèmes de perturbations singulières

We study the existence, uniqueness and stability of solutions of backward stochastic differential equations with random terminal time under new assumptions; then we establish a large deviation principle for the solutions of such equations, related to a family of Markov processes, the diffusion coefficient of which tends to zero. Finally we apply these results to the analysis of some singular perturbation problems for a class of nonlinear partial differential equations.     

Structures de dérivabilité

We introduce a very general framework in which Quillen's theorems of existence, composition and adjunction for derived functors can be proved. We thus generalize and unify previous results by Dwyer, Hirschhorn, Kan and Smith, obtained in their formalism of “homotopical categories,” and by Radulescu-Banu in the context of Cisinski's “derivable categories.”     

Mesures d'indépendance linéaire de carrés de périodes et quasi-périodes de courbes elliptiques

A linear independence measure over Q is obtained for values of some generalized hypergeometric functions at rational points and for the squares of real periods and quasi-periods of elliptic curves in Legendre form y²=x(x−1)(x−1/q) for almost every q∈Z.     

Construction de quasimodes de Rayleigh à grande durée de vie

For the Neumann problem in linear elasticity at the exterior of a convex body with analytic boundary in two dimension, we know that exists exactly two sequences of resonances (called Rayleigh resonances), such that . If the boundary is close enough to a circle and analytic in a wide enough complex band, we construct quasimodes associated to these resonances as far as possible from the obstacle. We study the exponential decay of these quasimodes according to the geometry of the obstacle.