Showing papers in "Comptes Rendus De L Academie Des Sciences Serie I-mathematique in 1999"

The Lorenz attractor exists

Abstract: We prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. The proof is based on a combination of normal form theory and rigorous numerical computations.

Etude théorique et numérique d'une hiérarchie de modèles aux moments pour le transfert radiatif

Abstract: Resume Nous appliquons ici le principe de minimisation d'entropie a l'equation de transfert radiatif. Comme dans le cadre des equations cinetiques classiques, nous obtenons a l'aide de ce principe une hierarchie de systemes aux moments fermes. Chaque systeme de la hierarchie est hyperbolique, possede un flux limite par la vitesse de la lumiere et dissipe localement l'entropie. Nous analysons plus particulierement les proprietes du premier systeme de la hierarchie.

Hölder continuity of the gradient of p(x)-harmonic mappings

Abstract: We prove that local minimizers u : R n → R N of the functional ∫ ∣D u ( x )∣ p(r) d x are of class C 1, α for some α > 0, provided p(x) > 1 is Holder continuous.

Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier

Abstract: Resume Dans cette Note, on considere l'equation de la chaleur lineaire u t — Δu = a(x)u dans (O, T ) x Ω, u = O sur (O, T ) et u (0) = u 0 sur Ω. Ici Ω est un domaine borne et regulier de ℝ N et on suppose a L loc 1 (Ω), a ≥ 0 et u 0 ≥ 0. Une condition simple sur le potentiel a est necessaire et suffisante pour l'existence de solutions faibles positives, globales en temps et a croissance en temps au plus exponentielle. On montre que cette condition, reposant sur l'existence d'une inegalite de type Hardy avec poids a(x) , est aussi ≪ presque ≫ necessaire pour l'existence de solutions faibles, positives et locales en temps. En appliquant ces resultats a quelques potentiels ≪ critiques ≫, on trouve des resultats nouveaux d'existence et d'explosion instantanee et totale.

Finite type invariants and n-equivalence of 3-manifolds

Abstract: We announce a new approach to a definition of finite type invariants and introduce a notion of n-equivalence for 3-manifolds with boundary. For the ℤHS we state equivalence of different definitions of finite type invariants. The theory for 3-manifolds is made completely parallel to the corresponding theory for links. We show how to reduce general classification problems of arbitrary 3-manifolds up to n-equivalences to the study of finite type invariants of string links and integer homology spheres.

Stabilisation de la poutre de Timoshenko

Abstract: Resume Dans cette Note on etudie la stabilisation de la poutre de Timoshenko. Pour ce faire on utilise une seule force de controle.

Dynamique des automorphismes des surfaces projectives complexes

Abstract: Resume Nous etudions la dynamique des automorphismes des surfaces complexes projectives dont l'entropie topologique est strictement positive. Siest un tel automorphisme nous construisons une mesure de probabilite canoniquement associee aet a la structure complexe de la surface. C'est l'unique mesure -invariante ergodique et d'entropie maximale. Les points periodiques desont equidistribues pour cette mesure.

Motion of a rigid body in a viscous fluid

Abstract: We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.

Hypergeometric period for a tame polynomial

Abstract: We analyze the Gauss-Manin system of differential equations (and its Fourier transform) attached to regular functions satisfying a tameness assumption on a smooth affine variety over C (e.g. tame polynomials on C n+1 ). We give a solution to the Birkhoff problem for this system and prove Hodge-type results analogous to those existing for germs of isolated hypersurface singularities. We deduce a formula for the determinant of the “Aomoto complex”.

Une approche locale de la limite incompressible

Abstract: Resume Nous etablissons dans cette Note un resultat local de convergence faible des solutions des equations de Navier-Stokes compressibles, quand le nombre de Mach tend vers 0, vers une solution a la Leray des equations de Navier-Stokes incompressibles.

Remark on the local ill-posedness for KdV equation

Abstract: On remarque que l'on peut construire un exemple tel que le probleme de Cauchy pour l'equation de KdV soit mal pose dans H s (R), s < -3/4, si on suppose que l'application qui a une donnee initiale associe la solution a l'instant t soit de classe C 2 (dans [2], une regularite C 3 est supposee).

The canonic diffusion above the diffeomorphism group of the circle

Abstract: Radonification of the heat equation associated to the Gelfand-Fuks cocycle is realized in a space of Holderian homeomorphism.

On Hurwitz numbers and Hodge integrals

Abstract: In this paper we find an explicit formula for the number of topologically different ramified coverings C → CP1 (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.

Sur le transport de mesures périodiques

Abstract: Resume On montre, en suivant la methode de McCann [8], qu'etant donne deux mesures periodiques μ, et ν sur ℝ d (suffisamment regulieres), on peut trouver une fonction convexe φ telle que ▽φ transporte μ sur ν.

Hausdorff dimension of the hairs without endpoints for λ exp z

Abstract: We consider the complex exponential maps Eλ (z) = λ ez where z ∈ ℂ and λ e (0, 1/e). It is known that the Julia set of Eλ is a Cantor bouquet of curves (“hairs”) extending from the set of their endpoints to ∞. We prove that the Hausdorff dimension of the set of these curves without the endpoints is equal to 1; in particular, it is smaller than the Hausdorff dimension of the set of endpoints alone (which is known to be equal to 2, see [5]).

A general formulation for a posteriori bounds for output functionals of partial differential equations; application to the eigenvalue problem*

Abstract: We present an a posteriori finite element procedure that provides rigorous, constant-free, asymptotic lower and upper bounds for smooth nonlinear-functional outputs of general elliptic partial differential equations. This new abstract framework includes not only our earlier bound procedures for coercive linear (symmetric or nonsymmetric), noncoercive linear (e.g., Helmholtz), and nonlinear (e.g., Burgers) equations, but also the (symmetric) generalized eigenvalue problem. The latter-which provides a posteriori bounds both for the eigenvalues and for functionals of the eigenvectors-is described in detail, and sustained by illustrative numerical results.

Polynomial upwind schemes for hyperbolic systems

Abstract: In this paper, we present an upwinding methodology for systems of conservation laws. Our aim is to construct upwind schemes which do not require the extraction of the eigensystem of the Jacobian matrix (but just the knowledge of the eigenvalues) and rely on the introduction of an appropriate polynomial approximation.

On second grade fluids with vanishing viscosity

Abstract: We consider in ℝ n ( n = 2, 3) the equation of a second grade fluid with vanishing viscosity, also known as Camassa-Holm equation. We prove local existence and uniqueness of solutions for smooth initial data. We also give a blow-up condition which implies that the solution is global for n = 2. Finally, we prove the convergence of the solutions of second grade fluid equation to the solution of the Camassa-Holm equation as the viscosity tends to zero.

Drap brownien fractionnaire

Abstract: Resume On definit un processus a deux indices par integration fractionnaire de parametres α et β d'un bruit blanc On demontre qu'il est auto-similaire et a accroissements stationnaires et l'on donne quelques proprietes de regularite des trajectoires et celle du processus par rapport aux deux parametres

Stabilisation frontière indirecte de systèmes faiblement couplés

Abstract: Resume On etudie le probleme de la stabilisation frontiere indirecte de systemes lineaires couples. Pour de tels systemes (dans le cas de deux equations couplees), seule une des deux equations comporte un ≪ feedback ≫. On se pose alors la question de savoir si le systeme complet peut etre stabilise. Ces questions ont ete introduites par Russell [9]. On montre pour differents systemes (ondes couplees, plaques de Kirchhoff couplees) qu'il y a stabilite forte. De plus, on montre que l'energie decroit polynomialement, avec des taux de decroissance explicites pour les solutions suffisamment regulieres. On obtient ainsi de nouveaux resultats d'unicite. On generalise ces resultats au cas d'equations couplees ayant des vitesses de propagation differentes, sous une condition sur leur rapport dans le cas de domaines qui sont des produits d'intervalles.

Maxwell's equations when the charge conservation is not satisfied

Abstract: Maxwell's equations are overdetermined when the charge conservation equation is not verified. In order to overcome this problem, different methods have been introduced. We notice that they fit into a framework in which a new formulation which we introduce also fits. These formulations can be classified according to the type of the resulting PDE-system as hyperbolic-elliptic, hyperbolic-parabolic and purely hyperbolic. We show that the resolution of Maxwell's equation through the potentials is always equivalent to the purely hyperbolic formulation and that the hyperbolic-parabolic and hyperbolic-elliptic formulations converge to the purely hyperbolic formulation when introducing a parameter which goes to 0.

A simple definition for locally compact quantum groups

Abstract: In this Note we propose a simple definition of a locally compact quantum group in reduced form. By the word “reduced” we mean that we suppose the Haar weight to be faithful, and hence we define in fact arbitrary locally compact quantum groups represented on the L 2 -space of the Haar weight. We construct the multiplicative unitary associated with our quantum group. We construct the antipode with its polar decomposition, and the modular element. We prove the unicity of the Haar weights, define the dual and prove a Pontryagin duality theorem.

Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces*

Abstract: Resume Nous etudions la convergence de la solution d'un probleme d'elasticite lineaire anisotrope heterogene pose dans un cylindre dont le diametre tend vers zero. Nous montrons en particulier que uɛ − (u + ɛv + ɛ2w) converge fortement vers zero (dans un sens que nous precisons), ou (u, v, w) est la solution unique d'un systeme elliptique d'equations aux derivees partielles.

On the stability of measure valued processes with applications to filtering

Abstract: The purpose of this work is to present a novel approach to study the asymptotic stability of a class of measure valued processes arising in biology, in the theory of genetic type algorithms and in advanced signal processing, A new aspect of this result, as compared to existing literature on filtering, is that our approach is not based on auxiliary Hilbert projective metrics but only on Dobrushin's ergodic coefficient. Another advantage is that it allows to treat time in-homogencous signals taking values in a polish space and it is not restricted to non-linear filtering settings. What also makes our results interesting and new is that these qualitative properties lead to our knownledge to a first uniform convergence result with respect to time for u class of interacting particle schemes.

Flow of shear dependent electrorheological fluids

Abstract: We prove the existence of weak and strong solutions to the steady and unsteady system of partial differential equations with non-standard growth conditions describing the flow of shear dependent electrorheological fluids.

A version of Pitman's 2M-X theorem for geometric Brownian motions

Abstract: We show that geometric Brownian motion with parameter μ, i.e., the exponential of linear Brownian motion with drift μ, divided by its quadratic variation process is a diffusion process. Taking logarithms and an appropriate scaling limit, we recover the Rogers-Pitman extension to Brownian motion with drift of Pitman's representation theorem for the three-dimensional Bessel process. Time inversion and generalized inverse Gaussian distributions play crucial roles in our proofs.

Suite spectrale d'Adams et invariants cohomologiques des formes quadratiques

Abstract: Resume Pour tout corps k de caracteristique 0, la suite spectrale d'Adams pour le spectre des spheres et a coefficients dans la cohomologie motivique modulo 2 de Suslin-Voevodsky [8] converge vers le gradue associe a la filtration de l'anneau de Grothendieck-Witt des formes quadratiques sur k par les puissances de l'ideal engendre par les formes de rang pair. De plus, une certaine propriete de la cohomologie motivique modulo 2 de k , consequence de la demonstration par Voevodsky [9] de la conjecture de Milnor sur la cohomologie galoisienne modulo 2 de k , implique la degenerescence de cette suite spectrale dans la zone critique. Ceci nous permet de donner une nouvelle demonstration de la conjecture de Milnor sur le gradue de l'anneau de Witt de k [4], differente de celle de [11].

Analyse spectrale et singularités d'un problème de transmission non coercif

Abstract: Cette Note est consacree a l'analyse spectrale d'un operateur non borne associe a un probleme de transmission bidimensionnel non coercif. Nous montrons par une methode d'equations integrales que si l'interface est reguliere, cet operateur est auto-adjoint a resolvante compacte. Si l'interface presente un coin, une etude des singularites par transformee de Mellin permet d'obtenir une condition necessaire et suffisante, portant sur le contraste entre les deux milieux, pour que l'operateur soit auto-adjoint. S'il ne l'est pas, nous donnons une caracterisation de ses extensions auto-adjointes.

On the Deligne-Simpson problem

Abstract: Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ∃(n, C) (or Cjλ(n, C) so that there exist irreducible (p + 1) -tuples of matrices Mj e Cj (or Aj e cj) satisfying the equality m1 … Mp+1 = I (or A1 + … + Ap+1 = 0). We solve the problem for generic eigenvalues; their set is defined by explicit algebraic inequalities. There exist no reducible (p + l)-tuples of matrices for generic eigenvalues. The matrices M1 and Aj are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere.

Uniqueness of mild solutions of the Navier-Stokes equation and maximal Lp-regularity

Abstract: In this Note, we give a new proof of the uniqueness of mild solutions of the Navier-Stokes equation in C ([0, T ) ; (L 3 (ℝ 3 )) 3 ). The main tool of the proof is the maximal L p -regularity of the Laplacian in (L 3 (ℝ 3 )) 3 .