Publications

2004

• A negatively curved Kähler threefold not covered by the ball
  
  • Deraux Martin
  Inventiones Mathematicae, Springer Verlag, 2004, 160 (3), pp.501-525. (10.1007/s00222-004-0414-z)
  DOI : 10.1007/s00222-004-0414-z
• Existence globale de solutions d'énergie infinie de l'équation de Navier-Stokes 2D
  
  • Germain Pierre
  , 2004. We study in this article the solutions of the Navier-Stokes equations, with initial data in the closure of the Schwartz class in BMO-1. For such intial data, we obtain the existence and uniqueness of a global solution, and an estimate on its norm in BMO-1.
• Study of dispersive phenomena in geophysical fluids mechanics
  
  • Charve Frédéric
  , 2004. The introduction is composed by two parts: after a presentation of the geophysical fluids and the principles leading to the primitive equations system and to the quasigeostrophic approximation, we focus on the works done for the primitive equations and the rotating fluids systems. In the second chapter, we formally obtain the asymptotic for the sequence of solutions of the primitive system when the small parameter epsilon goes to zero. This also allows to define the potential vorticity, which we will be crucial in this study. We then study the convergence in the case of the Leray solutions. The third chapter is devoted to the same convergence in the case of solutions in the sense of Fujita and Kato. The last chapter gives more precise informations about the speed of convergence and we also prove a convergence theorem in the case of the vortex patches.
• Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints I: un modèle
  
  • Hager Mildred
  , 2004. Dans ce travail, nous considérons un opérateur différentiel simple ainsi que des perturbations. Alors que le spectre de l'opérateur non-perturbé est confiné à une droite à l'intérieur du pseudospectre, nous montrons pour les opérateurs perturbés que les valeurs propres se distribuent à l'intérieur du pseudospectre d'après une loi de Weyl.
• Almost isomorphism for countable state Markov shifts
  
  • Boyle Mike
  • Buzzi Jerome
  • Gomez Ricardo
  , 2004. Countable state Markov shifts are a natural generalization of the well-known subshifts of finite type. They are the subject of current research both for their own sake and as models for smooth dynamical systems. In this paper, we investigate their almost isomorphism and entropy conjugacy and obtain a complete classification for the especially important class of strongly positive recurrent Markov shifts. This gives a complete classification up to entropy conjugacy of the natural extensions of smooth entropy expanding maps, including all smooth interval maps with non-zero topological entropy.
• Decay of correlations on towers with non-Holder continuous Jacobian and non-exponential return time
  
  • Buzzi Jerome
  • Maume-Deschamps Véronique
  , 2004. We establish upper bounds on the rate of decay of correlations of tower systems with summable variation of the Jacobian and integrable return time. That is, we consider situations in which the Jacobian is not Holder and the return time is only subexponentially decaying. We obtain a subexponential bound on the correlations, which is essentially the slowest of the decays of the variation of the Jacobian and of the return time.
• On the universal cover of certain exotic Kähler surfaces of negative curvature
  
  • Deraux Martin
  Mathematische Annalen, Springer Verlag, 2004, 329 (4), pp.653-683. (10.1007/s00208-004-0531-4)
  DOI : 10.1007/s00208-004-0531-4
• Limite non visqueuse pour le système de Navier-Stokes dans un espace critique
  
  • Hmidi Taoufik
  • Keraani Sahbi
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2004, 338 (9), pp.689-692. Dans un article récent [11], Vishik montre que le système d'Euler bidimensionnel est globalement bien posé dans l'espace de Besov critique $B^2_{2,1}$. Nous montrons ici que le système de Navier-Stokes est globalement bien posé dans $B^2_{2,1}$, avec des estimations uniformes par rapport à la viscosité. Nous prouvons également un résultat global de limite non visqueuse. Le taux de convergence dans $L^2$ est de l'ordre $\nu$. (10.1016/j.cma.2004.02.013)
  DOI : 10.1016/j.cma.2004.02.013
• Semiclassical nonlinear Schrödinger equations with potential and focusing initial data
  
  • Carles Rémi
  • Miller Luc
  Osaka Journal of Mathematics, Osaka University, 2004, 41, pp.693-725. We study the asymptotic behaviour of solutions to semi-classical nonlinear Schrödinger equations with a potential, for concentrating and oscillating initial data, when the nonlinearity is repulsive and the potential is a polynomial of degree at most two. We describe the separate roles of the nonlinearity and of the potential, with tools which seem to be specific to the class of potentials that we consider. We also discuss the case of more general subquadratic potentials.
• Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length
  
  • Moroianu Andrei
  • Ornea Liviu
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2004, 338, pp.561-564. We prove that on a compact n-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue λ of the Dirac operator satisfies some inequality involving the scalar curvature. In the limiting case the universal cover of the manifold is isometric to a cylinder RxN where N is a manifold admitting Killing spinors. (10.1016/j.crma.2004.01.030)
  DOI : 10.1016/j.crma.2004.01.030
• Critical exponents and rigidity in negative curvature
  
  • Courtois Gilles
  Séminaires et congrès, Société mathématique de France, 2009, 18, pp.293-320. The goal of this lecture is to describe a theorem of M. Bonk and B. Kleiner on the rigidity of discrete groups acting on CAT(-1)-spaces whose limit set's Hausdorff and topological dimensions coincide. We will give the proof of M. Bonk and B. Kleiner and also an alternative proof in a particular case.
• Fourier-Laplace transform of irreducible regular differential systems on the Riemann sphere
  
  • Sabbah Claude
  Russian Mathematical Surveys, Turpion, 2004, 59, pp.1165-1180. We show that the Fourier-Laplace transform of an irreducible regular differential system on the Riemann sphere underlies, when one only considers the part at finite distance, a polarizable regular twistor $\mathcal{D}$-module. The associated holomorphic bundle out of the origin is therefore equipped with a natural harmonic metric with a tame behaviour near the origin.
• How violent are fast controls for Schrödinger and plate vibrations ?
  
  • Miller Luc
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2004, 172 (3), pp.429-456. Given a time T>0 and a region Omega on a compact Riemannian manifold M, we consider the best constant, denoted C_{T,Omega}, in the observation inequality for the Schrödinger evolution group of the Laplacian Delta with Dirichlet boundary condition: for all f in L^2(M), ||f||_{L^2(M)} \leq C_{T,Omega} ||exp(itDelta)f||_{L^2((0,T)xOmega)}. We investigate the influence of the geometry of Omega on the growth of C_{T,Omega} as T tends to 0. By duality, C_{T,Omega} is also the controllability cost of the free Schrödinger equation on M with Dirichlet boundary condition in time T by interior controls on Omega. It relates to hinged vibrating plates as well. We emphasize a tool of wider scope: the control transmutation method. We prove that C_{T,Omega} grows at least like exp(d^2/8T), where d is the largest distance of a point in M from Omega, and at most like exp(alpha L^2/T), where L is the length of the longest generalized geodesic in M which does not intersect Omega, and alpha is a constant in ]0,4[ (it is the growth rate of the controllability cost in a similar one dimensional problem). We also deduce such upper bounds on product manifolds for some control regions which are not intersected by all geodesics. (10.1007/s00205-004-0312-y)
  DOI : 10.1007/s00205-004-0312-y
• A criterion for existence of solutions to the supercritical Bahri-Coron's problem
  
  • Khenissy Saima
  • Rey Olivier
  Houston Journal of Mathematics, 2004, 30 (2), pp.587-613. We consider the supercritical elliptic problem $-\Delta u=u^{5+\epsilon},$ $u>0$ in $\Omega $; $u=0$ on $\partial\Omega$ with $\Omega$ a smooth bounded domain in $\mathbb{R}^3$, and $\epsilon>0$ a small number. Denoting by $G$ the Green's function of $-\Delta$ on $\Omega$ with Dirichlet boundary conditions, and by $H$ its regular part, we show that a nontrivial relative homology between the level sets $\varphi^{b}$ and $\varphi^{a}$ of $\varphi$, $0>b>a$, $\varphi(x,y)=H(x,x)^{1/2}H(y,y)^{1/2}-G(x,y)$, implies the existence, for $\epsilon$ small enough, of a solution to the problem which blows up, as $\epsilon$ goes to 0, at two points $x,y$ such that $a\leq\varphi(x,y)\leq b,$ $\nabla\varphi(x,y)=0.$
• Motivic zeta functions of infinite dimensional Lie algebras
  
  • Sautoy M. Du
  • Loeser F.
  Selecta Mathematica (New Series), Springer Verlag, 2004, 10, pp.253-303. We associate motivic zeta functions to a large class of infinite dimensional Lie algebras
• Isomotifs de dimension inférieure ou égale à un
  
  • Orgogozo Fabrice
  Manuscripta mathematica, Springer Verlag, 2004, 115 (3), pp.339-360. Après avoir rappelé quelques résultats de V. Voevodsky sur sa catégorie de motifs (cf. [Voe00]), on démontre l'équivalence de catégories, annoncée dans loc. cit. § 3.4 (p. 218), entre la catégorie dérivée des 1-isomotifs de P. Deligne sur un corps parfait d'une part et la catégorie triangulée des motifs géométriques effectifs de dimension inférieure ou égale à un. (10.1007/s00229-004-0495-4)
  DOI : 10.1007/s00229-004-0495-4
• Hamiltonian 2-forms in Kähler geometry II : global classification
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  • Tonnesen-Friedman C.
  Journal of Differential Geometry, International Press, 2004, 68 (2), pp.277-345.
• Volume et courbure totale pour les hypersurfaces de l'espace euclidien
  
  • Oancea Alexandru
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2004, 54, pp.733-772. The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get semi-local variants of the inequality holding in any dimension that involve domains with non-vanishing Gauss-Kronecker curvature. The paper also contains inequalities of isoperimetric type involving the total curvature, as well as a "reverse" isoperimetric inequality for spaces with constant curvature.
• Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time
  
  • Miller Luc
  Journal of Differential Equations, Elsevier, 2004, 204 (1), pp.202-226. Given a control region $\Omega$ on a compact Riemannian manifold $M$, we consider the heat equation with a source term $g$ localized in $\Omega$. It is known that any initial data in $L^{2}(M)$ can be steered to $0$ in an arbitrarily small time $T$ by applying a suitable control $g$ in $L^{2}([0,T]\times\Omega)$, and, as $T$ tends to $0$, the norm of $g$ grows like $\exp(C/T)$ times the norm of the data. We investigate how $C$ depends on the geometry of $\Omega$. %% 72 words We prove $C\geq d^{2}/4$ where $d$ is the largest distance of a point in $M$ from $\Omega$. When $M$ is a segment of length $L$ controlled at one end, we prove $C\leq \alpha_{*}L^{2}$ for some $\alpha_{*}<2$. Moreover, this bound implies $C\leq\alpha_{*}L_{\Omega}^{2}$ where $L_{\Omega}$ is the length of the longest generalized geodesic in $M$ which does not intersect $\Omega$. The {\em control transmutation method} used in proving this last result is of a broader interest. (10.1016/j.jde.2004.05.007)
  DOI : 10.1016/j.jde.2004.05.007
• A pseudodifferential equation with damping for one-way wave propagation in inhomogeneous acoustic media
  
  • Stolk Christiaan C.
  Wave Motion, Elsevier, 2004, 40, pp.111-121. A one-way wave equation is an evolution equation in one of the space directions that describes (approximately) a wave field. The exact wave field is approximated in a high frequency, microlocal sense. Here we derive the pseudodifferential one-way wave equation for an inhomogeneous acoustic medium using a known factorization argument. We give explicitly the two highest order terms, that are necessary for approximating the solution. A wave front (singularity) whose propagation velocity has non-zero component in the special direction is correctly described. The equation can't describe singularities propagating along turning rays, i.e. rays along which the velocity component in the special direction changes sign. We show that incorrectly propagated singularities are suppressed if a suitable dissipative term is added to the equation.
• Blowing up solutions for an elliptic Neumann problem with sub- and supercritical nonlinearity. Part I: N=3
  
  • Rey Olivier
  • Wei Juncheng
  Journal of Functional Analysis, Elsevier, 2004, 212, pp.472-499. We consider the sub- or supercritical Neumann elliptic problem $-\Delta u+\mu u=u^{5+\epsilon}$, $u>0$ in $\Omega $; $\frac{\partial u}{\partial n}=0$ on $\partial\Omega$, $\Omega$ being a smooth bounded domain in $\mathbb{R}^{3}$, $\mu>0$ and $\epsilon\neq0$ a small number. $H_{\mu}$ denoting the regular part of the Green's function of the operator $-\Delta +\mu$ in $\Omega$ with Neumann boundary conditions, and $\varphi_{\mu}(x)=\mu^{\frac{1}{2}}+H_{\mu}(x,x)$, we show that a nontrivial relative homology between the level sets $\varphi_{\mu}^{c}$ and $\varphi_{\mu}^{b}$, $b_0$ small enough, of a solution to the problem, which blows up as $\epsilon$ goes to zero at a point $a\in\Omega$ such that $b\leq\varphi_{\mu}(a)\leq c$. The same result holds, for $\epsilon< 0$, assuming that $0