Publications

2010

• Le système d'Euler de Kato
  
  • Wang Shanwen
  , 2010. Cette texte est consacrée au système d'Euler de Kato, construit à partir des unités modulaires, et à son image par l'application exponentielle duale (loi de réciprocité explicite de Kato). La présentation que nous en donnons est sensiblement différente de la présentation originelle de Kato.
• Distributions propres invariantes sur la paire symétrique (gl(4,R)/gl(2,R)*gl(2,R))
  
  • Jacquet Nicolas
  , 2010. Nous construisons des distributions propres invariantes pour la paire symétrique (gl(4,R)/gl(2,R)*gl(2,R)). Pour ceci, j'ai dans un premier temps décrit les orbites de GL(2,R)*Gl(2,R) sur ce quotient. J'ai ensuite généralisé certains résultats sur les intégrales orbitales de rang un (de J.Faraut) au rang deux. Ainsi j'ai obtenu le comportement des intégrales orbitales au voisinage des points semi-réguliers. Je me suis restreint à l'étude des distributions invariantes, propres sous l'action des opérateurs différentiels invariants à coefficients constants. données par des fonctions localement intégrables. J'ai d'abord déterminé les fonctions propres invariantes sur l'ouvert dense des éléments réguliers. Ceci est rendu possible par l'expression des parties radiales des opérateurs différentiels considérés en terme des opérateurs de Dunkl. Le comportement des intégrales orbitales m'a permis de déterminer lesquelles de ces fonctions donnaient une distribution propre invariante sur l'ensemble des éléments privés des nilpotents. Nous obtenons un espace vectoriel de dimension 6 dont certaines se prolongent naturellement à tout l'espace.
• Des sons et des Quanta
  
  • Paul Thierry
  , 2010. Dans cet article nous voudrions montrer certains liens entre musique, en particulier notation musicale, et formalisme quantique. Il apparaît en effet que la notation notes/portées permet d'apprivoiser un peu le formalisme mathématique de la mécanique quantique, réputé abstrait - comme d'ailleurs l'est la notation musicale. La discussion s'articulera sur deux points : la notion d'intrication qui est une extension ``inaudible" (littéralement) de celle d'accord et l'aléatoire quantique qui, lui, est bien représenté dans la notation mathématique et dont le pendant musical, dans les \oe uvres ouvertes, n'a pas d'écriture appropriée. Plusieurs autres points, plus musicaux et moins techniques, seront discutés.
• On the propagation of oceanic waves driven by a strong macroscopic flow
  
  • Gallagher Isabelle
  • Paul Thierry
  • Saint-Raymond Laure
  , 2010. In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around a inhomogeneous (non zonal) stationary profile. This extends the study~\cite{CGPS}, where the profile was assumed to be zonal only and where explicit calculations were made possible due to the 1D setting. Here the diagonalization of the system, which allows to identify Rossby and Poincaré waves, is proved by an abstract semi-classical approach. The dispersion of Poincaré waves is also obtained by a more abstract and more robust method using Mourre estimates. Only some partial results however are obtained concerning the Rossby propagation, as the two dimensional setting complicates very much the study of the dynamical system.
• On the propagation of oceanic waves driven by a macroscopic current
  
  • Gallagher Isabelle
  • Paul Thierry
  • Saint-Raymond Laure
  , 2010. In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around a inhomogeneous (non zonal) stationary profile. This extends the study, where the profile was assumed to be zonal only and where explicit calculations were made possible due to the 1D setting. Here the diagonalization of the system, which allows to identify Rossby and Poincare waves, is proved by an abstract semi-classical approach based on normal forms. The dispersion of Poincare waves is also obtained by a more abstract and more robust method using Mourre estimates. Only some partial results however are obtained concerning the Rossby propagation, as the two dimensional setting complicates very much the study of the dynamical system.
• Attaching handles to Delaunay nodoids
  
  • Pacard Frank
  • Rosenberg Harold
  Pacific Journal of Mathematics, Mathematical Sciences Publishers, 2013, 266 (1), pp.129-183. (10.2140/pjm.2013.266.129)
  DOI : 10.2140/pjm.2013.266.129
• À propos des phrasés d'énoncés mathématiques
  
  • Paul Thierry
  , 2010. Nous envisageons la notion de phrasé dans les diverses façons d'exprimer un énoncé mathématique.
• Strong Semiclassical Approximation of Wigner Functions for the Hartree Dynamics
  
  • Athanassoulis A.
  • Paul T.
  • Pezzotti F.
  • Pulvirenti M.
  , 2010. We consider the Wigner equation corresponding to a nonlinear Schroedinger evolution of the Hartree type in the semiclassical limit $\hbar\to 0$. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in $L^2$ to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the $L^2$ norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which -- as it is well known -- is not pointwise positive in general.
• Dispersion and controllability for the Schrödinger equation on negatively curved manifolds
  
  • Anantharaman Nalini
  • Riviere Gabriel
  , 2010. We study the time-dependent Schrödinger equation $ \imath\frac{\partial u}{\partial t}=-\frac{1}{2}\Delta u,$ on a compact riemannian manifold on which the geodesic flow has the Anosov property. Using the notion of semiclassical measures, we prove various results related to the dispersive properties of the Schrödinger propagator, and to controllability.
• From the Kinetic Theory of Gases to Continuum Mechanics
  
  • Golse François
  AIP Conference Proceedings, American Institute of Physics, 2011, 1333, pp.15-27. Recent results on the fluid dynamic limits of the Boltzmann equation based on the DiPerna-Lions theory of renormalized solutions are reviewed in this paper, with an emphasis on regimes where the velocity field behaves to leading order like that of an incompressible fluid with constant density. (10.1063/1.3562621)
  DOI : 10.1063/1.3562621
• Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall
  
  • Tsuji Tetsuro
  • Aoki Kazuo
  • Golse François
  Journal of Statistical Physics, Springer Verlag, 2010, 140 (3), pp.518-543. A free-molecular gas contained in a static vessel with a uniform temperature is considered. The approach of the velocity distribution function of the gas molecules from a given initial distribution to the uniform equilibrium state at rest is investigated numerically under the diffuse reflection boundary condition. This relaxation is caused by the interaction of gas molecules with the vessel wall. It is shown that, for a spherical vessel, the velocity distribution function approaches the final uniform equilibrium distribution in such a way that their difference decreases in proportion to an inverse power of time. This is slower than the known result for a rarefied gas with molecular collisions. (10.1007/s10955-010-9997-5)
  DOI : 10.1007/s10955-010-9997-5
• Semi-classical Analysis of Spin Systems near Critical Energies
  
  • Ribeiro Pedro
  • Paul Thierry
  , 2010. The spectral properties of $su(2)$ Hamiltonians are studied for energies near the critical classical energy $\varepsilon_c$ for which the corresponding classical dynamics presents hyperbolic points (HP). A general method leading to an algebraic relation for eigenvalues in the vicinity of $\varepsilon_c$ is obtained in the thermodynamic limit, when the semi-classical parameter $n^{-1}=(2s)^{-1}$ goes to zero (where $s$ is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near $\varepsilon_c$, are also computed and shown to be in agreement with the numerical results.
• THE MATHEMATICS OF COMPUTING BETWEEN LOGIC AND PHYSICS
  
  • Longo Giuseppe
  • Paul Thierry
  , 2010. Do physical processes compute? And what is a computation? These questions have gained a revival of interest in recent years, due to new technologies in physics, new ideas in computer sciences (for example quantum computing, networks, non-deterministic algorithms) and new concepts in logic. In this paper we examine a few directions, as well as the problems they bring to the surface.
• Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data
  
  • Ambrosio Luigi
  • Figalli Alessio
  • Friesecke Gero
  • Giannoulis Johannes
  • Paul Thierry
  , 2010. In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for ``almost all'' initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we study the flow in the space of measures induced by the continuity equation: we prove existence, uniqueness and stability properties of the flow in this infinite-dimensional space, in the same spirit of the theory developed in the case when the state space is Euclidean.
• Structures de Weyl ALF
  
  • Vassal Guillaume
  , 2010. In this paper, we introduce the notions of ALF Weyl connection and of associated mass, and we prouve the positive mass theorem for the ALF Weyl structures.
• Homogenization of transport problems and semigroups
  
  • Bernard Etienne
  • Golse François
  • Salvarani Francesco
  Mathematical Methods in the Applied Sciences, Wiley, 2010, 33 (10), pp.1228-1234. In some cases, the homogenization of evolution differential equations whose solution is given by the action of a semigroup on their initial data may lead to evolution problems with a completely different structure, usually integro-differential equations whose dynamics is not defined by a semigroup, but involves memory effects. A typical example is the homogenization --- or discretization --- of opacities in radiative transfer. In this paper, we propose a formulation of homogenized equations in terms of a semigroup acting on an enlarged phase space (i.e. on functions involving more variables than in the original problem.) (10.1002/mma.1319)
  DOI : 10.1002/mma.1319
• INDETERMINISME QUANTIQUE ET IMPREDICTIBILITE CLASSIQUE
  
  • Paul Thierry
  , 2010. We present in an informal way some recent results concerning a possible overlapping between classical unpredictability and quantum indeterminism.
• BOHMIAN MEASURES AND THEIR CLASSICAL LIMIT
  
  • Markowich Peter
  • Paul Thierry
  • Sparber Christof
  , 2010. We consider a class of phase space measures, which naturally arise in the Bohmian interpretation of quantum mechanics (when written in a Lagrangian form). We study the so-called classical limit of these Bohmian measures, in dependence on the scale of oscillations and concentrations of the sequence of wave functions under consideration. The obtained results are consequently compared to those derived via semi-classical Wigner measures. To this end, we shall also give a connection to the theory of Young measures and prove several new results on Wigner measures themselves. We believe that our analysis sheds new light on the classical limit of Bohmian quantum mechanics and gives further insight on oscillation and concentration eects of semi-classical wave functions.
• The zero set of conformal vector fields
  
  • Moroianu Andrei
  • Ornea Liviu
  , 2010. We show that every connected component of the zero set of a conformal vector field on a Riemannian manifold is totally umbilical
• Autour des équations de Navier-Stokes
  
  • Gallagher Isabelle
  Images des mathématiques, CNRS, 2010. (10.60868/vr1b-0a10)
  DOI : 10.60868/vr1b-0a10
• Clifford structures on Riemannian manifolds
  
  • Moroianu Andrei
  • Semmelmann Uwe
  , 2010. We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds, several classes of 8-dimensional manifolds, families of real, complex and quaternionic Grassmannians, as well as Rosenfeld's elliptic projective planes, which are symmetric spaces associated to the exceptional simple Lie groups. As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.
• On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas
  
  • Caglioti Emanuele
  • Golse François
  Journal of Statistical Physics, Springer Verlag, 2010, 141 (2), pp.264-317. The two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, this dynamical system is studied in the Boltzmann-Grad limit, assuming that the obstacle radius r and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle's distribution function is slowly varying in the space variable. In this limit, the periodic Lorentz gas cannot be described by a linear Boltzmann equation (see [F. Golse, Ann. Fac. Sci. Toulouse 17 (2008), 735--749]), but involves an integro-differential equation conjectured in [E. Caglioti, F. Golse, C.R. Acad. Sci. Ser. I Math. 346 (2008) 477--482] and proved in [J. Marklof, A. Stroembergsson, preprint arXiv:0801.0612], set on a phase-space larger than the usual single-particle phase-space. The main purpose of the present paper is to study the dynamical properties of this integro-differential equation: identifying its equilibrium states, proving a H Theorem and discussing the speed of approach to equilibrium in the long time limit. In the first part of the paper, we derive the explicit formula for a transition probability appearing in that equation following the method sketched in [E. Caglioti, F. Golse, loc. cit.]. (10.1007/s10955-010-0046-1)
  DOI : 10.1007/s10955-010-0046-1
• Density of vector bundles periodic under the action of Frobenius
  
  • Ducrohet Laurent
  • Mehta Vikram
  Bulletin des Sciences Mathématiques, Elsevier, 2010, 134 (5), pp.454-460. (10.1016/j.bulsci.2009.11.001)
  DOI : 10.1016/j.bulsci.2009.11.001
• Equidistribution of Dense Subgroups on Nilpotent Lie Groups
  
  • Breuillard Emmanuel
  Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2010, 30 (1), pp.131-150. Let $\Gamma$ be a dense subgroup of a simply connected nilpotent Lie group $G$ generated by a finite symmetric set $S$. We consider the $n$-ball $S_n$ for the word metric induced by $S$ on $\Gamma$. We show that $S_n$ (with uniform measure) becomes equidistributed on $G$ with respect to the Haar measure as n tends to infinity. We give rates and also prove the analogous result for random walk averages (i.e. the local limit theorem). (10.1017/S0143385709000091)
  DOI : 10.1017/S0143385709000091
• Groups and Symmetries. From Finite Groups to Lie Groups
  
  • Kosmann-Schwarzbach Yvette
  , 2010, pp.194. (10.1007/978-0-387-78866-1)
  DOI : 10.1007/978-0-387-78866-1