Publications

2012

• Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model
  
  • Bernard Etienne
  • Salvarani Francesco
  , 2012. In this paper we study the decay to the equilibrium state for the solution of a generalized version of the Goldstain-Taylor system, posed in the one-dimensional torus $\T=\R/\Z$, by allowing that the non-negative cross section $\sigma$ can vanish in a subregion $X:=\{ x \in \T\, \vert \, \sigma(x)=0\}$ of the domain with $\text{meas}\,(X)\geq 0$ with respect to the Lebesgue measure. We prove that the solution converges in time, with espect to the strong $L^2$-topology, to its unique equilibrium with an exponential rate whenever $\text{meas}\,(\T \setminus X)\geq 0$ and we give an optimal estimate of the spectral gap.
• A splitting theorem for extremal Kaehler metrics
  
  • Apostolov Vestislav
  • Huang Hongnian
  , 2012. Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics on each factor, provided that the integral Futaki invariants of the polarized manifold vanish or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau about the splitting of a Kaehler-Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kaehler metrics.
• Rotational and translational bias estimation based on depth and image measurements
  
  • Zarrouati Nadège
  • Rouchon Pierre
  • Beauchard Karine
  , 2012, pp.6627- 6634. Constant biases associated to measured linear and angular velocities of a moving object can be estimated from measurements of a static environment by embedded camera and depth sensor. We propose here a Lyapunov-based observer taking advantage of the SO(3)-invariance of the partial differential equations satisfied by the measured brightness and depth fields. The resulting observer is governed by a nonlinear integro/partial differential system whose inputs are the linear/angular velocities and the brightness/depth fields. Convergence analysis is investigated under C3 regularity assumptions on the object motion and its environment. Technically, it relies on Ascoli-Arzela theorem and pre-compacity of the observer trajectories. It ensures asymptotic convergence of the estimated brightness and depth fields. Convergence of the estimated biases is characterized by constraints depending only on the environment. We conjecture that these constraints are automatically satisfied when the environment does not admit any rotational symmetry axis. Such asymptotic observers can be adapted to any realistic camera model. Preliminary simulations with synthetic image and depth data (corrupted by noise around 10%) indicate that such Lyapunov-based observers converge for much weaker regularity assumptions.
• Fluid Dynamic Limits of the Kinetic Theory of Gases
  
  • Golse François
  , 2014, 75, pp.viii+320 pp.. These three lectures introduce the reader to recent progress on the hydrodynamic limits of the kinetic theory of gases. Lecture 1 outlines the main mathematical results in this direction, and explains in particular how the Euler or Navier-Stokes equations for compressible as well as incompressible fluids, can be derived from the Boltzmann equation. It also presents the notion of renormalized solution of the Boltzmann equation, due to P.-L. Lions and R. DiPerna, together with the mathematical methods used in the proofs of the fluid dynamic limits. Lecture 2 gives a detailed account of the derivation by L. Saint-Raymond of the incompressible Euler equations from the BGK model with constant collision frequency [L. Saint-Raymond, Bull. Sci. Math. 126 (2002), 493-506]. Finally, lecture 3 sketches the main steps in the proof of the incompressible Navier-Stokes limit of the Boltzmann equation, connecting the DiPerna-Lions theory of renormalized solutions of the Boltzmann equation with Leray's theory of weak solutions of the Navier-Stokes system, following [F. Golse, L. Saint-Raymond, J. Math. Pures Appl. 91 (2009), 508-552]. As is the case of all mathematical results in continuum mechanics, the fluid dynamic limits of the Boltzmann equation involve some basic properties of isotropic tensor fields that are recalled in Appendices 1-2. (10.1007/978-3-642-54271-8_1)
  DOI : 10.1007/978-3-642-54271-8_1
• On quantum averaging, quantum KAM and quantum diffusion
  
  • Neishtadt A. I.
  • Kuksin Sergei
  , 2012. For nonautonomous Hamiltonian systems and their quantisations we discuss properties of the quantised systems, related to those of the corresponding classical systems, described by the KAM-related theories: the proper KAM, the averaging theory, the Nekhoroshev stability and the diffusion.
• Approximation of a simple Navier-Stokes model by monotonic rearrangement
  
  • Brenier Yann
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2014, 34 (4), pp.1285-1300. We consider a Navier-Stokes model for compressible fluids in one space dimension. We show that it can be approximated by a time-discrete scheme combining the discretization of a trivial stochastic differential equation and the application of a suitable monotonic rearrangement operator In addition, our result can be easily extended to a related Navier-Stokes-Poisson system. (10.3934/dcds.2014.34.1285)
  DOI : 10.3934/dcds.2014.34.1285
• Sampling-based proofs of almost-periodicity results and algorithmic applications
  
  • Ben-Sasson Eli
  • Ron-Zewi Noga
  • Tulsiani Madhur
  • Wolf Julia
  , 2012. We give new combinatorial proofs of known almost-periodicity results for sumsets of sets with small doubling in the spirit of Croot and Sisask, whose almost-periodicity lemma has had far-reaching implications in additive combinatorics. We provide an alternative (and L^p-norm free) point of view, which allows for proofs to easily be converted to probabilistic algorithms that decide membership in almost-periodic sumsets of dense subsets of F_2^n. As an application, we give a new algorithmic version of the quasipolynomial Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by the last two authors, this implies an algorithmic version of the quadratic Goldreich-Levin theorem in which the number of terms in the quadratic Fourier decomposition of a given function is quasipolynomial in the error parameter, compared with an exponential dependence previously proved by the authors. It also improves the running time of the algorithm to have quasipolynomial dependence instead of an exponential one. We also give an application to the problem of finding large subspaces in sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n contains a large subspace. Using Fourier analytic methods, Sanders proved that such a subspace must have dimension bounded below by a constant times the density times n. We provide an alternative (and L^p norm-free) proof of a comparable bound, which is analogous to a recent result of Croot, Laba and Sisask in the integers.
• Homogénéisation symplectique et Applications de la théorie des faisceaux à la topologie symplectique
  
  • Vichery Nicolas
  , 2012. Dans une première partie, nous développerons la théorie de l'homogénéisation symplectique ainsi que ses applications à la théorie de Mather et à la rigidité symplectique. Les invariants spectraux lagrangiens seront l'outil de base de ce travail. Dans une seconde partie, nous rappelerons les toutes nouvelles applications de la théorie des faisceaux aux problèmes de non déplaçabilité. Nous formulerons ce que nous pensons être l'équivalent de l'homologie de Floer dans ce cas là et les invariants spectraux. Puis, à l'aide de ces outils nous prouverons la non-déplaçabilité de sous-variétés lagrangiennes non exactes du cotangent. Ensuite, nous parlerons des applications à la topologie symplectique $C^0$ et à l'optimisation non lisse.
• Equation de Burgers g en eralis ée a force al éatoire et a viscosit é petite
  
  • Boritchev Alexandre
  , 2012. Cette thèse traite du comportement des solutions u de l'équation de Burgers généralisée sur le cercle: u_t+f'(u)u_x=\nu u_{xx}+\eta,\ x \in S^1=\R/\Z. Ici, f est lisse, fortement convexe et satisfait certaines conditions de croissance. La constante 0<\nu << 1 correspond à un coefficient de viscosité. Nous considérons le cas où \eta=0, ainsi que le cas où \eta est une force aléatoire, lisse en x et peu régulière (de type "kick" ou bruit blanc) en t. Nous obtenons des estimations sur les normes de Sobolev de u moyennées en temps et en probabilité de la forme C \nu^{-\delta}, \delta >= 0, avec les mêmes valeurs de \delta pour les bornes supérieures et inférieures. On en déduit des estimations précises pour les quantités à petite échelle caractérisant la turbulence qui confirment exactement les prédictions physiques. Nous nous intéressons également au comportement asymptotique des solutions. Nous obtenons un résultat d'hyperbolicité des minimiseurs pour l'action correspondant à l'équation de Hamilton-Jacobi stochastique, dont la dérivée en espace est l'équation de Burgers stochastique avec \nu=0.
• Note on Decaying Turbulence in a Generalised Burgers Equation
  
  • Boritchev Alexandre
  , 2012. We consider a generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1, where $f$ is strongly convex and $\nu$ is small and positive. Under mild assumptions on the initial condition, we obtain sharp estimates for small-scale quantities. In particular, upper and lower bounds only differ by a multiplicative constant. The quantities which we estimate are the dissipation length scale and averages for structure functions and the energy spectrum for solutions $u$, which characterise the "Burgulence". Our proof uses a quantitative version of arguments contained in Aurell, Frisch, Lutsko & Vergassola 1992. Our estimates remain true for the inviscid Burgers equation.
• A uniform open image theorem for -adic representations i
  
  • Cadoret Anna
  • Tamagawa Akio
  Duke Mathematical Journal, Duke University Press, 2012. Let k be a field finitely generated over Q and let X be a smooth, separated and geometrically connected curve over k. Fix a prime . A representation ρ : π 1 (X) → GLm(Z ) is said to be geometrically Lie perfect if the Lie algebra of ρ(π 1 (X k )) is perfect. Typical examples of such representations are those arising from the action of π 1 (X) on the generic -adic Tate module T (Aη) of an abelian scheme A over X or, more generally, from the action of π 1 (X) on the -adic etale cohomology groups H i et (Y η , Q ), i ≥ 0 of the geometric generic fiber of a smooth proper scheme Y over X. Let G denote the image of ρ. Any k-rational point x on X induces a splitting x :</p><p>The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation ρ : π 1 (X) → GLm(Z ), the set Xρ of all x ∈ X(k) such that Gx is not open in G is finite and there exists an integer Bρ ≥ 1 such that [G : Gx] ≤ Bρ for every x ∈ X(k) Xρ.
• On the dynamics of Bohmian measures
  
  • Markowich Peter
  • Paul Thierry
  • Sparber Christof
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2012, 205 (3), pp.1031-1054. We revisit the concept of Bohmian measures recently introduced by the authors in \cite{MPS}. We rigorously prove that for sufficiently smooth wave functions the corresponding Bohmian measure furnishes a distributional solution of a nonlinear Vlasov-type equation. Moreover, we study the associated defect measures appearing in the classical limit. In one space dimension, this yields a new connection between mono-kinetic Wigner and Bohmian measures. In addition, we shall study the dynamics of Bohmian measures associated to so-called semi-classical wave packets. For these type of wave functions, we prove local in-measure convergence of a rescaled sequence of Bohmian trajectories towards the classical Hamiltonian flow on phase space. Finally, we construct an example of wave functions whose limiting Bohmian measure is not mono-kinetic but nevertheless equals the associated Wigner measure. (10.1007/s00205-012-0528-1)
  DOI : 10.1007/s00205-012-0528-1
• On the exponential decay to equilibrium of the degenerate linear Boltzmann equation
  
  • Bernard Etienne
  • Salvarani Francesco
  , 2012. In this paper we study the decay to the equilibrium state for the solution of the linear Boltzmann equation in the torus $\T^d=\bR^d/\bZ^{d}$, $d \in \N$, by allowing that the non-negative cross section $\sigma$ can vanish in a subregion $X:=\{ x \in \T^d\, \vert \, \sigma(x)=0\}$ of the domain with $\text{meas}(X)\geq 0$ with respect to the Lebesgue measure. We show that the geometrical characterization of $X$ is the key property to produce exponential decay to equilibrium.
• Henri Poincaré : des mathématiques à la philosophie, entretien avec Gerhard Heinzmann.
  
  • Paul Thierry
  , 2012. Entretien avec Gerhard Heinzmann, directeur de la MSH Lorraine, USR 3261 du CNRS et fondateur des Archives Henri Poincaré, UMR 7117
• Convergence of the calabi flow on toric varieties and related Kaehler manifolds
  
  • Huang Hongnian
  , 2012. Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $ \int_{\partial P} u ~ d \sigma < C_1, $ then there exists a constant $C_2$ depending only on $C_1$ and $P$ such that $\max_P u < C_2$. As an application, we show that if $(X,P)$ is analytic uniform $K$-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Riemannian curvature is uniformly bounded along the Calabi flow. Also we provide a proof of a conjecture of Donaldson. Finally, assuming that the curvature is bounded along the Calabi flow, our method would provide a proof of a conjecture due to Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman.
• Claude Viterbo - Théorie des faisceaux et Topologie symplectique (Part 4)
  
  • Viterbo Claude
  • Binder Robert
  • Bastien Fanny
  , 2012. The use of methods from the Sheaf Theory (Kashiwara-Schapira) was developped recently by Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara and Schapira. We will try to give an insight of that, in order to prove classical results, such as the Arnold conjecture, and to obtain new results.
• Claude Viterbo - Théorie des faisceaux et Topologie symplectique (Part 3)
  
  • Viterbo Claude
  • Binder Robert
  • Bastien Fanny
  , 2012. The use of methods from the Sheaf Theory (Kashiwara-Schapira) was developped recently by Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara and Schapira. We will try to give an insight of that, in order to prove classical results, such as the Arnold conjecture, and to obtain new results.
• Claude Viterbo - Théorie des faisceaux et Topologie symplectique (Part 2)
  
  • Viterbo Claude
  • Binder Robert
  • Bastien Fanny
  , 2012. The use of methods from the Sheaf Theory (Kashiwara-Schapira) was developped recently by Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara and Schapira. We will try to give an insight of that, in order to prove classical results, such as the Arnold conjecture, and to obtain new results.
• Claude Viterbo - Théorie des faisceaux et Topologie symplectique (Part 1)
  
  • Viterbo Claude
  • Binder Robert
  • Bastien Fanny
  , 2012. L’utilisation de méthodes de théorie des faisceaux (Kashiwara-Schapira)a été dévelopée ces dernières années par Tamarkin, Nadler, Zaslow, Guillermou, Kashiwara et Schapira. Nous essaierons d’en donner un aperçu à la fois pour démontrer des résultats classiques, comme la conjecture d’Arnold, et pour des résultats nouveaux.
• Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models
  
  • Dudret Stéphane
  • Beauchard Karine
  • Ammouri Fouad
  • Rouchon Pierre
  , 2012, pp.3352 - 3358. Distillation column monitoring requires shortcut nonlinear dynamic models. On the basis of a classical wave-model and time-scale reduction techniques, we derive a one-dimensional partial differential equation describing the composition dynamics where convection and diffusion terms depend non-linearly on the internal compositions and the inputs. The Cauchy problem is well posed for any positive time and we prove that it admits, for any relevant constant inputs, a unique stationary solution. We exhibit a Lyapunov function to prove the local exponential stability around the stationary solution. For a boundary measure, we propose a family of asymptotic observers and prove their local exponential convergence. Numerical simulations indicate that these convergence properties seem to be more than local.
• Actions infinitésimales dans la correspondance de Langlands locale p-adique
  
  • Dospinescu Gabriel
  , 2012. Cette thèse s'inscrit dans le cadre de la correspondance de Langlands locale $p$-adique, imaginée par Breuil et établie par Colmez pour GL_2(Q_p). Soit L une extension finie de Q_p et soit V une L-représentation irréductible du groupe de Galois absolu de Q_p, de dimension 2. En utilisant la théorie des (phi,Gamma)-modules de Fontaine, Colmez associe à V une GL_2(Q_p)-représentation de Banach Pi(V), unitaire, admissible, topologiquement irréductible. On donne une nouvelle preuve, nettement plus simple, d'un théorème de Colmez, qui permet de décrire les vecteurs localement analytiques Pi(V)^an de Pi(V) en fonction du (phi,\Gamma)-module surconvergent attaché à V. Le résultat principal de cette thèse est une description simple de l'action infinitésimale de GL_2(Q_p) sur Pi(V)^an. En particulier, on montre que Pi(V)^an admet un caractère infinitésimal, que l'on peut calculer en fonction des poids de Hodge-Tate de V, ce qui répond à une question de Harris. En utilisant ces résultats, on montre aussi l'absence d'un analogue p-adique d'un théorème classique de Tunnell et Saito, répondant à une autre question de Harris. Nous étendons et précisons certains résultats de Colmez concernant le modèle de Kirillov des vecteurs U-finis de Pi(V) (U est l'unipotent supérieur de GL_2(Q_p)). En combinant cette étude avec la description de l'action infinitésimale, on obtient une démonstration simple d'un des résultats principaux de Colmez, caractérisant les représentations V telles que Pi(V) possède des vecteurs localement algébriques non nuls. Ce résultat permet de faire le pont avec la correspondance classique et est un des ingrédients clés de la preuve d'Emerton de la conjecture de Fontaine-Mazur en dimension 2. On étend nos méthodes pour démontrer l'analogue de ce résultat pour les déformations infinitésimales de V. Cela répond à une question de Paskunas et a des applications à la conjecture de Breuil-Mézard. Une autre application est l'étude du module de Jacquet de Pi(V)^an. On montre qu'il est non nul si et seulement si V est trianguline, ce qui permet de donner une preuve simple des conjectures de Berger, Breuil et Emerton. Enfin, dans un travail en collaboration avec Benjamin Schraen, nous démontrons le lemme de Schur pour les représentations de Banach et localement analytiques topologiquement irréductibles d'un groupe de Lie p-adique. Ce résultat basique n'était connu que pour des groupes de Lie commutatifs et pour GL_2(Q_p).
• The first Johnson subgroups act ergodically on SU_2-character varieties
  
  • Funar Louis
  • Marché Julien
  , 2012. We show that the first Johnson subgroup of the mapping class group of a surface S of genus greater than one acts ergodically on the moduli space of representations of the fundamental group of S in SU_2. Our proof relies on a local description of the latter space around the trivial representation and on the Taylor expansion of trace functions.
• Ricci surfaces
  
  • Moroianu Andrei
  • Moroianu Sergiu
  , 2012. A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in $\mathbb{R}^3$ is a Ricci surface of non-positive curvature. At the end of the 19th century Ricci-Curbastro has proved that conversely, every point $x$ of a Ricci surface has a neighborhood which embeds isometrically in $\mathbb{R}^3$ as a minimal surface, provided $K(x)<0$. We prove this result in full generality by showing that Ricci surfaces can be locally isometrically embedded either minimally in $\mathbb{R}^3$ or maximally in $\mathbb{R}^{2,1}$, including near points of vanishing curvature. We then develop the theory of closed Ricci surfaces, possibly with conical singularities, and construct classes of examples in all genera $g\geq 2$.
• Two Hartree-Fock models for the vacuum polarization
  
  • Gravejat Philippe
  • Hainzl Christian
  • Lewin Mathieu
  • Séré Eric
  Journées Équations aux Dérivées Partielles, 2012, 2012. We review recent results about the derivation and the analysis of two Hartree-Fock-type models for the polarization of vacuum. We pay particular attention to the variational construction of a self-consistent polarized vacuum, and to the physical agreement between our non-perturbative construction and the perturbative description provided by Quantum Electrodynamics.
• Stochastic CGL equations without linear dispersion in any space dimension
  
  • Kuksin Sergei
  • Nersesyan Vahagn
  , 2012. We consider the stochastic CGL equation $$ \dot u- \nu\Delta u+(i+a) |u|^2u =\eta(t,x),\;\;\; \text {dim} \,x=n, $$ where $\nu>0$ and $a\ge 0$, in a cube (or in a smooth bounded domain) with Dirichlet boundary condition. The force $\eta$ is white in time, regular in $x$ and non-degenerate. We study this equation in the space of continuous complex functions $u(x)$, and prove that for any $n$ it defines there a unique mixing Markov process. So for a large class of functionals $f(u(\cdot))$ and for any solution $u(t,x)$, the averaged observable $\E f(u(t,\cdot))$ converges to a quantity, independent from the initial data $u(0,x)$, and equal to the integral of $f(u)$ against the unique stationary measure of the equation.