Reference:LMENS-99-1 (January 99)

Source: ps

Abstract:In this paper we investigate uniqueness of solutions of the Ginzburg-Landau system for superconductivity in the regime where the thickness of the slab $2a$is small. We analyze the equations of first variation with respect to 2 shooting parameters and obtain estimates on all relevant quantities at the end of theslab where the boundary conditions are prescribed. Using these estimates, we prove that the bifurcation curve of symmetric solutions is given by a decreasing function of the order parameter when the film is thin enough. In addition, we prove that there is no curve of asymmetric solutions branching from the symmetric curve.

AMS Classification:34B15, 34C23, 93C15, 34C11

Keywords :Ginzburg-Landau system, uniqueness, symmetric, asymmetric, superconductivity, bifurcation.

Reference:LMENS-99-2 (January 99)

Source: dvi, ps

Abstract:This paper establishes various asymptotic limits of the Vlasov-Poisson equation with strong external magnetic field, some of which were announced in [GSR]. The so-called "guiding center approximation" is proved in the 2D case with a constant magnetic field orthogonal to the plane of motion, in various situations (non collisional or weakly collisional). The 3D case is studied on the time scale of the motion along the lines of the magnetic field, much shorter than that of the guiding center motion. We discuss in particular the effect of non constant external magnetic fields.

Reference:LMENS-99-3 (January 99)

Source: dvi, ps

Abstract:We consider the solutions to a kinetic equation which kinetic energy converges to zero fast enough. We prove that they concentrate near the speed zero and converge towards a measure which is a product of a measure on the spacial coordinates and a Dirac mass on the speed coordinates. The difficult point here is that the full solution converges since we do not know any characterisation of the limit problem for the spatial density. We give two results of this kind, depending on the regularity of the solution, and on the assumptions. Finally we present an example of equation which describes the interactions of particles in a flow and where these theorems apply.

Résumé:Nous démontrons ici que si une solution d'une quelconque équation cinétique a une énergie cinétique qui tend vers zero, alors toute la masse se concentre autour des vitesses nulles. Plus précisément la solution admet une limite en temps grand qui se d'ecompose en un produit d'une mesure sur les coordonn'ees spatiales et d'une masse de Dirac sur les coordonnées en vitesse. Nous détaillerons deux théorèmes avec leurs conditions d'application selon la régularité de la solution considérée et nous donnerons un exemple d'équation auquel ces théorèmes peuvent s'appliquer.

AMS Classification:35B40, 35B45, 35Q35, 76D07

Keywords :kinetic equation, long time asymptotic, Vlasov equation, systems of particles in a fluid

Reference:LMENS-99-4 (Janvier 99)

Source: ps

Abstract:Le sujet de cette thèse se situe à la frontière entre la théorie des lois de conservation et la théorie cinétique des gaz. On s'intéresse au système de la dynamique des gaz isentropiques avec $\gamma=3$ qui admet une bonne formulation cinétique. Une méthode de localisation, inspirée de méthodes utilisées pour étudier les singularités d'équations paraboliques, est développée dans ce cadre pour montrer différents résultats.

Dans une première partie, ces techniques nous permettent de démontrer un résultat de régularité en temps des solutions pour des données initiales quelconques. Plus précisément, on montre que toute solution uniformément bornée est continue en temps à valeurs dans $L^1_{\mathrm{loc}}(\R)$ en espace, ce qui est naturel au moins pour récupérer la donnée initiale dans un sens fort. Ce résultat est, à ma connaissance, le premier résultat de régularité de ce type pour un système sans théorie $BV$.

Dans une deuxième partie nous prouvons la convergence d'un schéma cinétique semi-discret en temps. La difficulté de ce résultat réside dans le fait qu'il apparaît des oscillations d'ordre $\Delta t$, le pas de discrétisation. Cependant, grâce à la méthode de localisation, on montre que l'amplitude de ces oscillations tend vers zéro, et que le schéma converge vers la bonne solution.

Enfin, dans une troisième partie nous nous intéressons à l'existence et aux propriétés des profils de chocs associés à ce schéma semi-discret.

Ces trois parties sont suivies de quatre appendices. Nous montrons notamment la convergence du schéma cinétique dans le cas scalaire sans utiliser la méthode de localisation. La preuve repose alors sur une étude précise de la variation d'entropie.

AMS Classification:35L65, 35L67, 65M12, 76P0

Keywords :Lois de conservation, solutions entropiques, formulation cinétique, pseudo-maxwellienne, lemmes de moyenne, oscillations, résultats de régularité, schémas numériques, profils de choc.

Reference:LMENS-99-5 (January 99)

Source: ps

Abstract:The aim of this paper is to introduce a method for restoring a sequence of time varying images. Our approach is based on the description of images by contour lines in a Bayesian context. The use of contour lines for image restoration has already been discussed in Catoni [Catoni92]. Here we adapt the methodology to our problem and consider the contour surfaces described by contour lines when the time dimension is added. We use a generalization of the noisy Ising model on the level sets representation of grey-level sequences of images. In order to allow boundary smoothing, we choose appropriately the connectivity of the model, so that we can make the Hamiltonian proportional to a satisfactory approximation of the Euclidean length of the boundaries. The noise model reflects the distance between the observation and the reconstructed sequence. We use a Gaussian noise model translated back to the level sets representation, to take into account the pixel contrast. We propose a generalization of the dichotomic Markov field model introduced in [Catoni92].

Bayesian methods of restoration are numerically intensive. We discuss the advantages of the contour lines dichotomic conditioning restoration compared to the usual global conditioning method of restoration. In order to speed-up the restoration, we implement the hierarchical multi-resolution algorithm [Cot98] for the dichotomic model of reconstruction of contour lines. A range of practical examples illustrate this approach.

AMS Classification:primary: 60J10, 60J20, 60J35, 60F10

secondary: 60-14, 93E14, 94A12

Keywords :Image restoration, Markov field, dichotomic Markov field model, Monte-Carlo algorithms

Reference:LMENS-99-6 (January 99)

Source : ps

Abstract:Using the stability results of Bressan and Colombo ([B-C]) for strictly hyperbolic ${2 \times 2}$ systems in one space dimension, we prove that the solutions of isentropic and non isentropic Euler equations in one space dimension with the respective initial data $(\rho _0, u_0)$ and $(\rho _0, u_0, \theta _0= \rho _0^{\gamma -1})$ remain close as soon as the total variation of $(\rho _0, u_0)$ is sufficiently small.

AMS Classification :76Nxx, 35Q30

Keywords :compressible Euler equations, isentropic Euler equations, hyperbolic system, Glimm's functional, macroscopic entropy, Riemann invariant.

Reference:LMENS-99-7 (February 99)

Source: dvi, ps

Abstract:In this note, we propose a formal argument identifying the hydrodynamic limit of a Fokker-Planck model for granular media appearing in [BCCP]. More precisely, in the limit of large background temperature and vanishing friction, this hydrodynamic limit is described by the classical system of isentropic gas dynamics with a nonstandard pressure law (specifically, the pressure is proportional to the cube root of the density). Finally, some qualitative properties of the hydrodynamic model are studied.

Reference:LMENS-99-8 (February 99)

Source: dvi, ps

Abstract:We give a detailed treatment of some recent developments of the theory of spatial branching processes, and their connections with certain partial differential equations. We first present a short self-contained approach to the measure-valued branching processes called superprocesses. In the special case of the quadratic branching mechanism, we explain the powerful construction based on the path-valued process called the Brownian snake. This construction relies on remarkable properties of branching trees embedded in Brownian excursions, which are of independent interest. In this connection, we discuss Aldous' continuum random tree and the random measure known as ISE (integrated super-Brownian excursion). We then investigate connections between the Brownian snake and the partial differential equation $\Delta u=u^2$. In particular, we present the Dynkin-Perkins characterization of polar sets, corrresponding to removable singularities for the p.d.e., we provide a necessary and sufficient condition for the existence in a domain of a solution with boundary blow-up, and we discuss the probabilistic classification of general nonnegative solutions in a smooth domain. Finally, we explain how the snake approach can be extended to more general branching mechanisms by introducing suitable functionals of L\'evy processes without negative jumps

AMS Classification:60J80, 60G57, 60J65, 60J55, 35J60, 35J65

Keywords :Branching process, superprocess, super-Brownian motion, Brownian snake, continuous-state branching process, tree, Galton-Watson tree, Brownian excursion, continuum random tree, integrated super-Brownian excursion, local time, exit measure, partial differential equation, Dirichlet problem, polar set, removable singularity, solution with boundary blow-up, boundary polar set, trace, L\'evy process, height process, exploration process.

Reference:LMENS-99-9 (February 99)

Source: ps

Abstract :The purpose of this note is to derive compactness properties for both incompressible and compressible viscous flows in a bounded domain interacting with a finite number of rigid bodies. We prove the global existence of weak solutions away from collisions.

AMS Classification :35Q10, 76D99, 73B99

Keywords :Fluid-structure interaction, rigid bodies, incompressible and compressible Navier-Stokes equations, weak solutions

Reference :LMENS-99-10 (February 99)

Abstract :One studies the subgroups of ${\rm GL}(m,{\bf R})$ which preserve a properly convex cone of ${\bf R}^{m}$ and whose action on ${\bf R}^{m}$ is irreducible. In particular, one describes the Zariski closure of these subgroups.

As an application, one describes the Zariski closure $G$ of the subgroups of ${\rm GL}(m,{\bf R})$ all of whose elements have nothing but positive eigenvalues. For instance, one can get the group $G={\rm GL}(m,{\bf R})$ if and only if $m\not\equiv 2\; \mbox{{\rm modulo $4$}}$.

Résumé :On étudie les sous-groupes de ${\rm GL}(m,{\bf R})$ qui préservent un cône convexe saillant de ${\bf R}^{m}$ et dont l'action sur ${\bf R}^{m}$ est irréductible. En particulier, on décrit les adhérences de Zariski possibles pour ces sous-groupes.

Comme application, on décrit les adhérences de Zariski $G$ possibles pour les sous-groupes de ${\rm GL}(m,{\bf R})$ dont tous les éléments ont toutes leurs valeurs propres positives. Par exemple, le groupe $G={\rm GL}(m,{\bf R})$ convient si et seulement si $m\not\equiv 2\; \mbox{{\rm modulo $4$}}$.

AMS Classification :20H15, 22E40, 22E46, 53C99

Keywords :groupes discrets, cônes convexes, ensemble limite, valeurs propres, proximalité

Reference:LMENS-99-11 (February 99)

Source: dvi, ps

Abstract:This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension $n$ is associated a closed subgroup of $SO(n)$, the holonomy group; this is one of the most basic invariants of the metric. A famous theorem of Berger gives a complete (and rather small) list of the groups which can appear. Surprisingly, the compact manifolds with holonomy $\not=SO(n)$ are all related in some way to Algebraic Geometry. This leads to the study of special algebraic varieties (Calabi-Yau, complex symplectic or complex contact manifolds) for which Riemannian geometry rises interesting questions.

AMS Classification :53C25, 32C17, 53C55

Keywords :holonomy, hyperkähler, quaternion-Kähler, Calabi-Yau manifolds, contact structures

Reference:LMENS-99-12 (April 99)

Reference:LMENS-99-13 (April 99)

Reference:LMENS-99-14 (May 99)

Source: dvi, ps

Abstract:The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L1) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L2 initial data of the acoustic equations. The associated local conservation laws are recovered in the limit.

Reference :LMENS-99-15 (May 99)

Abstract :We prove a Liouville Theorem for the following heat system whose nonlinearity has no gradient structure \[ \partial_t u =\Delta u+v^p,\;\;\partial_t v=\Delta v +u^q, \] where $pq>1$, $p\ge 1$, $q\ge 1$ and $|p-q|$ small.\\ We then deduce a localization property and uniform $L^\infty$ estimates of blowing-up solutions of this system.

AMS Classification :35K40, 35K55, 35A20

Keywords :Heat system, Liouville theorem, blow-up, localization.

Reference :LMENS-99-16 (May 99)

Reference :LMENS-99-17 (June 99)

Source : dvi, ps

Abstract :We use Bochner's subordination to give a representation of the genealogical structure associated with general continuous-state branching processes. We then apply this representation to connections between a branching process introduced by Neveu, and the coalescent process recently investigated by Bolthausen-Sznitman and others.

AMS Classification :Primary 60 J 80; secondary 60 J 30

Keywords :Continuous-state branching processes, genealogy, subordination, coalescent

Reference :LMENS-99-18 (Juin 99)

Source : dvi, ps

Reference :LMENS-99-19 (June 99)

Source : dvi, ps

Abstract :The purpose of this paper is to investigate the limit of some kinetic equa- tions with a strong force. Due to friction, the solution concentrates to a monokinetic distribution so as to keep the total of force bounded and in the limit we recover a macroscopic system. This kind of asymptotics is a natural question when the mass of the particles is very small or their inertia is neglected. After that we also study the properties of the limit system and especially the uniqueness of solutions which provides the full convergence of the family of solutions to the kinetic equation.

Résumé :Cet article se propose d'étudier la limite de solutions d'une équation cinétique avec frottement lorsque les termes de force deviennent prédominants. A cause du frottement, les solutions se concentrent progressivement en vitesse de manière à ce que la somme des forces reste bornée ; à la limite cette concentration nous oblige à remplacer l'équation cinétique par un système macroscopique. Cette problème apparait notamment quand on fait tendre vers zéro la masse des particules ou quand on néglige leur inertie. Enfin certaines propriétés du système, et particulièrement l'unicité, seront détaillées afin d'obtenir une convergence de toute la suite des solutions et pas seulement d'une suite extraite.

Reference :LMENS-99-20 (June 99)

Source : dvi, ps

Abstract :We introduce a new approach to prove the regularity of solutions to transport equations of the Vlasov type. Our approach is mainly based on the proof of propagation of velocity moments, as in a previous paper by Lions and Perthame [LP2 ]. We combine it with Moment Lemmas which assert that, locally in space, velocity moments can be gained from the kinetic equation itself. We apply our theory to two cases. First, to the Vlasov-Poisson system, and we solve a long standing conjecture, namely the propagation of any moment larger than two. Next, to the Vlasov-Stokes system where we prove the same result for fairly singular kernels.

Reference:DMA-99-22 (November 1999)

Source:dvi,ps

Abstract:We define a suitable weak formulation of the Boltzmann equation with a cross-section singular in both the relative velocity and the angular variables. This formulation allows us to extend the DiPerna-Lions theory to physically realistic long-range interactions, and to prove the validity of the Landau approximation in plasma physics assuming only natural physical bounds.

Reference:LMENS-99-23 (October 1999)

Source:dvi,ps

Abstract:We show that transport inequalities, similar to the one derived by Talagrand for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial differential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information.

AMS Classification :****

Keywords :****

Reference:LMENS-99-24 (October 1999)

Source:dvi,ps

Abstract:We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails. We compensate the lack of uniform in time estimates by the use of precise logarithmic Sobolev-type inequalities. Our method not only gives explicit results on the times of convergence, but is also able to cover situations in which compactness arguments apparently do not apply.

AMS Classification :****

Keywords :****

Reference:DMA-99-25 (July 2000)

Source:ps

Abstract:We derive the high frequency limit of the Helmholtz equations in terms of quadratic observables. We prove that it can be written as a stationary Liouville equation with source terms. Our method is based on the Wigner Transform, which is a classical tool for evolution dispersive equations. We extend its use to the stationary case after an appropriate scaling of the Helmholtz equation. Several specific difficulties arise here; first, the identification of the source term (which does not share the quadratic aspect) in the limit, then, the lack of $L^2$ bounds which can be handled with homogeneous Morrey-Campanato estimates, and finally the problem of uniqueness which, at several stage of the proof, is related to outgoing conditions at infinity.

AMS Classification :****

Keywords :****

Reference:DMA-99-26 (November 1999)

Source:ps

Abstract:In this paper, we study the Ginzburg-Landau equations for a 2 dimensional domain which has small size. We prove that if the domain is small, then the solution has no zero; that is no vortex. Additionnally, we obtain that if the domain is a disc of small radius, then the solution is symmetric.Then, in the case of a slab, that is a one dimensional domain, we use the same method to derive that solutions are symmetric. The proof uses a priori estimates and the Poincaré inequality.

AMS Classification :****

Keywords :****

Reference:DMA-99-27 (October 1999)

Source:dvi,ps

Abstract:A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed in a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier-Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in Knudsen layer, are included in the system of the Navier-Stakes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.

AMS Classification :****

Keywords :rarefied gas, Boltzmann equation, slip flow, kinetic theory

Reference:DMA-99-28 (November 1999)

Source:ps

Abstract:We give some extensions of the classical chain rule $\nabla[g(u)]=g'(u)\nabla u$ and of the chain rule for transport operators $a\cdot\nabla[g(u)]=g'(u)\,a\cdot\nabla u$, $a\in W^{1,1}$, via an inverse Sard lemma. Coefficients $a$ of bounded variation are also treated in the case of a Vlasov operator.

AMS Classification :46F10, 35R05, 35F99, 35L99

Keywords :chain-rule, inverse Sard lemma, transport equations with unsmooth coefficients

Reference:DMA-99-29 (November 1999)

Source:dvi,ps

Abstract:caractéristique résiduelle $p>0$, nous montrons qu'il n'existe pas de représentation irréductible supercuspidale inertiellement autoduale de $GL_N(F)$ dès que le produit $pN$ est impair ; nous en donnons divers exemples dans le cas inverse. A l'aide de {\bf [Le]}, nous construisons ensuite pour $p \not=2$ des paires couvrantes scindées de types simples de niveau $>0$ dans les groupes $Sp_{2N}(F)$, $SO_{2N}(F)$ et $SO_{2N+1}(F)$ relativement au parabolique de Siegel ; en particulier, nous les obtenons toutes si $N$ est impair.

AMS Classification :****

Keywords :****

Reference:DMA-99-30 (November 1999)

Source:dvi,ps

Abstract:This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.

AMS Classification :11M06, 60J65, 60E07

Keywords :Infinitely divisible laws, sums of independent exponential variables, Bessel process, functional equation

Reference:DMA-99-31 (January 2000)

Source:ps

Abstract:Consider the domain $$ Z_\eps=\{x\in\R^n\,|\,\hbox{dist}(x,\eps\Z^n)>\eps^\g\} $$ and let the free path length be defined as $$ \tau_\eps(x,v)=\inf\{t>0\,|\,x-tv\in\d Z_\eps\}\,. $$ In the Boltzmann-Grad scaling corresponding to $\g={n\over n-1}$, it is shown that the limiting distribution $\phi_\eps$ of $\tau_\eps$ is bounded from below by an expression of the form $C/t$, for some $C>0$. % A numerical study indicate that asymptotically for large $t$, $\tau_\eps \sim C/t$. % This is an extension of a previous work [BGW]. As a consequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describe the Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usual case of a Poisson distribution of scatterers.

AMS Classification :****

Keywords :****

Reference:DMA-99-32 (November 1999)

Source:dvi,ps

Abstract:We study the parametrization of the moduli space Bun_2(C)_L of rank 2 %bundles over a curve C with fixed determinant, provided by Hecke %modifications at fixed points of the trivial bundle. This %parametrization is closely related to the Tyurin parametrization of %vector bundles over curves. We use it to parametrize the Hitchin and %KZB systems, as well as lifts of the Beilinson-Drinfeld D-modules. We %express a generating series for the lifts of the Beilinson-Drinfeld %operators in terms of a "quantum L-operator" \ell(z). We explain the %relation to earlier joint work with G. Felder, based on parametrization %by flags of bundles (math/9807145) and introduce filtrations on conformal %blocks, related with the Hecke modifications.

AMS Classification :****

Keywords :****

Reference:DMA-99-33 (November 1999)

Source:dvi,ps

Abstract:Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations $dX_t=dS_t-{1\over 2}f(X_t)dt$ where $S_t$ is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as $t\to\infty$, towards the master field (i.e. the Gibbs state).

AMS Classification :****

Keywords :****