Gérard–Levelt membranes

We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical Gérard–Levelt lattice saturation procedure can be geometrically understood in terms of a projection on the tropical linear space attached to a subset of the local affine Bruhat–Tits building, which we call the Gérard–Levelt membrane. This provides a way to compute the true Poincaré rank, but also the Katz rank of a meromorphic connection without having to perform either gauge transforms or ramifications of the variable. We finally present an efficient algorithm to compute this tropical projection map, generalising Ardila’s method for Bergman fans to the case of the tight-span of a valuated matroid.     

Une formule de Landau–Zener pour un croisement non dégénéré et involutif de codimension 3Landau–Zener formula for non-degenerate involutive codimension 3 crossings

In this Note, we study a 2×2 system of evolution equations with some codimension 3 crossing. We derive two conditions of non-degeneracy. We focus on one of them and reduce our system to some Landau–Zener's type system. Using this reduction, we describe the energy transfer at the crossing by Landau–Zener formula for 2-scales semi-classical measures. To cite this article: C. Fermanian Kammerer, P. Gérard, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 915–920.     

Stabilization and Control for the Nonlinear Schrödinger Equation on a Compact Surface

In this paper, we study the stabilization property and the exact controllability for the nonlinear Schrödinger equation on a two dimensional compact Riemannian manifold, without boundary. We use a pseudo-differential dissipation. The proofs are based on a result of propagation of singularities and on recent dispersion estimates (Strichartz type inequalities) due to N. Burq, P. Gérard and N. Tzvetkov.     

Homogenization with an oscillating drift: from L 2-bounded to unbounded drifts, 2D compactness results, and 3D nonlocal effects

This paper extends results obtained by Tartar (1977, 1986) and revisited in Briane and Gérard (Ann Scuola Norm Sup Pisa, to appear), on the homogenization of a Stokes equation perturbed by an oscillating drift. First, a N-dimensional scalar equation, for N ≥ 3, and a tridimensional Stokes equation are considered in the periodic framework only assuming the L ²-boundedness of the drift and so relaxing the equi-integrability condition of Briane and Gérard (Ann Scuola Norm Sup Pisa, to appear). Then, it is proved that the L ²-boundedness can be removed in dimension two, provided that the divergence of the drift has a sign. On the contrary, nonlocal effects are derived in dimension three with a free divergence drift that is only bounded in L ¹.     

Instabilité Spectrale Semiclassique d’Opérateurs Non-Autoadjoints II

In this work, we consider analytic (pseudo-)differential operators as well as random perturbations. We show for the perturbed operators that with probability almost 1, the eigenvalues inside a subdomain of the pseudospectrum are distributed according to a bidimensional Weyl law.

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Résumé. Dans ce travail, nous considérons des opérateurs (pseudo-)différentiels analytiques ainsi que des perturbations aléatoires. Nous montrons pour les opérateurs perturbés qu’avec une probabilité proche de 1, les valeurs propres dans un sous-ensemble du pseudospectre sont distribuées d’après une loi de Weyl.

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Communicated by Christian Gérard

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Submitted: July 7, 2005; Accepted: February 8, 2006     

The AC Stark Effect, Time-Dependent Born–Oppenheimer Approximation, and Franck–Condon Factors

We study the quantum mechanics of a simple molecular system that is subject to a laser pulse. We model the laser pulse by a classical oscillatory electric field, and we employ the Born–Oppenheimer approximation for the molecule. We compute transition amplitudes to leading order in the laser strength. These amplitudes contain Franck–Condon factors that we compute explicitly to leading order in the Born–Oppenheimer parameter. We also correct an erroneous calculation in the mathematical literature on the AC Stark effect for molecular systems. Communicated by Christian Gérard. Submitted: August 15, 2005; Accepted: October 13, 2005     

On the Holomorphic Solution of Non–Linear Totally Characteristic Equations

The paper deals with a non–linear singular partial differential equation in the holomorphic category. When (E) is of Fuchsian type, the existence of the unique holomorphic solution was established by Gérard –Tahara [2]. In this paper, under the assumption that (E) is of totally characteristic type, the authors give a sufficient condition for (E) to have a unique holomorphic solution. The result is extended to higher order case.     

Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in three dimensional exterior domains

We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schrödinger and Gross–Pitaevskii equations on the exterior of a nontrapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate [R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, arxiv:math.AP/0512639, Bull. Soc. Math. France, submitted for publication] with a smoothing effect on exterior domains [N. Burq, P. Gérard, N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. I.H.P. (2004) 295–318].     

Parabolic variant of H-measures in homogenisation of a model problem based on Navier–Stokes equation

H-measures, as originally introduced by Luc Tartar and (independently) Patrick Gérard are well suited for hyperbolic problems. For parabolic problems, some variants should be considered, which would be better adapted to parabolic problems.Recently, we introduced a few parabolic scalings and corresponding variant H-measures, including the existence results, investigating their applicability. Here, we present an application of such a variant in homogenisation, for a model based on nonstationary Stokes (sometimes called linearised Navier–Stokes) system.Besides expressing the homogenised coefficients directly in the terms of variant H-measures corresponding to the oscillating coefficients, we also prove that the homogenised coefficients are symmetric, as originally conjectured by Tartar.     

A parabolic variant of H-measures

H-measures, as originally introduced by Luc Tartar and Patrick Gérard, are suited to hyperbolic problems. However, they turned out not to be well adjusted to the study of parabolic equations. A variant of H-measures is proposed, which is much better adapted to such kind of problems. We present the new parabolic scaling and the main ingredients for the proof of existence of the new variant. Some applications to the Schrödinger equation and vibrating plate equation are shown, together with an outlook to possible applications in other problems.     

Nonlinear generalized Dunkl-wave equations and applications

In this paper, we introduce a class of nonlinear wave equations associated with the generalized Dunkl-Laplace operator, we study local and global well-posedness. Next we establish the linearization of bounded energy solutions in the spirit of Gérard (1996) [9]. The proof uses Strichartz type inequalities and the energy estimate.     

Scattering Poles near the Real Axis for Two Strictly Convex Obstacles

To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator M along the trapped ray between the two obstacles. Using this method Gérard (cf. [7]) obtained complete asymptotic expansions for the poles in a strip Im z ≤ c as Re z tends to infinity. He established the existence of parallel rows of poles close to Assuming that the boundaries are analytic and the eigenvalues of Poincaré map are non-resonant we use the Birkhoff normal form for M to improve his result and to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis. Submitted: May 17, 2006. Accepted: September 19, 2006.     

Unconditional global well-posedness for the 3D Gross-Pitaevskii equation for data without finite energy

The Cauchy problem for the Gross-Pitaevskii equation in three space dimensions is shown to have an unconditionally unique global solution for data of the form 1 + H ^s for 5/6 < s < 1, which do not have necessarily finite energy. The proof uses the I-method which is complicated by the fact that no L ²-conservation law holds. This shows that earlier results of Bethuel-Saut for data of the form 1 + H ¹ and Gérard for finite energy data remain true for this class of rough data.     

A note on Prandtl boundary layers

This note concerns nonlinear ill‐posedness of the Prandtl equation and an invalidity of asymptotic boundary layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear illposedness results established by Gérard‐Varet and Dormy and an analysis by Guo and Tice. We show that the asymptotic boundary layer expansion is not valid for nonmonotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well‐posedness and prove that the nonlinear Prandtl equation is not well‐posed in this sense near nonstationary and nonmonotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well‐posed. © 2011 Wiley Periodicals, Inc.     

Semiclassical resonances associated with a periodic orbit

We consider resonances for a h-pseudo-differential operator H(x, hD _x; h) induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincarémap, and look for so-called semi-excited resonances with imaginary part of magnitude ?h log h, or h ^δ, with 0 < δ < 1.     

Asymptotics for the Low-Lying Eigenstates of the Schrödinger Operator with Magnetic Field near Corners

The Neumann realization for the Schr?dinger operator with magnetic field is considered in a bounded two-dimensional domain with corners. This operator is associated with a small semi-classical parameter h or, equivalently, with a large magnetic field.We investigate the behavior of its eigenpairs as h tends to zero, like in a semi-classical limit. We prove, in the situation where the domain is a polygon and the magnetic field is constant, that the lowest eigenvalues are exponentially close to those of model problems associated with the corners. We approximate the corresponding eigenvectors by linear combinations of functions concentrated in corners at the scale If the domain has curved sides and the magnetic field is smoothly varying, we exhibit a full asymptotics for eigenpairs in powers of Communicated by Christian Gérard Submitted: October 13, 2005 Accepted: December 19, 2005     

Combinatorial Tangent Space and Rational Smoothness of Schubert Varieties

Abstract

Following Contou-Carrère (Contou-Carrère,C. (1983). Géométrie des Groupes Semi-Simples,Résolutions équivariantes et Lieu Singulier de Leurs Variétés de Schubert. Thèse d’état,Université Montpellier II (published partly as,Le Lieu singulier des variétés de Schubert (1988). Adv. Math.,71:186–221)),we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the Tits building associated to the situation. Using Carrell and Peterson's characterization (Carrell,J. B. (1994). The Bruhat graph of a Coxeter group,a conjecture of Deodhar,and rational smoothness of Schubert varieties. Proc. Symp. in Pure Math. 56(Part I):53–61),we prove that rational smoothness of a Schubert variety can be expressed in terms of a subspace of the Zariski tangent space called,the combinatorial tangent space.     

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.     

Stabilization and control for the subcritical semilinear wave equation

In this paper, we analyze the exponential decay property of solutions of the semilinear wave equation in with a damping term which is effective on the exterior of a ball. Under suitable and natural assumptions on the nonlinearity we prove that the exponential decay holds locally uniformly for finite energy solutions provided the nonlinearity is subcritical at infinity. Subcriticality means, roughly speaking, that the nonlinearity grows at infinity at most as a power p<5. The method of proof combines classical energy estimates for the linear wave equation allowing to estimate the total energy of solutions in terms of the energy localized in the exterior of a ball, Strichartz's estimates and results by P. Gérard on microlocal defect measures and linearizable sequences. We also give an application to the stabilization and controllability of the semilinear wave equation in a bounded domain under the same growth condition on the nonlinearity but provided the nonlinearity has been cut-off away from the boundary.