Publications

2016

• On the controllability of the 2-D Vlasov-Stokes system
  
  • Moyano Iván
  , 2016. In this paper we prove an exact controllability result for the Vlasov-Stokes system in the two-dimensional torus with small data by means of an internal control. We show that one can steer, in arbitrarily small time, any initial datum of class C 1 satisfying a smallness condition in certain weighted spaces to any final state satisfying the same conditions. The proof of the main result is achieved thanks to the return method and a Leray-Schauder fixed-point argument.
• Maximal sets with no solution to x + y = 3z
  
  • Plagne Alain
  • de Roton Anne
  Combinatorica, Springer Verlag, 2016, 36 (2), pp.229-248. In this paper, we are interested in a generalization of the notion of sum-free sets. We address a conjecture first made in the 90s by Chung and Goldwasser. Recently, after some computer checks, this conjecture was formulated again by Matolcsi and Ruzsa, who made a first significant step towards it. Here, we prove the full conjecture by giving an optimal upper bound for the Lebesgue measure of a 3-sum-free subset A of [0, 1], that is, a set containing no solution to the equation x + y = 3z where x, y and z are restricted to belong to A. We then address the inverse problem and characterize precisely, among all sets with that property, those attaining the maximal possible measure. (10.1007/s00493-015-3100-4)
  DOI : 10.1007/s00493-015-3100-4
• Analytic continuation on Shimura varieties with μ-ordinary locus
  
  • Bijakowski Stéphane
  Algebra & Number Theory, Mathematical Sciences Publishers, 2016, 10 (4), pp.843 - 885. (10.2140/ant.2016.10.843)
  DOI : 10.2140/ant.2016.10.843
• Actions of the group of homeomorphisms of the circle on surfaces
  
  • Militon Emmanuel
  Fundamenta Mathematicae, Instytut Matematyczny, Polskiej Akademii Nauk,, 2016, 233 (2), pp.143-172. In this article, we describe all the group morphisms from the group of orientation-preserving homeomorphisms of the circle to the group of homeomorphisms of the annulus or of the torus.
• From the Dynamics of Charged Particles to a Regularized Vlasov-Maxwell System
  
  • Golse François
  Frontiers in Science and Engineering (international journal), 2016, 6 (1), pp.27-40. The present paper establishes the mean-field limit of the Abraham system for a system of identical charged particles subject to the electromagnetic interaction. This mean-field limit leads to a regularized variant of the relativistic Vlasov-Maxwell system. The result obtained here is similar to the one obtained in [F. Golse, Commun. Math. Phys. 310 (2012), 789-816], but is obtained by a purely Eulerian description of the particle dynamics.
• Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II
  
  • Huneau Cécile
  Asymptotic Analysis, IOS Press, 2016. We solve the Einstein constraint equations for a 3 + 1 dimensional vacuum space-time with a space-like translational Killing field in the asymptotically flat case. The presence of a space-like translational Killing field allows for a reduction of the 3 + 1 dimensional problem to a 2 + 1 dimensional one. The aim of this paper is to go further in the asymptotic expansion of the solutions than in [14]. In particular the expansion we construct involves quantities which are the 2-dimensional equivalent of the global charges.
• Classicité de formes modulaires de Hilbert
  
  • Bijakowski Stéphane
  Asterisque, Société Mathématique de France, 2016.
• On the Dynamics of Large Particle Systems in the Mean Field Limit
  
  • Golse François
  , 2016, 3, pp.1-144. This course explains how the usual mean field evolution partial differential equations (PDEs) in Statistical Physics - such as the Vlasov-Poisson system, the vorticity formulation of the two-dimensional Euler equation for incompressible fluids, or the time-dependent Hartree equation in quantum mechanics - can be rigorously derived from first principles, i.e. from the fundamental microscopic equations that govern the evolution of large, interacting particle systems. The emphasis is put on the mathematical methods used in these derivations, such as Dobrushin's stability estimate in the Monge-Kantorovich distance for the empirical measures built on the solution of the N-particle motion equations in classical mechanics, or the theory of BBGKY hierarchies in the case of classical as well as quantum problems. We explain in detail how these different approaches are related; in particular we insist on the notion of chaotic sequences and on the propagation of chaos in the BBGKY hierarchy as the number of particles tends to infinity. (10.1007/978-3-319-26883-5_1)
  DOI : 10.1007/978-3-319-26883-5_1
• Nonlinear Instability of Vlasov--Maxwell Systems in the Classical and Quasineutral Limits
  
  • Han-Kwan Daniel
  • Nguyen Toan
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2016, 48 (5), pp.3444 - 3466. (10.1137/15M1028765)
  DOI : 10.1137/15M1028765
• Pencils of quadrics and Gromov-Witten-Welschinger invariants of $\mathbb C P^3$
  
  • Brugallé Erwan
  • Georgieva Penka
  Mathematische Annalen, Springer Verlag, 2016, 365 (1-2), pp.363-380. We establish a formula for the Gromov-Witten-Welschinger invariants of $\mathbb CP^3$ with mixed real and conjugate point constraints. The method is based on a suggestion by J. Koll\'ar that, considering pencils of quadrics, some real and complex enumerative invariants of $\mathbb CP^3$ could be computed in terms of enumerative invariants of $\mathbb CP^1\times\mathbb CP^1$ and of elliptic curves.
• Recovering the Hamiltonian from spectral data
  
  • Heriveaux Cyrille
  • Paul Thierry
  Transactions of the American Mathematical Society, American Mathematical Society, 2016, 368 (7239-7279). We show that the contributions to the Gutzwiller formula with observable associated to the iterates of a given elliptic nondegenerate periodic trajectory $\gamma$ and to certain families of observables localized near $\gamma$ determine the quantum Hamiltonian in a formal neighborhood of the trajectory $\gamma$, that is the full Taylor expansion of its total symbol near $\gamma$. We also treat the ''bottom of a well" case both for general and Schrödinger operators.
• Additive properties of sequences of pseudo $s$-th powers
  
  • Cilleruelo Javier
  • Deshouillers Jean-Marc
  • Lambert Victor
  • Plagne Alain
  Mathematische Zeitschrift, Springer, 2016, 284, pp.175-193. In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erdös and Rényi in 1960. In 1975, Goguel proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order s + x for any x > 0. We then study the s-fold sumset sA = A + ... + A (s times) and in particular the minimal size of an additive complement, that is a set B such that sA + B contains all large enough integers. With respect to this problem, we prove quite precise theorems which are tantamount to asserting that a threshold phenomenon occurs. (10.1007/s00209-016-1651-8)
  DOI : 10.1007/s00209-016-1651-8
• Incompressible immiscible multiphase flows in porous media: a variational approach
  
  • Cancès Clément
  • Gallouët Thomas
  • Monsaingeon Leonard
  , 2016. We describe the competitive motion of (N + 1) incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization schemè a la [R. Jordan, D. Kinder-lehrer & F. Otto, SIAM J. Math. Anal, 29(1):1–17, 1998]. This allow to obtain a new existence result for a physically well-established system of PDEs consisting in the Darcy-Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure.
• Quantum singular complete integrability
  
  • Paul Thierry
  • Stolovitch Laurent
  Journal of Functional Analysis, Elsevier, 2016, 271, pp.1377-1443. We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schrödinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diagonalised by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uniform in the Planck constant near $\hbar= 0$. The unperturbed frequencies satisfy a small divisors condition %(Bruno type condition (including the Diophantine case) and we explicitly estimate how this condition can be released when the family tends to the unperturbed one.
• Global Solutions of the Boltzmann Equation over R D near Global Maxwellians with Small Mass
  
  • Bardos Claude
  • Gamba Irene
  • Golse François
  • Levermore C.David
  Communications in Mathematical Physics, Springer Verlag, 2016, 346 (2), pp.435-467. We study the dynamics defined by the Boltzmann equation set in the Euclidean space RD in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory. (10.1007/s00220-016-2687-7)
  DOI : 10.1007/s00220-016-2687-7
• Constraint equations for 3 + 1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case
  
  • Huneau Cécile
  Annales Henri Poincaré, Springer Verlag, 2016. We solve the Einstein constraint equations for a 3 + 1 dimensional vacuum space-time with a space-like translational Killing field. The presence of a space-like transla-tional Killing field allows for a reduction of the 3 + 1 dimensional problem to a 2 + 1 dimensional one. Vacuum Einstein equations with a space-like translational Killing field have been studied by Choquet-Bruhat and Moncrief in the compact case. In the case where an additional rotational symmetry is added, the problem has a long history (see [3], [1], [4]). In this paper we consider the asymptotically flat case. This corresponds to solving a nonlinear elliptic system on R 2. The main difficulty in that case is due to the delicate inversion of the Laplacian on R 2. In particular, we have to work in the non-constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.
• CLASSICITÉ DE FORMES MODULAIRES SURCONVERGENTES
  
  • Bijakowski Stéphane
  • Pilloni Vincent
  • Stroh Benoit
  Annals of Mathematics, Princeton University, Department of Mathematics, 2016, 183 (3), pp.975-1014. Nous généralisons le critère de classicité de formes modulaires surconvergentes sur les courbes modulaires dû à Coleman à certaines variétés de Shimura PEL de type (A) et (C) associées à des groupes réductifs non ramifiés sur Qp. Notre démonstration s’inspire de la méthode de prolongement analytique de Buzzard et Kassaei. (10.4007/annals.2016.183.3.5)
  DOI : 10.4007/annals.2016.183.3.5
• The Dynamical Manin-Mumford Problem for Plane Polynomial Automorphisms
  
  • Dujardin Romain
  • Favre Charles
  Journal of the European Mathematical Society, European Mathematical Society, 2016. Let f be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve C. We conjecture that this hap- pens if and only if f admits a time-reversal symmetry; in particular the Jacobian Jac(f) must be a root of unity. As a step towards this conjecture, we prove that its Jacobian, together with all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed Jac(f) is a root of unity. We use these results to show in various cases that any two automor- phisms sharing an infinite set of periodic points must have a common it- erate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.
• Null-controllability of the Kolmogorov equation in the whole phase space
  
  • Le Rousseau Jérôme
  • Moyano Iván
  Journal of Differential Equations, Elsevier, 2016, 260, pp.3193-3233. We prove the null controllability, in arbitrary positive time, of the Kolmogorov equation ∂t + v · ∇x − ∆v with (x, v) ∈ R d × R d , with a control region of the form ω = ωx × ωv, where both ωx and ωv are open subsets of R d that are sufficiently spread out throughout the whole space R d. The proof is based on, on the one hand, a spectral inequality in R d with an observation on ωx, and, on the other hand, a Carleman-based observability inequality for a family of parabolic operators, ∂t − iv · ξ − ∆v, coupled with a knowledge of the decay rate of the free solutions of the Kolmogorov equation. (10.1016/j.jde.2015.09.062)
  DOI : 10.1016/j.jde.2015.09.062
• Actions of groups of homeomorphisms on one-manifolds
  
  • Militon Emmanuel
  Groups, Geometry, and Dynamics, European Mathematical Society, 2016, 10 (1). In this article, we describe all the group morphisms from the group of compactly-supported homeomorphisms isotopic to the identity of a manifold to the group of homeomorphisms of the real line or of the circle.
• Quasineutral limit for Vlasov-Poisson with Penrose stable data
  
  • Han-Kwan Daniel
  • Rousset Frédéric
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2016, 49 (6), pp.1445-1495. We study the quasineutral limit of a Vlasov-Poisson system that describes the dynamics of ions in a plasma. We handle data with Sobolev regularity under the sharp assumption that the profile of the initial data in the velocity variable satisfies a Penrose stability condition. As a by-product of our analysis, we obtain a well-posedness theory for the limit equation (which is a Vlasov equation with Dirac distribution as interaction kernel) for such data.
• On the Mean Field and Classical Limits of Quantum Mechanics
  
  • Golse François
  • Mouhot Clément
  • Paul Thierry
  Communications in Mathematical Physics, Springer Verlag, 2016, 343, pp.165-205. The main result in this paper is a new inequality bearing on solutions of the $N$-body linear Schrödinger equation and of the mean field Hartree equations. This inequality implies that the mean field limit of the quantum mechanics of $N$ identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of $C^{1,1}$ interaction potentials. The quantity measuring the approximation of the $N$-body quantum dynamics by its mean field limit is analogous to the Monge-Kantorovich (or Wasserstein) distance with exponent $2$. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13 (1979), 115-123]. Our approach of this problem is based on a direct analysis of the $N$-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization. (10.1007/s00220-015-2485-7)
  DOI : 10.1007/s00220-015-2485-7
• On the structure of subsets of an orderable group with some small doubling properties
  
  • Plagne Alain
  • Freiman G. A.
  • Herzog M.
  • Longobardi P.
  • Maj M.
  • Robinson D. J. S.
  • Stanchescu Y. V.
  • Freiman A.
  • Robinson D.J.S.
  Journal of Algebra, Elsevier, 2016, 445, pp.307 - 326. The aim of this paper is to present a complete description of the structure of subsets S of an orderable group G satisfying |S^2| = 3|S|-2 and <S> is non-abelian. (10.1016/j.jalgebra.2015.07.038)
  DOI : 10.1016/j.jalgebra.2015.07.038