Publications

2017

• A JKO SPLITTING SCHEME FOR KANTOROVICH-FISHER-RAO GRADIENT FLOWS
  
  • Gallouët Thomas
  • Monsaingeon Leonard
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2017, 49(2). In this article we set up a splitting variant of the JKO scheme in order to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric , recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge-Kantorovich distance, and then for the Hellinger/Fisher-Rao distance. Exploiting some inf-convolution structure of the metric we show convergence of the whole process for the standard class of energy functionals under suitable compactness assumptions, and investigate in details the case of internal energies. The interest is double: On the one hand we prove existence of weak solutions for a certain class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers.
• From conservative to dissipative systems through quadratic change of time, with application to the curve-shortening flow
  
  • Brenier Yann
  • Duan Xianglong
  , 2017. We provide several examples of dissipative systems that can be obtained from conservative ones through a simple, quadratic, change of time. A typical example is the curve-shortening flow in R^d, which is a particular case of mean-curvature flow with co-dimension higher than one (except in the case d=2). Through such a change of time, this flow can be formally derived from the conservative model of vibrating strings obtained from the Nambu-Goto action. Using the concept of ``relative entropy" (or ``modulated energy"), borrowed from the theory of hyperbolic systems of conservation laws, we introduce a notion of generalized solutions, that we call dissipative solutions, for the curve-shortening flow. For given initial conditions, the set of generalized solutions is convex, compact, if not empty. Smooth solutions to the curve-shortening flow are always unique in this setting.
• Limiting motion for the parabolic Ginzburg-Landau equation with infinite energy data
  
  • Côte Delphine
  • Côte Raphaël
  Communications in Mathematical Physics, Springer Verlag, 2017, 350 (2), pp.507-568. We study a class of solutions to the parabolic Ginzburg-Landau equation in dimension 2 or higher, with ill-prepared infinite energy initial data. We show that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. Then, we prove that in the plane, point vortices do not move in the original time scale. These results extend the work of Bethuel, Orlandi and Smets [8, 9] for infinite energy data; they allow to consider the point vortices on a lattice (in dimension 2), or filament vortices of infinite length (in dimension 3). (10.1007/s00220-016-2736-2)
  DOI : 10.1007/s00220-016-2736-2
• Sharp asymptotics for the minimal mass blow up solution of the critical gKdV equation
  
  • Combet Vianney
  • Martel Yvan
  Bulletin des Sciences Mathématiques, Elsevier, 2017, 141 (2), pp.20 - 103. (10.1016/j.bulsci.2017.01.001)
  DOI : 10.1016/j.bulsci.2017.01.001
• Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5
  
  • Jendrej Jacek
  Journal of Functional Analysis, Elsevier, 2017, 272 (3), pp.866-917. (10.1016/j.jfa.2016.10.019)
  DOI : 10.1016/j.jfa.2016.10.019
• DEGREES OF ITERATES OF RATIONAL MAPS ON NORMAL PROJECTIVE VARIETIES
  
  • Dang Nguyen-Bac
  , 2017. Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rational selfmap of X, recovering former results by Dinh, Sibony [DS05b], and by Truong [Tru16]. Precisely, we give a new proof of the submultiplicativity properties of these degrees and of its birational invariance. Our approach exploits intensively positivity properties in the space of numerical cycles of arbitrary codimension. In particular, we prove an algebraic version of an inequality first obtained by Xiao [Xia15] and Popovici [Pop16], which generalizes Siu’s inequality to algebraic cycles of arbitrary codimension. This allows us to show that the degree of a map is controlled up to a uniform constant by the norm of its action by pull-back on the space of numerical classes in X.
• Overconvergent modular forms, ramification and classicity
  
  • Bijakowski Stéphane
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2017, 67 (6), pp.2463 - 2518. (10.5802/aif.3140)
  DOI : 10.5802/aif.3140
• On the universal CH_0 group of cubic threefolds in positive characteristic
  
  • Mboro Rene
  Manuscripta mathematica, Springer Verlag, 2017. We adapt for algebraically closed fields k of characteristic greater than 2 two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface X of dimension 3 over ℂ, namely: the equivalence between Chow-theoretic and cohomological decompositions of the diagonal of those hypersurfaces and the fact that the algebraicity (with ℤ2-coefficients) of the minimal class θ4/4! of the intermediate jacobian J(X) of X implies the Chow-theoretic decomposition of the diagonal of X. Using the second result, the Tate conjecture for divisors on surfaces defined over finite fields predicts, via a theorem of Schoen, that every smooth cubic hypersurface of dimension 3 over the algebraic closure of a finite field of characteristic >2 admits a Chow-theoretic decomposition of the diagonal. (10.1007/s00229-016-0912-5)
  DOI : 10.1007/s00229-016-0912-5
• Collision of almost parallel vortex filaments
  
  • Banica Valeria
  • Faou Erwan
  • Miot Evelyne
  Communications on Pure and Applied Mathematics, Wiley, 2017, 70 (2), pp.378-405. We investigate the occurrence of collisions in the evolution of vortex filaments through a system introduced by Klein, Majda and Damodaran [KMD95] and Zakharov [Z88, Z99]. We first establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. The collision mechanism is based on the one of the self-similar solutions of the model, described in [BFM14]. In the second part of this paper we extend this construction to the case of an arbitrary number of filaments, with poly-gonial symmetry, that are perturbations of a configuration of parallel vortex filaments forming a polygon, with or without its center, rotating with constant angular velocity. (10.1002/cpa.21637)
  DOI : 10.1002/cpa.21637
• A DERIVATION OF THE VLASOV-NAVIER-STOKES MODEL FOR AEROSOL FLOWS FROM KINETIC THEORY
  
  • Bernard Etienne
  • Desvillettes Laurent
  • Golse François
  • Ricci Valeria
  Communications in Mathematical Sciences, International Press, 2017, 15 (6), pp.1703-1741. This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of the model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the collision kernels, we prove that the sequences of solutions to the multiphase Boltzmann system converge to distributional solutions to the Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit. Specifically, we assume (a) that the mass ratio of the gas molecules to the dust particles/droplets is small, (b) that the thermal speed of the dust particles/droplets is much smaller than that of the gas molecules and (c) that the mass density of the gas and of the dispersed phase are of the same order of magnitude. (10.4310/CMS.2017.v15.n6.a11)
  DOI : 10.4310/CMS.2017.v15.n6.a11
• The Grunwald problem and approximation properties for homogeneous spaces
  
  • Demarche Cyril
  • Lucchini Arteche Giancarlo
  • Neftin Danny
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2017, 67 (3), pp.1009-1033. (10.5802/aif.3104)
  DOI : 10.5802/aif.3104
• Wonderful compactifications of Bruhat-Tits buildings
  
  • Remy Bertrand
  • Thuillier Amaury
  • Werner Annette
  Épijournal de Géométrie Algébrique, EPIGA, 2017, 1. Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.
• The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial
  
  • Ardila Federico
  • Brugallé Erwan
  International Mathematics Research Notices, Oxford University Press (OUP), 2017 (2), pp.614-641. We define the double Gromov-Witten invariants of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their 1-dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory. (10.1093/imrn/rnv379)
  DOI : 10.1093/imrn/rnv379
• Additive bases in groups
  
  • Plagne Alain
  • Lambert Victor
  • Le Thai Hoang
  Israel Journal of Mathematics, Springer, 2017, 217, pp.383--411. In this paper, we study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions $X$, $S$ and $E$ (defined in the case of the integers) and obtain bounds on them. Our estimates on the functions $S_G$ and $E_G$ are valid for general abelian groups $G$ while in the case of $X_G$ we show that distinct types of behaviours may occur depending on the group $G$.
• Normalization in Lie algebras via mould calculus and applications
  
  • Paul Thierry
  • Sauzin David
  Regular and Chaotic Dynamics, MAIK Nauka/Interperiodica, 2017, 22 (6), pp.616--649. We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians. (10.1134/S1560354717060041)
  DOI : 10.1134/S1560354717060041
• Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs
  
  • Boucksom Sébastien
  • Hisamoto Tomoyuki
  • Jonsson Mattias
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2017, 67 (2), pp.743 - 841. (10.5802/aif.3096)
  DOI : 10.5802/aif.3096
• REMARKS ON THE CH_2 OF CUBIC HYPERSURFACES
  
  • Mboro Rene
  , 2017. This paper presents two approaches to reducing problems on 2-cycles on a smooth cubic hypersurface X over an algebraically closed field of characteristic = 2, to problems on 1-cycles on its variety of lines F (X). The first one relies on bitangent lines of X and Tsen-Lang theorem. It allows to prove that CH 2 (X) is generated, via the action of the universal P 1-bundle over F (X), by CH 1 (F (X)). When the characteristic of the base field is 0, we use that result to prove that if dim(X) ≥ 7, then CH 2 (X) is generated by classes of planes contained in X and if dim(X) ≥ 9, then CH 2 (X) Z. Similar results, with slightly weaker bounds, had already been obtained by Pan([27]). The second approach consists of an extension to subvarieties of X of higher dimension of an inversion formula developped by Shen ([30], [31]) in the case of curves of X. This inversion formula allows to lift torsion cycles in CH 2 (X) to torsion cycles in CH 1 (F (X)). For complex cubic 5-folds, it allows to prove that the birational invariant provided by the group CH 3 (X) tors,AJ of homologically trivial, torsion codimension 3 cycles annihilated by the Abel-Jacobi morphism is controlled by the group CH 1 (F (X)) tors,AJ which is a birational invariant of F (X), possibly always trivial for Fano varieties.
• p-divisible groups with Pappas-Rapoport condition and Hasse invariants
  
  • Bijakowski Stéphane
  • Hernandez Valentin
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2017, 4, pp.935 - 972. (10.5802/jep.60)
  DOI : 10.5802/jep.60
• Construction of two-bubble solutions for the energy-critical NLS
  
  • Jendrej Jacek
  Analysis & PDE, Mathematical Sciences Publishers, 2017, 10 (8), pp.1923-1959. (10.2140/apde.2017.10.1923)
  DOI : 10.2140/apde.2017.10.1923
• Relative Riemann–Hilbert correspondence in dimension one
  
  • Fernandes Teresa Monteiro
  • Sabbah Claude
  Portugaliae Mathematica, European Mathematical Society Publishing House, 2017, 74 (2), pp.149 - 159. (10.4171/PM/1997)
  DOI : 10.4171/PM/1997
• Small doubling in ordered groups: generators and structure
  
  • Plagne Alain
  • Freiman Gregory A.
  • Herzog Marcel
  • Longobardi Patrizia
  • Maj Mercede
  • Stanchescu Yonutz V.
  Groups, Geometry, and Dynamics, European Mathematical Society, 2017, 11 (2), pp.585-612. We prove several new results on the structure of the subgroup generated by a small doubling subset of an ordered group, abelian or not. We obtain precise results generalizing Freiman's 3k-3 and 3k-2 theorems in the integers and several further generalizations. (10.4171/GGD/409)
  DOI : 10.4171/GGD/409
• Nonexistence of small, odd breathers for a class of nonlinear wave equations
  
  • Kowalczyk Michał
  • Martel Yvan
  • Muñoz Claudio
  Letters in Mathematical Physics, Springer Verlag, 2017, 107 (5), pp.921-931. In this note, we show that for a large class of nonlinear wave equations with odd nonlinearities, any globally defined odd solution which is small in the energy space decays to 0 in the local energy norm. In particular, this result shows nonexistence of small, odd breathers for some classical nonlinear Klein Gordon equations, such as the sine-Gordon equation and $\phi ^4$ and $\phi ^6$ models. It also partially answers a question of Soffer and Weinstein (Invent Math 136(1): 9–74, p 19 1999) about nonexistence of breathers for the cubic NLKG in dimension one. (10.1007/s11005-016-0930-y)
  DOI : 10.1007/s11005-016-0930-y
• The quasineutral limit of the Vlasov–Poisson equation in Wasserstein metric
  
  • Han-Kwan Daniel
  • Iacobelli Mikaela
  Communications in Mathematical Sciences, International Press, 2017, 15 (2), pp.481 - 509. (10.4310/CMS.2017.v15.n2.a8)
  DOI : 10.4310/CMS.2017.v15.n2.a8
• Blow-up phenomena for gradient flows of discrete homogeneous functionals
  
  • Calvez Vincent
  • Gallouët Thomas O.
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2017. We investigate gradient flows of some homogeneous functionals in R^N , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong self-interaction, the functional possesses a cone of negative energy. It is immediate to see that solutions with negative energy at some time become singular in finite time, meaning that a subset of particles concentrate at a single point. Here, we establish that all solutions become singular in finite time for the class of functionals under consideration. The paper is completed with numerical simulations illustrating the striking non linear dynamics when initial data have positive energy. (10.1007/s00245-017-9443-z)
  DOI : 10.1007/s00245-017-9443-z
• Normalization in Banach scale Lie algebras via mould calculus and applications
  
  • Paul Thierry
  • Sauzin David
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2017, 37, pp.4461 - 4487. We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper. (10.48550/arXiv.1607.00780)
  DOI : 10.48550/arXiv.1607.00780