Publications

2015

• Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting
  
  • Tristani Isabelle
  , 2015. In this paper, we consider the spatially inhomogeneous diffusively driven inelastic Boltzmann equation in different cases: the restitution coefficient can be constant or can depend on the impact velocity (which is a more physically relevant case), including in particular the case of viscoelastic hard spheres. In the weak thermalization regime, i.e. when the diffusion parameter is sufficiently small, we prove existence of global solutions considering the close-to-equilibrium regime as well as the weakly inhomogeneous regime in the case of a constant restitution coefficient. It is the very first existence theorem of global solution in an inelastic ``collision regime'' (that is excluding \cite{AR} where an existence theorem is proven in a near to the vacuum regime). We also study the long-time behavior of these solutions and prove a convergence to equilibrium with an exponential rate. The basis of the proof is the study of the linearized equation. We obtain a new result on it, we prove existence of a spectral gap in weighted (stretched exponential and polynomial) Sobolev spaces and a result of exponential stability for the semigroup generated by the linearized operator. To do that, we develop a perturbative argument around the spatially inhomogeneous equation for elastic hard spheres and we take advantage of the recent paper \cite{GMM} where this equation has been considered. We then link the linearized theory with the nonlinear one in order to handle the full non-linear problem thanks to new bilinear estimates on the collision operator that we establish. As far as the case of a constant coefficient is concerned, the present paper largely improves similar results obtained in \cite{MM2} in a spatially homogeneous framework. Concerning the case of a non-constant coefficient, this kind of results is new and we use results on steady states of the linearized equation from \cite{AL3}.
• On the Classical Limit of the Schrödinger Equation
  
  • Bardos Claude
  • Golse François
  • Markowich Peter
  • Paul Thierry
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (12), pp.5689-5709. This paper provides an elementary proof of the classical limit of the Schrödinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schrödinger type equations [A. Laptev, I. Sigal, Review of Math. Phys. 12 (2000), 749-766]. We also explain in detail how the phase shifts across caustics obtained when using the Laptev-Sigal parametrix are related to the Maslov index. (10.3934/dcds.2015.35.5689)
  DOI : 10.3934/dcds.2015.35.5689
• Asymptotic stability in the energy space for dark solitons of the Landau-Lifshitz equation
  
  • Bahri Yakine
  , 2015. We prove the asymptotic stability in the energy space of non-zero speed solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy ∂ t m + m × (∂ xx m − m 3 e 3) = 0 for a map m = (m 1 , m 2 , m 3) : R × R → S 2 , where e 3 = (0, 0, 1). More precisely, we show that any solution corresponding to an initial datum close to a soliton with non-zero speed, is weakly convergent in the energy space as time goes to infinity, to a soliton with a possible different non-zero speed, up to the invariances of the equation. Our analysis relies on the ideas developed by Martel and Merle for the generalized Korteweg-de Vries equations. We use the Madelung transform to study the problem in the hydrodynamical framework. In this framework, we rely on the orbital stability of the solitons and the weak continuity of the flow in order to construct a limit profile. We next derive a monotonicity formula for the momentum, which gives the localization of the limit profile. Its smoothness and exponential decay then follow from a smoothing result for the localized solutions of the Schrödinger equations. Finally, we prove a Liouville type theorem, which shows that only the solitons enjoy these properties in their neighbourhoods.
• From the N-Body Schrödinger Equation to the Vlasov Equation
  
  • Golse François
  , 2017, 209, pp.199-219. This paper describes a method for obtaining an estimate of the convergence rate for the joint mean-field and semiclassical limit of the N-particle Schrödinger equation leading to the Vlasov equation. The interaction force is assumed to be Lipschitz continuous. This is an account of a recent work in collaboration with T. Paul [Arch. Ration. Mech. Anal. 223 (2017), 57-94]. (10.1007/978-3-319-66839-0_10)
  DOI : 10.1007/978-3-319-66839-0_10
• On the irregular Hodge filtration of exponentially twisted mixed Hodge modules
  
  • Sabbah Claude
  • Yu Jeng-Daw
  Forum of Mathematics, Sigma, Cambridge University press, 2015, 3. (10.1017/fms.2015.8)
  DOI : 10.1017/fms.2015.8
• Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons
  
  • Côte Raphaël
  • Muñoz Claudio
  • Pilod Didier
  • Simpson Gideon
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 220 (2), pp.639-710. (10.1007/s00205-015-0939-x)
  DOI : 10.1007/s00205-015-0939-x
• On the Soliton resolution for equivariant wave maps to the sphere
  
  • Côte Raphaël
  Communications on Pure and Applied Mathematics, Wiley, 2015, 68 (11), pp.1946-2004. We consider finite energy corotationnal wave maps with target manifold $\m S^2$. We prove that for a sequence of times, they decompose as a sum of decoupled harmonic maps in the light cone, and a smooth wave map (in the blow case) or a linear scattering term (in the global case), up to an error which tends to 0 in the energy space.
• Distribution of postcritically finite polynomials
  
  • Favre Charles
  • Gauthier Thomas
  Israel Journal of Mathematics, Springer, 2015. We prove that Misiurewicz parameters with prescribed combinatorics and hyperbolic parameters with (d − 1) distinct attracting cycles with given multipliers are equidistributed with respect to the bifurcation measure in the moduli space of degree d complex polynomials. Our proof relies on Yuan's equidistribution results of points of small heights, and uses in a crucial way Epstein's transversality results. (10.1007/s11856-015-1218-0)
  DOI : 10.1007/s11856-015-1218-0
• Singular semipositive metrics in non-Archimedean geometry
  
  • Boucksom S.
  • Favre Charles
  • Jonsson M.
  Journal of Algebraic Geometry, American Mathematical Society, 2015. Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, endowed with an ample line bundle L. We introduce a general notion of (possibly singular) semipositive (or plurisubharmonic) metrics on L, and prove the analogue of the following two basic results in the complex case: the set of semipositive metrics is compact modulo constants, and each semipositive metric is a decreasing limit of smooth semipositive ones. In particular, for continuous metrics our definition agrees with the one by S.-W. Zhang. The proofs use multiplier ideals and the construction of suitable models of X over the valuation ring of K. (10.1090/jag/656)
  DOI : 10.1090/jag/656
• Presentations: from Kac-Moody groups to profinite and back
  
  • Capdeboscq Inna
  • Lubotzky Alexander
  • Rémy Bertrand
  , 2015. We go back and forth between, on the one hand, presentations of arithmetic and Kac-Moody groups and, on the other hand, presentations of profinite groups, deducing along the way new results on both.
• FLATNESS FOR A STRONGLY DEGENERATE 1-D PARABOLIC EQUATION
  
  • Moyano Iván
  , 2015. We consider the degenerate equation $$\partial_t f(t,x) - \partial_x \left( x^{\alpha} \partial_x f \right)(t,x) =0,$$ on the unit interval $x\in(0,1)$, in the strongly degenerate case $\alpha \in [1,2)$ with adapted boundary conditions at $x=0$ and boundary control at $x=1$. We use the flatness approach to construct explicit controls in some Gevrey classes steering the solution from any initial datum $f_0 \in L^2(0,1)$ to zero in any time $T>0$.
• Uniform semigroup spectral analysis of the discrete, fractional & classical Fokker-Planck equations
  
  • Mischler Stéphane
  • Tristani Isabelle
  , 2015. In this paper, we investigate the spectral analysis (from the point of view of semi-groups) of discrete, fractional and classical Fokker-Planck equations. Discrete and fractional Fokker-Planck equations converge in some sense to the classical one. As a consequence, we first deal with discrete and classical Fokker-Planck equations in a same framework, proving uniform spectral estimates using a perturbation argument and an enlargement argument. Then, we do a similar analysis for fractional and classical Fokker-Planck equations using an argument of enlargement of the space in which the semigroup decays. We also handle another class of discrete Fokker-Planck equations which converge to the fractional Fokker-Planck one, we are also able to treat these equations in a same framework from the spectral analysis viewpoint, still with a semigroup approach and thanks to a perturbative argument combined with an enlargement one. Let us emphasize here that we improve the perturbative argument introduced in [7] and developed in [11], relaxing the hypothesis of the theorem, enlarging thus the class of operators which fulfills the assumptions required to apply it.
• Quelques problèmes additifs : bases, pseudo-puissances et ensembles k-libres
  
  • Lambert Victor
  , 2015. Habituellement étudié dans N ou Z, on s'intéresse aux bases additives dans les groupes abéliens infinis. On obtient des résultats sur les fonctions E, X et S, qui caractérisent le comportement d'une base lorsqu'on lui enlève un élément. On étudie également l'ensemble A des pseudo-puissances s-ièmes. Celles-ci forment presque sûrement une base additive d'ordre s+1. On cherche à affiner ce résultat en déterminant des compléments additifs de sA de taille minimale, c'est-à-dire des ensembles B tels que sA+B contient presque sûrement tout entier suffisamment grand. Enfin, nous montrons quelle est la taille maximale d'un ensemble k-libre dans Z/nZ. La contrainte modulaire joue ici un rôle prépondérant. Les méthodes employées sont très différentes, selon la relation arithmétique entre k et n. En particulier, nous démontrons un résultat sur des arbres combinatoires, dans l'étude du cas général.
• Hölder Regularity for Hypoelliptic Kinetic Equations with Rough Diffusion Coefficients
  
  • Golse François
  • Vasseur Alexis F.
  , 2015. This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical DeGiorgi method via the averaging lemmas. The result can be seen as a Hölder regularity version of the classical averaging lemmas.
• Webs invariant by rational maps on surfaces
  
  • Favre Charles
  • Pereira Jorge Vitorio
  Rendiconti del Circolo Matematico di Palermo, Springer-Verlag Italia, 2015. We prove that under mild hypothesis rational maps on a surface preserving webs are of Lattès type. We classify endomorphisms of P^2 preserving webs, extending former results of Dabija-Jonsson. (10.1007/s12215-015-0207-9)
  DOI : 10.1007/s12215-015-0207-9
• Knot state asymptotics II, Witten conjecture and irreducible representations
  
  • Charles Laurent
  • Marche Julien
  Publications Mathématiques de L'IHÉS, IHÉS, 2015, 121 (1), pp.323-361. This article pursues the study of the knot state asymptotics in the large level limit initiated in "Knot sate Asymptotics I". As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2)-character manifold of the peripheral torus. In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the Chern-Simons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot. Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from q-differential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know. (10.1007/s10240-015-0069-x)
  DOI : 10.1007/s10240-015-0069-x
• Knot state asymptotics I, AJ Conjecture and abelian representations
  
  • Charles Laurent
  • Marche Julien
  Publications Mathématiques de L'IHÉS, IHÉS, 2015, 121 (1), pp.279-322. Consider the Chern-Simons topological quantum field theory with gauge group SU(2) and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the SU(2)-character variety of the peripheral torus, the knot state may be viewed as a section defined over this character variety. We first conjecture that the knot state concentrates in the large level limit to the character variety of the knot. This statement may be viewed as a real and smooth version of the AJ conjecture. Our second conjecture says that the knot state in the neighborhood of abelian representations is a Lagrangian state. Using microlocal techniques, we prove these conjectures for the figure eight and torus knots. The proof is based on q-difference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two conjectures. (10.1007/s10240-015-0068-y)
  DOI : 10.1007/s10240-015-0068-y
• Calculabilité de la cohomologie étale modulo $\ell$
  
  • Madore David
  • Orgogozo Fabrice
  Algebra & Number Theory, Mathematical Sciences Publishers, 2015, 9 (7), pp.1647-1739. Nous démontrons la calculabilité, au sens de Church-Turing, de la cohomologie étale modulo ℓ. Après avoir considéré le cas des schémas algébriques sur un corps algébriquement clos de caractéristique différente de ℓ, et notamment la fonctorialité, nous étudions les images directes de faisceaux constructictibles de Z/ℓ par un morphisme propre. Nous utilisons une variante pro-ℓ de la théorie des bons voisinages de Michael Artin. (10.2140/ant.2015.9.1647)
  DOI : 10.2140/ant.2015.9.1647
• Vanishing sequences and Okounkov bodies
  
  • Boucksom Sébastien
  • Küronya Alex
  • Maclean Catriona
  • Szemberg Tomasz
  Mathematische Annalen, Springer Verlag, 2015, 361 (3-4), pp.811-834. (10.1007/s00208-014-1081-z)
  DOI : 10.1007/s00208-014-1081-z
• A DOUBLE LARGE DEVIATION PRINCIPLE FOR MONGE-AMPERE GRAVITATION
  
  • Brenier Yann
  , 2015. Monge-Ampere gravitation is a nonlinear modification of classical Newtonian gravitation, when the Monge-Ampere equation substitutes for the Poisson equation. We establish, through two applications of the large deviation principle, that the MA gravitation for a finite number of particles can be reduced, through a double application of the large deviation principle, to the simplest possible stochastic model: a collection of independent Brownian motions with vanishing noise.
• FREE BOUNDARY MINIMAL SURFACES IN THE UNIT 3-BALL
  
  • Folha Abigail
  • Pacard Frank
  • Zolotareva Tatiana
  , 2015. In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $\Sigma_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $ n \geq 3$. For large $n$, we give an independent construction of $\Sigma_n$ and prove the existence of free boundary minimal surfaces $\tilde \Sigma_n$ in $B^3$ which have genus $1$ and $n$ boundary components. As $n$ tends to infinity, the sequence $\Sigma_n$ converges to a double copy of the unit horizontal (open) disk, uniformly on compacts of $B^3$ while the sequence $\tilde \Sigma_n$ converges to a double copy of the unit horizontal (open) punctured disk, uniformly on compacts of $B^3-\{0\}$.
• HIGHER CODIMENSION ISOPERIMETRIC PROBLEMS
  
  • Mazzeo Rafe
  • Pacard Frank
  • Zolotareva Tatiana
  , 2015. We consider a variational problem for submanifolds Q ⊂ M with nonempty boundary ∂Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective when dim Q < dim M . We then construct small nearly-spherical solutions of this higher codimension CMC prob-lem; these concentrate near the critical points of a certain curvature function.
• Coherence between geodetic and seismic deformation in a context of slow tectonic activity (SW Alps, France)
  
  • Walpersdorf A.
  • Sue Christian
  • Baize S.
  • Cotte N.
  • Bascou P.
  • Beauval C.
  • Collard Philippe
  • Daniel G.
  • Dyer Hellen
  • Grasso J.-R.
  • Hautecoeur Olivier
  • Helmstetter A.
  • Hok S.
  • Langlais M.
  • Menard Gabrielle
  • Mousavi Z.
  • Ponton F.
  • Rizza Magali
  • Rolland L.
  • Souami D.
  • Thirard L.
  • Vaudey P.
  • Voisin C.
  • Martinod J.
  Journal of Geodynamics, Elsevier, 2015, pp.20. A dense, local network of 30 geodetic markers covering a 50 × 60 km2 area in the southwestern European Alps (Briançon region) has been temporarily surveyed in 1996, 2006 and 2011 by GPS. The aim is to measure the current deformation in this seismically active area. The study zone is characterized by a majority of extensional and dextral focal mechanisms, along north–south to N160 oriented faults. The combined analysis of the three measurement campaigns over 15 years and up to 16 years of permanent GPS data from the French RENAG network now enables to assess horizontal velocities below 1 mm/year within the local network. The long observation interval and the redundancy of the dense campaign network measurement help to constrain a significant local deformation pattern in the Briançon region, yielding an average E–W extension of 16 ± 11 nanostrain/year. We compare the geodetic deformation field to the seismic deformation rate cumulated over 37 years, and obtain good coherencies both in amplitude and direction. Moreover, the horizontal deformation localized in the Briançon region represents a major part of the Adriatic-European relative plate motion. However, the average uplift of the network in an extensional setting needs the presence of buoyancy forces in addition to plate tectonics. (10.1016/j.jog.2015.02.001)
  DOI : 10.1016/j.jog.2015.02.001
• Countability properties of some Berkovich spaces
  
  • Favre Charles
  , 2015. We prove that any compact Berkovich space over the field of Laurent series over an arbitrary field is angelic. In particular, is it sequentially compact.
• Measuring a Cherenkov ring in the radio emission from air showers at 110–190MHz with LOFAR
  
  • Nelles A.
  • Schellart P.
  • Buitink S.
  • Corstanje A.
  • de Vries K.D.
  • Enriquez J.E.
  • Falcke H.
  • Frieswijk W.
  • Hörandel J.R.
  • Scholten O.
  • ter Veen S.
  • Thoudam S.
  • van den Akker M.
  • Anderson J.
  • Asgekar A.
  • Bell M.E.
  • Bentum M.J.
  • Bernardi G.
  • Best P.
  • Bregman J.
  • Breitling F.
  • Broderick J.
  • Brouw W.N.
  • Brüggen M.
  • Butcher H.R.
  • Ciardi B.
  • Deller A.
  • Duscha S.
  • Eislöffel J.
  • Fallows R.A.
  • Garrett M.A.
  • Gunst A.W.
  • Hassall T.E.
  • Heald G.
  • Horneffer A.
  • Iacobelli M.
  • Juette E.
  • Karastergiou A.
  • Kondratiev V.I.
  • Kramer M.
  • Kuniyoshi M.
  • Kuper G.
  • Maat P.
  • Mann G.
  • Mevius M.
  • Norden M.J.
  • Paas H.
  • Pandey-Pommier M.
  • Pietka G.
  • Pizzo R.
  • Polatidis A.G.
  • Reich W.
  • Röttgering H.
  • Scaife A.M.M.
  • Schwarz D.
  • Smirnov O.
  • Stappers B.W.
  • Steinmetz M.
  • Stewart A.
  • Tagger Michel
  • Tang Y.
  • Tasse C.
  • Vermeulen R.
  • Vocks C.
  • van Weeren R.J.
  • Wijnholds S.J.
  • Wucknitz O.
  • Yatawatta S.
  • Zarka P.
  Astroparticle Physics, Elsevier, 2015, 65, pp.11–21. Measuring radio emission from air showers offers a novel way to determine properties of the primary cosmic rays such as their mass and energy. Theory predicts that relativistic time compression effects lead to a ring of amplified emission which starts to dominate the emission pattern for frequencies above ∼100∼100 MHz. In this article we present the first detailed measurements of this structure. Ring structures in the radio emission of air showers are measured with the LOFAR radio telescope in the frequency range of 110–190 MHz. These data are well described by CoREAS simulations. They clearly confirm the importance of including the index of refraction of air as a function of height. Furthermore, the presence of the Cherenkov ring offers the possibility for a geometrical measurement of the depth of shower maximum, which in turn depends on the mass of the primary particle. (10.1016/j.astropartphys.2014.11.006)
  DOI : 10.1016/j.astropartphys.2014.11.006