Publications

2015

• Global Stability and Local Bifurcations in a Two-Fluid Model for Tokamak Plasma
  
  • Zhelyazov D.
  • Han-Kwan D.
  • Rademacher J.
  SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2015, 14 (2), pp.730 - 763. (10.1137/130912384)
  DOI : 10.1137/130912384
• Asymptotic stability of the black soliton for the Gross-Pitaevskii equation
  
  • Gravejat Philippe
  • Smets Didier
  Proceedings of the London Mathematical Society, London Mathematical Society, 2015, 111 (2), pp.305-353. We introduce a new framework for the analysis of the stability of solitons for the one-dimensional Gross-Pitaevskii equation. In particular, we establish the asymptotic stability of the black soliton with zero speed. (10.1112/plms/pdv025)
  DOI : 10.1112/plms/pdv025
• Time-Averaging for Weakly Nonlinear CGL Equations with Arbitrary Potentials
  
  • Huang Guan
  • Kuksin Sergei
  • Maiocchi Alberto
  , 2015, 75, pp.323-349. Consider weakly nonlinear complex Ginzburg-Landau (CGL) equation of the form: u t C i. 4u C V.x/u/ D " u C "P.ru; u/; x 2 R d ; (*) under the periodic boundary conditions, where 0 and P is a smooth function. Let f 1 .x/; 2 .x/; : : : g be the L 2-basis formed by eigenfunctions of the operator 4 C V.x/. For a complex function u.x/, write it as u.x/ D P k 1 v k k .x/ and set I k .u/ D 1 2 jv k j 2. Then for any solution u.t; x/ of the linear equation. / "D0 we have I.u.t; // D const. In this work it is proved that if equation. / with a sufficiently smooth real potential V.x/ is well posed on time-intervals t " 1 , then for any its solution u " .t; x/, the limiting behavior of the curve I.u " .t; // on time intervals of order " 1 , as " ! 0, can be uniquely characterized by a solution of a certain well-posed effective equation: u t D " 4u C "F.u/; where F.u/ is a resonant averaging of the nonlinearity P.ru; u/. We also prove similar results for the stochastically perturbed equation, when a white in time and (10.1007/978-1-4939-2950-4_11)
  DOI : 10.1007/978-1-4939-2950-4_11
• Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations
  
  • Morancey Morgan
  • Nersesyan Vahagn
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2015, 103 (1). We consider a system of an arbitrary number of \textsc{1d} linear Schrödinger equations on a bounded interval with bilinear control. We prove global exact controllability in large time of these $N$ equations with a single control. This result is valid for an arbitrary potential with generic assumptions on the dipole moment of the considered particle. Thus, even in the case of a single particle, this result extends the available literature. The proof combines local exact controllability around finite sums of eigenstates, proved with Coron's return method, a global approximate controllability property, proved with Lyapunov strategy, and a compactness argument. (10.1016/j.matpur.2014.04.002)
  DOI : 10.1016/j.matpur.2014.04.002
• Level one algebraic cusp forms of classical groups of small rank
  
  • Chenevier Gaëtan
  • Renard David
  , 2015, 237 (1121), pp.128 pp.. We determine the number of level 1, self-dual, half-algebraic regular, cuspidal automorphic representations of GL(n) over Q of any given infinitesimal character, and essentially all n less than or equal to 8. For this, we compute the dimensions of the spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO(7), SO(8), SO(9) (and G_2) and determine Arthur's endoscopic partition of these spaces in all cases. We also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GL(n) with trivial infinitesimal character, and to vector valued Siegel modular forms of genus 3. A part of our results are conditional.
• On the nonexistence of pure multi-solitons for the quartic gKdV equation
  
  • Martel Yvan
  • Merle Frank
  International Mathematics Research Notices, Oxford University Press (OUP), 2015, 3, pp.688-739. (10.1093/imrn/rnt214)
  DOI : 10.1093/imrn/rnt214
• De Newton à Boltzmann et Einstein : validation des modèles cinétiques et de diffusion,
  
  • Golse François
  Asterisque, Société Mathématique de France, 2015, 367-368, pp.285-326. La théorie cinétique des gaz de Maxwell et Boltzmann s’est trouvée au cœur de controverses scientifiques majeures. L’incompatibilité supposée entre le caractère réversible des équations de la mécanique classique et l’augmentation de l’entropie, qui, dans le cadre de la théorie cinétique des gaz, est une propriété mathématique de l’équation de Boltzmann connue sous le nom de théorème H, était l’un des arguments couramment utilisés contre la validité de cette théorie. Il a fallu attendre environ un siècle pour que O. Lanford propose, en 1974, une stratégie de preuve permettant de démontrer que l’équation de Boltzmann décrit une certaine limite asymptotique des équations de Newton de la mécanique classique pour un système formé d’un très grand nombre N de particules sphériques identiques n’interagissant qu’au cours de collisions élastiques. Un travail récent de I. Gallagher, L. Saint-Raymond et B. Texier précise la preuve de Lanford et l’étend au cas où l’interaction entre particules est décrite par un potentiel à très courte portée. Un article ultérieur de T. Bodineau, I. Gallagher et L. Saint-Raymond étudie ensuite la dynamique d’une particule marquée parmi N dans la même limite asymptotique, établissant ainsi la validité de l’équation de Boltzmann linéaire sur un intervalle de temps dont la longueur tend vers l’infini avec N. En utilisant des résultats aujourd’hui classiques sur la théorie asymptotique de l’équation de Boltzmann linéaire, les mêmes auteurs démontrent que le processus stochastique connu sous le nom de mouvement brownien décrit une certaine limite de la dynamique déterministe de particules en interaction.
• Characterization of large energy solutions of the equivariant wave map problem: I
  
  • Côte Raphaël
  • Kenig Carlos
  • Lawrie Andrew
  • Schlag Wilhelm
  American Journal of Mathematics, Johns Hopkins University Press, 2015, 137 (1), pp.139-207. We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic projection. We deduce this result via the concentration compactness/rigidity method developed by the second author and Merle. In particular, we establish a classification of equivariant wave maps with trajectories that are pre-compact in the energy space up to the scaling symmetry of the equation. Indeed, a wave map of this type can only be either 0 or Q up to a rescaling. This gives a proof in the equivariant case of a refined version of the threshold conjecture adapted to the degree zero theory where the true threshold is 2E(Q), not E(Q). The aforementioned global existence and scattering statement can also be deduced by considering the work of Sterbenz and Tataru in the equivariant setting. For wave maps of topological degree one, we establish a classification of solutions blowing up in finite time with energies less than three times the energy of Q. Under this restriction on the energy, we show that a blow-up solution of degree one is essentially the sum of a rescaled Q plus a remainder term of topological degree zero of energy less than twice the energy of Q. This result reveals the universal character of the known blow-up constructions for degree one, 1-equivariant wave maps of Krieger, the fourth author, and Tataru as well as Raphael and Rodnianski.
• Dynamics of periodic Toda chains with a large number of particles
  
  • Bambusi Dario
  • Kappeler Thomas
  • Paul Thierry
  Journal of Differential Equations, Elsevier, 2015, 258 (12), pp.4103-4490. For periodic Toda chains with a large number $N$ of particles we consider states which are $N^{-2}-$close to the equilibrium and constructed by discretizing any given $C^2-$functions with mesh size $N^{-1}$. For such states we derive asymptotic expansions of the Toda frequencies $(\omega^N_n)_{0 < n < N}$ and the actions $(I^N_n)_{0 < n < N},$ both listed in the standard way, in powers of $N^{-1}$ as $N \to \infty$. %listed in accordance with the ordering of the frequencies at the equilibrium, %$(2 \sin \frac{n\pi } {N})_{0 < n < N}$. At the two edges $n \sim 1$ and $N -n \sim 1$, the expansions of the frequencies are computed up to order $N^{-3}$ with an error term of higher order. Specifically, the coefficients of the expansions of $\omega^N_n$ and $\omega^N_{N-n}$ at order $N^{-3}$ are given by a constant multiple of the n'th KdV frequencies $\omega^-_n$ and $\omega^+_n$ of two periodic potentials, $q_{-}$ respectively $q_+$, constructed in terms of the states considered. The frequencies $\omega^N_n$ for $n$ away from the edges are shown to be asymptotically close to the frequencies of the equilibrium. For the actions $(I^N_n)_{0 < n < N},$ asymptotics of a similar nature are derived.
• Blow up for the critical gKdV equation. II: Minimal mass dynamics
  
  • Martel Yvan
  • Merle Frank
  • Raphaël Pierre
  Journal of the European Mathematical Society, European Mathematical Society, 2015, 17 (8), pp.1855 - 1925. (10.4171/JEMS/547)
  DOI : 10.4171/JEMS/547
• Blow up for the critical gKdV equation III: exotic regimes
  
  • Martel Yvan
  • Merle Frank
  • Raphaël Pierre
  Annali della Scuola Normale Superiore di Pisa, 2015, XIV, pp.575-631. We consider the blow up problem in the energy space for the critical (gKdV) equation in the continuation of part I and part II. We know from part I that the unique and stable blow up rate for solutions close to the solitons with strong decay on the right is $1/t$. In this paper, we construct non-generic blow up regimes in the energy space by considering initial data with explicit slow decay on the right in space. We obtain finite time blow up solutions with speed $t^{-\nu}$ where $ \nu>11/13,$ as well as global in time growing up solutions with both exponential growth or power growth. These solutions can be taken with initial data arbitrarily close to the ground state solitary wave.
• Hamiltonian Evolution of Monokinetic Measures with Rough Momentum Profile
  
  • Bardos Claude
  • Golse François
  • Markowich Peter
  • Paul Thierry
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 217 (1), pp.71-111. Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp. (10.1007/s00205-014-0829-7)
  DOI : 10.1007/s00205-014-0829-7
• The Steady Boltzmann and Navier-Stokes Equations
  
  • Aoki Kazuo
  • Golse François
  • Kosuge Shingo
  Bulletin of the Institute of Mathematics, Academia Sinica (New Series), Institute of Mathematics, Academia sinica, 2015, 10 (2), pp.205-257. The paper discusses the similarities and the differences in the mathematical theories of the steady Boltzmann and incompressible Navier-Stokes equations posed in a bounded domain. First we discuss two different scaling limits in which solutions of the steady Boltzmann equation have an asymptotic behavior described by the steady Navier-Stokes Fourier system. Whether this system includes the viscous heating term depends on the ratio of the Froude number to the Mach number of the gas flow. While the steady Navier-Stokes equations with smooth divergence-free external force always have at least one smooth solutions, the Boltzmann equation with the same external force set in the torus, or in a bounded domain with specular reflection of gas molecules at the boundary may fail to have any solution, unless the force field is identically zero. Viscous heating seems to be of key importance in this situation. The nonexistence of any steady solution of the Boltzmann equation in this context seems related to the increase of temperature for the evolution problem, a phenomenon that we have established with the help of numerical simulations on the Boltzmann equation and the BGK model.
• Toeplitz operators in TQFT via skein theory
  
  • Marché Julien
  • Paul Thierry
  Transactions of the American Mathematical Society, American Mathematical Society, 2015, 367 (3669-3704). Topological quantum field theory associates to a punctured surface $\Sigma$, a level $r$ and colors $c$ in $\{1,\ldots,r-1\}$ at the marked points a finite dimensional hermitian space $V_r(\Sigma,c)$. Curves $\gamma$ on $\Sigma$ act as Hermitian operator $T_r^\gamma$ on these spaces. In the case of the punctured torus and the 4 times punctured sphere, we prove that the matrix elements of $T_r^\gamma$ have an asymptotic expansion in powers of $\frac{1}{r}$ and we identify the two first terms using trace functions on representation spaces of the surface in $\su$. We conjecture a formula for the general case. Then we show that the curve operators are Toeplitz operators on the sphere in the sense that $T_r^{\gamma}=\Pi_r f^\gamma_r\Pi_r$ where $\Pi_r$ is the Toeplitz projector and $f^\gamma_r$ is an explicit function on the sphere which is smooth away from the poles. Using this formula, we show that under some assumptions on the colors associated to the marked points, the sequence $T^\gamma_r$ is a Toeplitz operator in the usual sense with principal symbol equal to the trace function and with subleading term explicitly computed. We use this result and semi-classical analysis in order to compute the asymptotics of matrix elements of the representation of the mapping class group of $\Sigma$ on $V_r(\Sigma,c)$. We recover in this way the result of \cite{tw} on the asymptotics of the quantum 6j-symbols and treat the case of the punctured S-matrix. We conclude with some partial results when $\Sigma$ is a genus 2 surface without marked points. \end{abstract}
• Valuation spaces and multiplier ideals on singular varieties
  
  • Boucksom Sébastien
  • de Fernex Tommaso
  • Favre Charles
  • Urbinati Stefano
  , 2015.
• Stability in the energy space for chains of solitons of the Landau-Lifshitz equation
  
  • de Laire André
  • Gravejat Philippe
  Journal of Differential Equations, Elsevier, 2015, 258 (1), pp.1-80. We prove the orbital stability of sums of solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy, under the assumptions that the (non-zero) speeds of the solitons are different, and that their initial positions are sufficiently separated and ordered according to their speeds. (10.1016/j.jde.2014.09.003)
  DOI : 10.1016/j.jde.2014.09.003
• Characterization of large energy solutions of the equivariant wave map problem: II
  
  • Côte Raphaël
  • Kenig Carlos
  • Lawrie Andrew
  • Schlag Wilhelm
  American Journal of Mathematics, Johns Hopkins University Press, 2015, 137 (1), pp.209-250. We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere of finite energy. We establish a classification of all degree 1 global solutions whose energies are less than three times the energy of the harmonic map Q. In particular, for each global energy solution of topological degree 1, we show that the solution asymptotically decouples into a rescaled harmonic map plus a radiation term. Together with our companion article, where we consider the case of finite-time blow up, this gives a characterization of all 1-equivariant, degree 1 wave maps in the energy regime [E(Q), 3E(Q)). (10.1353/ajm.2015.0003)
  DOI : 10.1353/ajm.2015.0003
• The local Laplace transform of an elementary irregular meromorphic connection
  
  • Hien Marco
  • Sabbah Claude
  Rendiconti del Seminario Matematico della Università di Padova, University of Padua / European Mathematical Society, 2015, 134, pp.133 - 196. (10.4171/RSMUP/134-4)
  DOI : 10.4171/RSMUP/134-4
• Valuation spaces and multiplier ideals on singular varieties. Recent advances in algebraic geometry, 29–51, London Math. Soc. Lecture Note Ser., 417, Cambridge Univ. Press, Cambridge, 2015
  
  • Boucksom Sébastien
  • de Fernex Tommaso
  • Favre Charles
  • Urbinati Stefano
  , 2015.
• Brief introduction to tropical geometry
  
  • Brugallé Erwan
  • Itenberg Ilia
  • Mikhalkin Grigory
  • Shaw Kristin
  Proceedings of the Gökova Geometry-Topology conference, International press, 2015, pp.1-75. The paper consists of lecture notes for a mini-course given by the authors at the G\"okova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.
• Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium
  
  • Han-Kwan Daniel
  • Léautaud Matthieu
  Annals of PDE, Springer, 2015. This work is devoted to the analysis of the linear Boltzmann equation in a bounded domain, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity. We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We handle several cases of phase spaces, including those associated to specular reflection in a bounded domain, or to a compact Riemannian manifold. (10.1007/s40818-015-0003-z)
  DOI : 10.1007/s40818-015-0003-z
• A Gradient Flow Approach to Quantization of Measures
  
  • Caglioti Emanuele
  • Golse François
  • Iacobelli Mikaela
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2015, 25 (10), pp.1845-1885. In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$ , we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics. (10.1142/S0218202515500475)
  DOI : 10.1142/S0218202515500475
• Stabilité des solitons de l'équation de Landau-Lifshitz à anisotropie planaire
  
  • de Laire André
  • Gravejat Philippe
  Séminaire Laurent Schwartz - EDP et applications, Centre de mathématiques Laurent Schwartz, 2015. Cet exposé présente plusieurs résultats récents quant à la stabilité des solitons sombres de l'équation de Landau-Lifshitz à anisotropie planaire, en particulier, quant à la stabilité orbitale des trains (bien préparés) de solitons gris et à la stabilité asymptotique de ces mêmes solitons.
• IMPACT OF A SAHARAN DUST OUTBREAK ON PM 10 GROUND LEVELS IN SOUTHEASTERN FRANCE
  
  • Michelot Nicolas
  • Endlicher Wilfried
  • Carrega Pierre
  • Martin Nicolas
  • Favez Olivier
  • Langner Marcel
  Climatologie, Association internationale de climatologie, 2015. Southeastern France is often subject to thermal breezes and inversions that are partly responsible for the dispersion behavior of air pollutants in this region. Generally, the coastal urban zone is the main contributor to PM 10 emissions. However, a southerly wind, commonly known as Sirocco, occasionally generates dust advections from the Sahara desert, resulting in poor air quality in the study area. This work demonstrates the quick rise of PM 10 levels on the French coastline under the influence of such a weather outbreak. Measurements were performed during a Saharan dust episode which occurred end of April 2013 and caused the tripling of PM 10 daily averages at the regional scale in about 24 hours. In Vence, located in the Alpes-Maritimes department, the highest daily average was 7 times greater during the peak than before the dust outbreak. In Venaco (Corsica) off-line chemical characterizations for filter samples show that about 50% of the PM 10 mass was composed of terrigenous dust, which confirms that they played a central role in the degradation of air quality and in the exceeding of the EU daily limit value of 50 µg/m 3 at a regional scale. Résumé : Impact d'une advection de poussières sahariennes sur les niveaux de PM 10 dans le Sud-Est de la France
• Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation
  
  • Bethuel Fabrice
  • Gravejat Philippe
  • Smets Didier
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2015, 48 (6), pp.1327-1381. We pursue our work on the dynamical stability of dark solitons for the one-dimensional Gross-Pitaevskii equation. In this paper, we prove their asymptotic stability under small perturbations in the energy space. In particular, our results do not require smallness in some weighted spaces or a priori spectral assumptions. Our strategy is reminiscent of the one used by Martel and Merle in various works regarding generalized Korteweg-de Vries equations. The important feature of our contribution is related to the fact that while Korteweg-de Vries equations possess unidirectional dispersion, Schrödinger equations do not.