Publications

2018

• A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations
  
  • Gallouët Thomas O
  • Mérigot Quentin
  Foundations of Computational Mathematics, Springer Verlag, 2018, 18, pp.835–865. We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold's interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D. (10.1007/s10208-017-9355-y)
  DOI : 10.1007/s10208-017-9355-y
• WAVE PACKETS AND THE QUADRATIC MONGE-KANTOROVICH DISTANCE IN QUANTUM MECHANICS
  
  • Golse François
  • Paul Thierry
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2018, 356, pp.177-197. In this paper, we extend the upper and lower bounds for the " pseudo-distance " on quantum densities analogous to the quadratic Monge-Kantorovich(-Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016) 165–205] to positive quantizations defined in terms of the family of phase space translates of a density operator, not necessarily of rank one as in the case of the Töplitz quantization. As a corollary , we prove that the uniform (for vanishing h) convergence rate for the mean-field limit of the N-particle Heisenberg equation holds for a much wider class of initial data than in [F. Golse, C. Mouhot, T. Paul, loc. cit.]. We also discuss the relevance of the pseudo-distance compared to the Schatten norms for the purpose of metrizing the set of quantum density operators in the semiclassical regime. (10.1016/j.crma.2017.12.007)
  DOI : 10.1016/j.crma.2017.12.007
• A DERIVATION OF THE VLASOV-STOKES SYSTEM FOR AEROSOL FLOWS FROM THE KINETIC THEORY OF BINARY GAS MIXTURES
  
  • Bernard Etienne
  • Desvillettes Laurent
  • Golse François
  • Ricci Valeria
  Kinetic and Related Models, AIMS, 2018, 11 (1), pp.43-69. In this short paper, we formally derive the thin spray equation for a steady Stokes gas (i.e. the equation consists in a coupling between a kinetic — Vlasov type — equation for the dispersed phase and a — steady — Stokes equation for the gas). Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard-Desvillettes-Golse-Ricci, arXiv:1608.00422[math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation. (10.3934/krm.2018003)
  DOI : 10.3934/krm.2018003
• Erratum to Hodge Theory of the Middle Convolution
  
  • Dettweiler Michael
  • Sabbah Claude
  Publications of the Research Institute for Mathematical Sciences, European Mathematical Society, 2018, 54 (2), pp.427-431. (10.4171/PRIMS/54-2-8)
  DOI : 10.4171/PRIMS/54-2-8
• Multi-travelling waves for the nonlinear klein-gordon equation
  
  • Côte Raphaël
  • Martel Yvan
  Transactions of the American Mathematical Society, American Mathematical Society, 2018. For the nonlinear Klein-Gordon equation in R 1+d , we prove the existence of multi-solitary waves made of any number N of decoupled bound states. This extends the work of Côte and Muñoz (Forum Math. Sigma 2 (2014)) which was restricted to ground states, as were most previous similar results for other nonlinear dispersive and wave models. (10.1090/tran/7303)
  DOI : 10.1090/tran/7303
• Irregular Hodge theory. Avec la collaboration de Jeng-Daw Yu
  
  • Sabbah Claude
  , 2018, 156. We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\exp\varphi$ for any meromorphic function $\varphi$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also show that those rigid irreducible holonomic D-modules on the complex projective line whose local formal monodromies have eigenvalues of absolute value one, are equipped with such an irregular Hodge filtration in a canonical way, up to a shift of the filtration. In a chapter written jointly with Jeng-Daw~Yu, we make explicit the case of irregular mixed Hodge structures, for which we prove in particular a Thom-Sebastiani formula.
• On the derivation of the Hartree equation in the mean field limit: Uniformity in the Planck constant
  
  • Golse François
  • Paul Thierry
  • Pulvirenti Mario
  Journal of Functional Analysis, Elsevier, 2018, 275 (7), pp.1603-1649. In this paper the Hartree equation is derived from the $N$-body Schr\"odinger equation in the mean-field limit, with convergence rate estimates that are uniform in the Planck constant $\hbar$. Specifically, we consider the two following cases: (a) T\"oplitz initial data and Lipschitz interaction forces, and (b) analytic initial data and interaction potential, over short time intervals independent of $\hbar$. The convergence rates in these two cases are $1/\sqrt{\log\log N}$ and $1/N$ respectively. The treatment of the second case is entirely self-contained and all the constants appearing in the final estimate are explicit. It provides a derivation of the Vlasov equation from the $N$-body classical dynamics using BBGKY hierarchies instead of empirical measures. (10.1016/j.jfa.2018.06.008)
  DOI : 10.1016/j.jfa.2018.06.008
• Transport et diffusion
  
  • Allaire Grégoire
  • Blanc Xavier
  • Després Bruno
  • Golse François
  , 2018, pp.328. Ce livre est issu d'un cours enseigné par les auteurs en troisième année de l'Ecole Polytechnique, ce qui correspond à un niveau de première année de Master. Le sujet en est l'étude mathématique et numérique de modèles d'équations aux dérivées partielles, dits de transport et diffusion. Ces équations modélisent l'évolution d'une densité de particules ou d'individus en interaction avec leur milieu. L'origine de ces modèles est très variée. Ils proviennent classiquement de la physique et servent à décrire des particules neutres comme les neutrons et les photons. Dans le premier cas on parle aussi de neutronique alors que le dernier cas est appelé transfert radiatif. La biologie fait aussi appel aux équations de transport pour modéliser la dynamique de populations structurées. Citons aussi pour mémoire la dynamique des gaz raréfiés, le transport d'électrons dans les semi-conducteurs ou encore la physique des plasmas qui sont des phénomènes modélisés par des équations où l'opérateur de transport est une brique de base essentielle.
• Quantization of Measures and Gradient Flows: a Perturbative Approach in the 2-Dimensional Case
  
  • Caglioti Emanuele
  • Golse François
  • Iacobelli Mikaela
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2018, 35, pp.1531-1555. In this paper we study a perturbative approach to the problem of quantization of measures in the plane. Motivated by the fact that, as the number of points tends to infinity, hexagonal lattices are asymptotically optimal from an energetic point of view (see [Morgan, Bolton: Amer. Math. Monthly 109 (2002), 165-172]), we consider configurations that are small perturbations of the hexagonal lattice and we show that: (1) in the limit as the number of points tends to infinity, the hexagonal lattice is a strictly minimizer of the energy; (2) the gradient flow of the limiting functional allows us to evolve any perturbed configuration to the optimal one exponentially fast. In particular, our analysis provides a solid mathematical justification of the asymptotic optimality of the hexagonal lattice among its nearby configurations. (10.1016/j.anihpc.2017.12.003)
  DOI : 10.1016/j.anihpc.2017.12.003
• Null-controllability of hypoelliptic quadratic differential equations
  
  • Beauchard Karine
  • Pravda-Starov Karel
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2018, 5, pp.1-43. We study the null-controllability of parabolic equations associated to a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated to these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated to hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated to any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space. (10.5802/jep.62)
  DOI : 10.5802/jep.62
• The Vlasov-Navier-Stokes system in a 2D pipe: existence and stability of regular equilibria
  
  • Moussa Ayman
  • Glass Olivier
  • Han-Kwan Daniel
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2018, 230 (2), pp.593–639. In this paper, we study the Vlasov-Navier-Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase. (10.1007/s00205-018-1253-1)
  DOI : 10.1007/s00205-018-1253-1