Publications

2017

• 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
• 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
• 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
• THE SCHRÖDINGER EQUATION IN THE MEAN-FIELD AND SEMICLASSICAL REGIME
  
  • Golse François
  • Paul Thierry
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2017, 223, pp.57-94. In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the N-body linear Schrödinger equation uniformly in N leading to the N-body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N-body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of exponent 2 adapted to the classical limit, reminiscent of, but different from the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. As a by-product, we also provide bounds on the quadratic Monge-Kantorovich distances between the classical densities and the Husimi functions of the quantum density matrices. (10.1007/s00205-016-1031-x)
  DOI : 10.1007/s00205-016-1031-x
• Energy decay for a locally undamped wave equation
  
  • Léautaud Matthieu
  • Lerner Nicolas
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2017, 26 (1), pp.157 - 205. (10.5802/afst.1528)
  DOI : 10.5802/afst.1528