Nicolas LERNER

Institut de Mathématiques de Jussieu,
Université Pierre et Marie Curie (Paris VI)
boîte 186
4, Place Jussieu
75252 Paris cedex 05

Projet analyse fonctionnelle
Bureau 16-26-423
01 44 27 85 70   (from abroad 331 44 27 85 70)

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Recent Papers

Lecture Notes


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  • Unique continuation through transversal characteristic hypersurfaces, file on arXiv, to appear in the Journal d'Analyse Mathématique. We prove a unique continuation result for an ill-posed characteristic problem. A model problem of this type occurs in A.D. Ionescu & S. Klainerman article (Theorem 1.1 in [MR2470908]) and we extend their model-result using only geometric assumptions. The main tools are Carleman estimates and Hörmander's pseudo-convexity conditions.

  • Energy decay for a locally undamped wave equation, with Matthieu Léautaud, file on arXiv, to appear in the Annales de la Faculté des Sciences de Toulouse. We study the decay rate for the energy of solutions of a damped wave equation in a situation where the Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function. Our method relies on a refined microlocal analysis linked to a second microlocalization procedure to cut the phase space into tiny regions respecting the uncertainty principle but way too small to enter a standard semi-classical analysis localization. Using a multiplier method, we obtain the energy estimates in each region and we then patch the microlocal estimates together.

  • The onset of instability in first-order systems, with Toan T. Nguyen and Benjamin Texier, file on arXiv, to appear in the Journal of the European Mathematical Society. We study in this paper the Cauchy problem for first-order quasi-linear systems of partial differential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., when hyperbolicity is violated at initial time, then the Cauchy problem is strongly unstable, in the sense of Hadamard. This phenomenon, which extends the linear Lax-Mizohata theorem, was explained by G. Métivier in Remarks on the well-posedness of the nonlinear Cauchy problem, (Contemp. Math. 2005). In this article, we are interested in the transition from hyperbolicity to non-hyperbolicity, that is the limiting case where hyperbolicity holds at initial time, but is violated at positive times: under such an hypothesis, we generalize a recent work by N. Lerner, Y. Morimoto and C.-J. Xu, Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems, (American J. Math. 2010), on complex scalar systems, as we prove that even a weak defect of hyperbolicity implies a strong Hadamard instability. Our examples include Burgers systems, Van der Waals gas dynamics, and Klein-Gordon-Zakharov systems. Our analysis relies on an approximation result for pseudo-differential flows, introduced by B. Texier in Approximations of pseudo-differential flows, file on arXiv.

  • Gelfand-Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation, with Yoshinori Morimoto, Karel Pravda-Starov and Chao-Jiang Xu, published by the Journal of Functional Analysis, Volume 269, Issue 2, 15 July 2015, pages 459-535, link to the file. We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.

  • Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff, with Yoshinori Morimoto, Karel Pravda-Starov and Chao-Jiang Xu, Journal of Differential Equations, 256 (2014), no. 2, 797-831. We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.

  • Carleman estimates for anisotropic elliptic operators with jumps at an interface, file on arxiv, with Jérôme Le Rousseau, Analysis & PDE 6-7 (2013), 1601--1648. We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.

  • Fonctions classiques, Cours de master 1.

  • Eléments d'analyse pour le M2, Cours introductif de master 2.

  • Some links

    The Paris-London Analysis Seminar

    Analyse non-linéaire et EDP

    Société mathématique de France

    American Mathematical Society

    Notices of the AMS

    European Mathematical Society

    EMS Newsletter


    Journal of the IMJ

    Master de mathématiques, UPMC

    Mathematics Genealogy

    History of Mathematics


    The MacTeX Distribution

    Wolfram Alpha

    updated August 29, 2016