Nicolas LERNER

Institut de Mathématiques de Jussieu
Sorbonne Université
(formerly Université Pierre et Marie Curie (Paris VI))
Campus Pierre et Marie Curie
4, Place Jussieu - Boîte Courrier 247
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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  • An energy method for averaging lemmas, with Diogo Arsénio, file on arXiv, This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approach---based upon the classical energy method---provides a powerful duality principle in kinetic transport equations which allows for a natural extension of classical averaging lemmas to previously unknown cases where the density and the source term belong to dual spaces. More generally, this kinetic duality principle produces regularity results where one can trade a loss of regularity or integrability somewhere in the kinetic transport equation for a suitable opposite gain elsewhere. Also, it looks simpler and more robust to rely on proving inequalities instead of constructing exact parametrices. The results in this article are introduced from a functional analytic point of view and they are motivated by the abstract regularity theory of kinetic transport equations.

  • Carleman Inequalities: an Introduction and More, a new book, published in 2019 in the Springer-Verlag Series Grundlehren der Mathematischen Wissenschaften . This is a 576-page book, dealing with various aspects of Carleman inequalities. On that link, you will find a short description of the contents of the book as well as some previews opportunities.

  • On the anti-Wick symbol as a Gelfand-Shilov generalized function, with Laurent Amour and Jean Nourrigat, file on arXiv, to appear in the Proceedings of the American Mathematical Society. The purpose of this article is to prove that the anti-Wick symbol of an operator mapping S(ℝn) into S'(ℝn), which is generally not a tempered distribution, can still be defined as a Gelfand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gelfand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.

  • On integrals over a convex set of the Wigner distribution, with Bérangère Delourme and Thomas Duyckaerts, Journal of Fourier Analysis and Applications, volume 26, February 2020. We provide an example of a normalized L2(ℝ) function u such that its Wigner distribution W(u,u) has an integral >1 on the square [0,a]×[0,a] for a suitable choice of a. This provides a negative answer to a question raised by P. Flandrin in 1988. Our arguments are based upon the study of the Weyl quantization of the indicatrix of ℝ+×ℝ+ along with a precise numerical analysis of its discretization.

  • Mehler's formula and functional calculus, Science China Mathematics, 62 (2019), no. 6, 1143-1166. We show that Mehler's formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the Hermite basis the Weyl quantization of the characteristic function of several domains of the phase space.

  • Unique continuation through transversal characteristic hypersurfaces, Journal d'Analyse Mathématique ,138, (2019), no.1, 135-156. 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.

  • Some natural subspaces and quotient spaces of L1, with Gilles Godefroy, Advances in Operator Theory, 3 (2018), no.1, 73-86, paper online.

  • The onset of instability in first-order systems, with Toan T. Nguyen and Benjamin Texier, Journal of the European Mathematical Society, 20, 6, 1303-1373, (2018), paper on the JEMS website. 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.

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    updated July 1rst, 2020