Publications

2019

• p-Laplacian Keller–Segel Equation: Fair Competition and Diffusion Dominated Cases
  
  • Lafleche Laurent
  • Salem Samir
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2019, 357 (4), pp.360–365. This work deals with the aggregation diffusion equation ∂t/∂ρ = ∆_p(ρ) + λ div ((K_a * ρ)ρ) , where K_a(x) = x/|x|^a is an attraction kernel and ∆_p is the so called p-Laplacian. We show that the domain a < p(d + 1) − 2d is subcritical with respect to the competition between the aggregation and diffusion by proving that there is existence unconditionally with respect to the mass. In the critical case we show existence of solution in a small mass regime for an L ln L initial condition. (10.1016/j.crma.2019.03.002)
  DOI : 10.1016/j.crma.2019.03.002
• Constructibilité et modération uniformes en cohomologie étale
  
  • Orgogozo Fabrice
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2019, 155 (4), pp.711-757. Let S be a Noetherian scheme and f:X -> S a proper morphism. By SGA 4 XIV, for any constructible sheaf F of Z/nZ-modules on X, the sheaves of Z/nZ-modules R^if_*F obtained by direct image (for the etale topology) are also constructible: there is a stratification of S on whose strata these sheaves are locally constant constructible. After previous work of N. Katz and G. Laumon, or L. Illusie, on the special case in which S is generically of characteristic zero or the sheaves F are constant (with invertible torsion on S), here we study the dependency of the stratification on F. We show that a natural "uniform" tameness and constructibility condition satisfied by constant sheaves, which was introduced by O. Gabber, is stable under the functors R^if_*. If f is not proper, this result still holds assuming tameness at infinity, relatively to S. We also prove the existence of uniform bounds on Betti numbers, in particular for the stalks of the sheaves R^if_*Z/lZ, where l ranges through all prime numbers invertible on S.
• GLOBAL WELL-POSEDNESS OF THE AXISYMMETRIC NAVIER-STOKES EQUATIONS WITH MEASURE-VALUED INITIAL DATA
  
  • Lévy Guillaume
  , 2019.
• A multidimensional generalization of Symanzik polynomials
  
  • Piquerez Matthieu
  , 2019. Symanzik polynomials are defined on Feynman graphs and they are used in quantum field theory to compute Feynman amplitudes. They also appear in mathematics from different perspective. For example, recent results show that they allow to describe asymptotic limits of geometric quantities associated to families of Riemann surfaces. In this paper, we propose a generalization of Symanzik polynomials to the setting of higher dimensional simplicial complexes and study their basic properties and applications. We state a duality relation between these generalized Symanzik polynomials and what we call Kirchhoff polynomials, which have been introduced in recent generalizations of Kirchhoff's matrix-tree theorem to simplicial complexes. Moreover, we obtain geometric invariants which compute interesting data on triangulable manifolds. As the name indicates, these invariants do not depend on the chosen triangulation. We furthermore prove a stability theorem concerning the ratio of Symanzik polynomials which extends a stability theorem of Amini to higher dimensional simplicial complexes. In order to show that theorem, we will make great use of matroids, and provide a complete classification of the connected components of the exchange graph of a matroid, a result we hope could be of independent interest. Finally, we give some idea on how to generalize Symanzik polynomials to the setting of matroids over hyperfields defined recently by Baker and Bowler.
• MEAN FIELD LIMIT FOR CUCKER-SMALE MODELS
  
  • Natalini R.
  • Paul Thierry
  , 2019. In this very short note, we consider the Cucker-Smale dynamical system and we derive rigorously the Vlasov-type equation introduced in [4] in the mean-field limit. The vector field we consider is bounded at infinity in the velocity variables, and Lipschitz continuous in the space variables.
• Existence of multi-solitary waves with logarithmic relative distances for the NLS equation
  
  • Nguyễn Tiến Vinh
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2019, 357, pp.13 - 58. (10.1016/j.crma.2018.11.012)
  DOI : 10.1016/j.crma.2018.11.012
• Existence of solutions of a non-linear eigenvalue problem with a variable weight
  
  • Hadiji Rejeb
  • Vigneron François
  Journal of Differential Equations, Elsevier, 2019, 266 (2-3), pp.1488-1513. (10.1016/j.jde.2018.08.001)
  DOI : 10.1016/j.jde.2018.08.001
• Planar Tropical Cubic Curves of Any Genus, and Higher Dimensional Generalisations
  
  • Bertrand Benoît
  • Brugallé Erwan
  • de Medrano Lucía López
  , 2019. We prove that there exist planar tropical cubic curves of genus $g$ for any non-negative integer $g$. More generally, we study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in $\mathbb{T}P^n$. We provide a lower bound for the maximal value of the top Betti number, which naturally depends on the dimension and degree, but also on the codimension. In particular, when the codimension is large enough, this lower bound is larger than the maximal value of the corresponding Hodge number of complex algebraic projective varieties of the given dimension and degree. In the case of surfaces, we extend our study to all tropical homology groups.
• Analyse fréquentielle du signal
  
  • Bahouri Hajer
  • Vigneron François
  Images des mathématiques, CNRS, 2019. (10.60868/wrqs-xz66)
  DOI : 10.60868/wrqs-xz66
• Geometric side of a local relative trace formula
  
  • Delorme Patrick
  • Harinck Pascale
  • Souaifi Sofiane
  Transactions of the American Mathematical Society, American Mathematical Society, 2019, 371 (3), pp.1815-1857. Following a scheme suggested by B. Feigon, we investigate a local relative trace formula in the situation of a reductive p-adic group G relative to a symmetric subgroup H " HpF q where H is split over the local field F of characteristic zero and G " GpF q is the restriction of scalars of H IE relative to a quadratic unramified extension E of F. We adapt techniques of the proof of the local trace formula by J. Arthur in order to get a geometric expansion of the integral over H ˆ H of a truncated kernel associated to the regular representation of G. Mathematics Subject Classification 2000: 11F72, 22E50. (10.1090/tran/7360)
  DOI : 10.1090/tran/7360
• Riemann–Hilbert correspondence for mixed twistor D-modules
  
  • Monteiro Fernandes Teresa
  • Sabbah Claude
  Journal of the Institute of Mathematics of Jussieu, Cambridge University Press, 2019, 18 (3), pp.629-672. (10.1017/S1474748017000184)
  DOI : 10.1017/S1474748017000184
• Finite-time blowup for a Schr\"odinger equation with nonlinear source term
  
  • Cazenave Thierry
  • Martel Yvan
  • Zhao Lifeng
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2019, 39 (2), pp.1171-1183. We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical assumptions $\alpha\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finite set of points of ${\mathbb R}^N$. The construction involves explicit functions $U$, solutions of the ordinary differential equation $U_t=|U|^\alpha U$. In the simplest case, $U(t,x)=(|x|^k-\alpha t)^{-\frac 1\alpha}$ for $t<0$, $x\in {\mathbb R}^N$. For $k$ sufficiently large, $U$ satisfies $|\Delta U|\ll U_t$ close to the blow-up point $(t,x)=(0,0)$, so that it is a suitable approximate solution of the problem. To construct an actual solution $u$ close to $U$, we use energy estimates and a compactness argument. (10.3934/dcds.2019050)
  DOI : 10.3934/dcds.2019050
• Automorphic Forms and Even Unimodular Lattices
  
  • Chenevier Gaëtan
  • Lannes Jean
  , 2019, 69, pp.428 p.. In this memoir, we study the even unimodular lattices of rank at most 24, as well as a related collection of automorphic forms of the orthogonal, symplectic and linear groups of small rank. Our guide is the question of determining the number of p-neighborhoods, in the sense of M. Kneser, between two isometry classes of such lattices. We prove a formula for this number, in which occur certain Siegel modular forms of genus 1 and 2. It has several applications, such as the proof of a conjecture of G. Nebe and B. Venkov about the linear span of the higher genus theta series of the Niemeier lattices, the computation of the p-neighborhoods graphs of the Niemeier lattices (the case p = 2 being due to Borcherds), or the proof of a congruence conjectured by G. Harder. Classical arguments reduce the problem to the description of the automorphic representations of a suitable integral form of the Euclidean orthogonal group of R^24 which are unramified at each finite prime and trivial at the archimedean prime. The recent results of J. Arthur suggest several new approaches to this type of questions. This is the other main theme that we develop in this memoir. We give a number of other applications, for instance to the classification of Siegel modular cuspforms of weight at most 12 for the full Siegel modular group.
• Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation
  
  • Golse F F
  • Imbert Cyril
  • Mouhot Clément
  • Vasseur A F
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2019, XIX (issue 1), pp.PP. 253-295. We extend the De Giorgi–Nash–Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the Hölder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations " of type II " , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law γ ∈ [−d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the Hölder regularity of these solutions. (10.2422/2036-2145.201702_001)
  DOI : 10.2422/2036-2145.201702_001
• Fractional Keller–Segel Equation: Global Well-posedness and Finite Time Blow-up
  
  • Lafleche Laurent
  • Salem Samir
  Communications in Mathematical Sciences, International Press, 2019, 17 (8), pp.2055–2087. This article studies the aggregation diffusion equation ∂ρ/∂t = ∆^(α/2) ρ + λ div((K * ρ)ρ), where ∆^(α/2) denotes the fractional Laplacian and K = x/|x|^a is an attractive kernel. This equation is a generalization of the classical Keller-Segel equation, which arises in the modeling of the motion of cells. In the diffusion dominated case a < α we prove global well-posedness for an L^1_k initial condition, and in the fair competition case a = α for an L^1_k ∩ L ln L initial condition. In the aggregation dominated case a > α, we prove global or local well posedness for an L^p initial condition, depending on some smallness condition on the L^p norm of the initial condition. We also prove that finite time blow-up of even solutions occurs, under some initial mass concentration criteria. (10.4310/CMS.2019.v17.n8.a1)
  DOI : 10.4310/CMS.2019.v17.n8.a1
• Solutions with prescribed local blow-up surface for the nonlinear wave equation
  
  • Cazenave Thierry
  • Martel Yvan
  • Zhao Lifeng
  Advanced Nonlinear Studies, Walter de Gruyter GmbH, 2019. We prove that any sufficiently differentiable space-like hypersurface of ${\mathbb R}^{1+N} $ coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation $\partial_{tt} u - \Delta u=|u|^{p-1} u$ on ${\mathbb R} \times {\mathbb R} ^N$, for any $1\leq N\leq 4$ and $1 < p \le \frac {N+2} {N-2}$. We follow the strategy developed in our previous work [arXiv 1812.03949] on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blowup on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at $t=0$ for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at $t=0$. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with $H^2\times H^1$ solutions for the transformed problem. (10.1515/ans-2019-2059)
  DOI : 10.1515/ans-2019-2059
• Uniform K-stability and asymptotics of energy functionals in Kähler geometry
  
  • Boucksom Sébastien
  • Hisamoto Tomoyuki
  • Jonsson Mattias
  Journal of the European Mathematical Society, European Mathematical Society, 2019, 21 (9), pp.2905-2944. (10.4171/JEMS/894)
  DOI : 10.4171/JEMS/894
• Degenerations of SL(2,C) representations and Lyapunov exponents
  
  • Dujardin Romain
  • Favre Charles
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2019. We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity which can be interpreted as the non-Archimedean Lyapunov exponent of the family. We also describe the limit of the corresponding family of stationary measures on P 1 (C). (10.5802/ahl.24)
  DOI : 10.5802/ahl.24
• POINCARÉ INEQUALITY ON COMPLETE RIEMANNIAN MANIFOLDS WITH RICCI CURVATURE BOUNDED BELOW
  
  • Besson Gérard
  • Courtois Gilles
  • Ar Hersonsky Sa '
  Mathematical Research Letters, International Press, 2019, 18, pp.10001 - 100. We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary. 0. Introduction Statements of the main results. In this paper, we will establish that complete Rie-mannian manifolds with Ricci curvature bounded below and having polynomial growth, admit a family of uniform Poincaré inequalities. To begin with, let (M n , g) be a complete n-dimensional Riemannian manifold. Henceforth, we will assume that (M n , g) satisfies the Ricci curvature lower bound (0.1) Ricci (M n ,g) ≥ −(n − 1)κ, for some κ ≥ 0. We will also assume that (M n , g) has α-polynomial growth; this means that there exist constants v > 0, α > 0 and R 0 ≥ 0 such that for any m ∈ M n and R > R 0 , the ball of radius R centered at m satisfies (0.2) vol B(m, R) ≤ vR α , where vol denotes the canonical measure on (M n , g). Recall that Bishop's Comparison Theorem (cf. [15, Section IV]) implies that when κ = 0, (M n , g) satisfies polynomial growth with α = n. We will finally assume that the local geometry of (M n , g) is controled in the sense that for any m ∈ M n , (0.3) vol B(m, 1) ≥ ω The triple (M n , dist, vol) with dist being the standard metric induced by the Rie-mannian metric is an example of a metric measure space. Throughout this paper (X, ρ, µ) will denote a metric space endowed with a Borel measure µ. We will use the notation (0.4) u A = 1 µ(A) A udµ, (10.4310/mrl.2018.v25.n6.a3)
  DOI : 10.4310/mrl.2018.v25.n6.a3
• Construction of two-bubble solutions for energy-critical wave equations
  
  • Jendrej Jacek
  American Journal of Mathematics, Johns Hopkins University Press, 2019, 141 (1), pp.55-118. (10.1353/ajm.2019.0002)
  DOI : 10.1353/ajm.2019.0002