Publications

2021

• Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data
  
  • Lafleche Laurent
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2021, 38 (6), pp.1739–1762. We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension d ≥ 3, including the case of a Coulomb singularity in dimension d = 3. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative semiclassical bounds on the space and velocity moments of even order and the asymptotic behaviour of the spatial density due to dispersion effects. (10.1016/j.anihpc.2021.01.004)
  DOI : 10.1016/j.anihpc.2021.01.004
• On the characterization of equilibria of nonsmooth minimal-time mean field games with state constraints
  
  • Sadeghi Arjmand Saeed
  • Mazanti Guilherme
  , 2021, pp.5300–5305. In this paper, we consider a first-order deterministic mean field game model inspired by crowd motion in which agents moving in a given domain aim to reach a given target set in minimal time. To model interaction between agents, we assume that the maximal speed of an agent is bounded as a function of their position and the distribution of other agents. Moreover, we assume that the state of each agent is subject to the constraint of remaining inside the domain of movement at all times, a natural constraint to model walls, columns, fences, hedges, or other kinds of physical barriers at the boundary of the domain. After recalling results on the existence of Lagrangian equilibria for these mean field games and the main difficulties in their analysis due to the presence of state constraints, we show how recent techniques allow us to characterize optimal controls and deduce that equilibria of the game satisfy a system of partial differential equations, known as the mean field game system. (10.1109/CDC45484.2021.9683104)
  DOI : 10.1109/CDC45484.2021.9683104
• Description and Classification of 2-Solitary Waves for Nonlinear Damped Klein–Gordon Equations
  
  • Côte Raphaël
  • Martel Yvan
  • Yuan Xu
  • Zhao Lifeng
  Communications in Mathematical Physics, Springer Verlag, 2021, 388 (3), pp.1557-1601. (10.1007/s00220-021-04241-5)
  DOI : 10.1007/s00220-021-04241-5
• Tropical Hodge theory and applications
  
  • Piquerez Matthieu
  , 2021. In this thesis, we prove that the tropical cohomology of a smooth projective tropical variety verifies several symmetry properties: namely, tropical analogs of the Kähler package composed of the Poincaré duality, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations and the monodromy-weight conjecture. We also give some applications.In the local case, we construct a wide family of fans, called tropically shellable fans, whose canonical compactifications verify the Kähler package. We show that the tropical cohomology computes their Chow rings and some quotients of the Stanley-Reisner rings of simplicial complexes which are of particular interest in combinatorics.In the global case, the proof of the main theorem mentioned above uses interesting objects as the existence of some good triangulations and specific versions of tropical analogs of the Deligne spectral sequence, the Steenbrink spectral sequence and the monodromy operator also known as the tropical eigenwave operator.As an application of our results, we get a generalization of the work of De Concini-Procesi and Feichtner-Yuzvinski about wonderful compactifications to the case of toric compactifications induced by unimodular subfans of Bergman fans.In another direction, we prove a tropical Hodge conjecture for smooth projective varieties admitting a rational triangulation: the tropical Hodge classes coincide with the kernel of the monodromy restricted to parts of bidegree (p,p).Finally, we provide a generalization of Symanzik polynomials in higher dimensions. In dimension one, these polynomials appear in combinatorics, in physics and recently in asymptotic Hodge theory. They have many known properties that are still valid in our generalization. This is a first step to understand the asymptotic of some data on degenerating families of complex varieties in any dimension. We also provide a complete description of the exchange graph of independent sets of any matroid.
• Behaviour of some Hodge invariants by middle convolution
  
  • Martin Nicolas
  Bulletin de la société mathématique de France, Société Mathématique de France, 2021, 149 (3), pp.479-500. Following an article of Dettweiler and Sabbah, this article studies the behaviour of various Hodge invariants by middle additive convolution with a Kummer module. The main result gives the behaviour of the nearby cycle local Hodge numerical data at infinity. We also give expressions for Hodge numbers and degrees of some Hodge bundles without making the hypothesis of scalar monodromy at infinity, which generalizes the results of Dettweiler and Sabbah. (10.24033/bsmf.2835)
  DOI : 10.24033/bsmf.2835
• Middle multiplicative convolution and hypergeometric equations
  
  • Martin Nicolas
  Journal of Singularities, Worldwide Center of Mathematics, LLC, 2021, 23, pp.194-204. Using a relation due to Katz linking up additive and multiplicative convolutions, we make explicit the behaviour of some Hodge invariants by middle multiplicative convolution, following [DS13] and [Mar18a] in the additive case. Moreover, the main theorem gives a new proof of a result of Fedorov computing the Hodge invariants of hypergeometric equations. (10.5427/jsing.2021.23k)
  DOI : 10.5427/jsing.2021.23k
• Spectral interpretations of dynamical degrees and applications
  
  • Dang Nguyen-Bac
  • Favre Charles
  Annals of Mathematics, Princeton University, Department of Mathematics, 2021. We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space of b-classes of higher codimension cycles, and endow this space with various Banach norms. Building on these constructions, we design a natural extension to higher dimension of the Picard-Manin space introduced by Cantat and Boucksom-Favre-Jonsson in the case of surfaces. We prove a version of the Hodge index theorem, and a surprising compactness result in this Banach space. We use these two theorems to infer a precise control of the sequence of degrees of iterates of a map under the assumption that the square of the first dynamical degree is strictly larger than the second dynamical degree. As a consequence, we obtain that the dynamical degrees of an automorphism of the affine 3-space are all algebraic numbers. (10.4007/annals.2021.194.1.5)
  DOI : 10.4007/annals.2021.194.1.5
• Long time dynamics for nonlinear wave-type equations with or without damping
  
  • Yuan Xu
  , 2021. In this thesis, we study the qualitative behavior of solutions of nonlinear wave-type equations, with or without damping, putting a special emphasis on the description of the long-time dynamics of solutions in the energy space. Through the typical examples of the nonlinear Klein-Gordon equation (NLKG) with or without damping and the nonlinear wave equation (NLW), we study the behavior of solutions that decompose into a single soliton or into sums of solitons, when time goes to infinity.First, we show the conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity. Second, for the energy-critical NLW equation, we prove the existence of different types of multi-solitons based on the ground state or on suitable excited states, under various conditions on the space dimension and Lorentz speeds. Finally, we study the long-time dynamics of solitons and multi-solitons of the damped energy sub-critical NLKG equation.
• Semiclassical Evolution With Low Regularity
  
  • Golse François
  • Paul Thierry
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2021, 153, pp.257-311. We prove semiclassical estimates for the Schr\"odinger-von Neumann evolution with $C^{1,1}$ potentials and density matrices whose square root have either Wigner functions with low regularity independent of the dimension, or matrix elements between Hermite functions having long range decay. The estimates are settled in different weak topologies and apply to initial density operators whose square root have Wigner functions $7$ times differentiable, independently of the dimension. They also apply to the $N$ body quantum dynamics uniformly in $N$. In a appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderon-Vaillancourt Theorem. (10.1016/j.matpur.2021.02.008)
  DOI : 10.1016/j.matpur.2021.02.008
• Integral $p$-adic \'etale cohomology of Drinfeld symmetric spaces
  
  • Colmez Pierre
  • Dospinescu Gabriel
  • Nizioł Wiesława
  Duke Mathematical Journal, Duke University Press, 2021, 170 (3). We compute the integral $p$-adic \'etale cohomology of Drinfeld symmetric spaces of any dimension. This refines the computation of the rational $p$-adic \'etale cohomology from Colmez-Dospinescu-Nizio{\l}. The main tools are: the computation of the integral de Rham cohomology from CDN and the integral $p$-adic comparison theorems of Bhatt-Morrow-Scholze and \v{C}esnavi\v{c}ius-Koshikawa which replace the quasi-integral comparison theorem of Tsuji used in CDN. (10.1215/00127094-2020-0084)
  DOI : 10.1215/00127094-2020-0084
• Finitely presented simple groups and measure equivalence
  
  • López Neumann Antonio Rodrigo Ignacio
  , 2021.
• Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation
  
  • Côte Raphaël
  • Martel Yvan
  • Yuan Xu
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2021, 239 (3), pp.1837-1874. (10.1007/s00205-020-01605-4)
  DOI : 10.1007/s00205-020-01605-4
• Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains
  
  • Ertzbischoff Lucas
  • Han-Kwan Daniel
  • Moussa Ayman
  Nonlinearity, IOP Publishing, 2021, 34 (10), pp.6843. We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system set on $\Omega \times \R^3$, for a smooth bounded domain $\Omega$ of $\R^3$, with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to $0$ while the distribution function concentrates towards a Dirac mass in velocity centered at $0$, with an exponential rate. The proof, which follows the methods introduced in \cite{HKMM}, requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviors for the kinetic density, from total absorption to no absorption at all. (10.1088/1361-6544/ac1558)
  DOI : 10.1088/1361-6544/ac1558
• Explicit generators of some pro-p groups via Bruhat-Tits theory
  
  • Loisel Benoit
  Bulletin de la société mathématique de France, Société Mathématique de France, 2021, 149 (2), pp.309-388. Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p subgroups. The maximal pro-p subgroups of quasisplit simply connected semisimple groups can be described in the combinatorial terms of a valued root groups datum, thanks to the Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p subgroups thanks to parametrizations of a suitable maximal torus and of the corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of a suitable root system. (10.24033/bsmf.2831)
  DOI : 10.24033/bsmf.2831
• Asymptotic stability of equilibria for screened Vlasov-Poisson systems via pointwise dispersive estimates
  
  • Han-Kwan Daniel
  • Nguyen Toan T.
  • Rousset Frédéric
  Annals of PDE, Springer, 2021. We revisit the proof of Landau damping near stable homogenous equilibria of Vlasov-Poisson systems with screened interactions in the whole space $\mathbb{R}^d$ (for $d\geq3$) that was first established by Bedrossian, Masmoudi and Mouhot. Our proof follows a Lagrangian approach and relies on precise pointwise in time dispersive estimates in the physical space for the linearized problem that should be of independent interest. This allows to cut down the smoothness of the initial data required in Bedrossian at al. (roughly, we only need Lipschitz regularity). Moreover, the time decay estimates we prove are essentially sharp, being the same as those for free transport, up to a logarithmic correction. (10.1007/s40818-021-00110-5)
  DOI : 10.1007/s40818-021-00110-5
• From Newton's second law to Euler's equations of perfect fluids
  
  • Han-Kwan Daniel
  • Iacobelli Mikaela
  Proceedings of the American Mathematical Society, American Mathematical Society, 2021. Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty. (10.1090/proc/15349)
  DOI : 10.1090/proc/15349
• Integrable deformations and degenerations of some irregular singularities
  
  • Sabbah Claude
  Publications of the Research Institute for Mathematical Sciences, European Mathematical Society, 2021, 57 (3-4), pp.755-794. Inspired by an article of Cotti, Dubrovin and Guzzetti arXiv:1706.04808, we extend to a degenerate case a result of Malgrange on integrable deformations of irregular singularities. We give an application to integrable deformations of the solution of some Birkhoff problem and apply it to the construction of Frobenius manifolds. (10.4171/PRIMS/57-3-2)
  DOI : 10.4171/PRIMS/57-3-2
• Time dependent Quantum Perturbations uniform in the semiclassical regime
  
  • Golse François
  • Thierry Paul
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2021, 72 (3), pp.1273-1296. We present a time dependent quantum perturbation result, uniform in the Planck constant, for perturbations of potentials whose gradients are Lipschitz continuous by potentials whose gradients are only bounded a.e.. Though this low regularity of the full potential is not enough to provide the existence of the classical underlying dynamics, at variance with the quantum one, our result shows that the classical limit of the perturbed quantum dynamics remains in a tubular neighbourhood of the classical unperturbed one of size of order of the square root of the size of the perturbation. We treat both Schrödinger and von Neumann-Heisenberg equations. (10.1512/iumj.2023.72.9363)
  DOI : 10.1512/iumj.2023.72.9363
• On the linearized Vlasov-Poisson system on the whole space around stable homogeneous equilibria
  
  • Han-Kwan Daniel
  • Nguyen Toan T.
  • Rousset Frédéric
  Communications in Mathematical Physics, Springer Verlag, 2021. We study the linearized Vlasov-Poisson system around suitably stable homogeneous equilibria on $\mathbb{R}^d\times \mathbb{R}^d$ (for any $d \geq 1$) and establish dispersive $L^\infty$ decay estimates in the physical space. (10.1007/s00220-021-04228-2)
  DOI : 10.1007/s00220-021-04228-2
• Blowup on an arbitrary compact set for a Sch\"odinger equation with nonlinear source term
  
  • Cazenave Thierry
  • Han Zheng
  • Martel Yvan
  Journal of Dynamics and Differential Equations, Springer Verlag, 2021, 33 (2), pp.941-960. We consider the nonlinear Schr\"odinger equation on ${\mathbb R}^N $, $N\ge 1$, \begin{equation*} \partial _t u = i \Delta u + \lambda | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \end{equation*} with $\lambda \in {\mathbb C}$ and $\Re \lambda >0$, for $H^1$-subcritical nonlinearities, i.e. $\alpha >0$ and $(N-2) \alpha < 4$. Given a compact set $K \subset {\mathbb R}^N $, we construct $H^1$ solutions that are defined on $(-T,0)$ for some $T>0$, and blow up on $K $ at $t=0$. The construction is based on an appropriate ansatz. The initial ansatz is simply $U_0(t,x) = ( \Re \lambda )^{- \frac {1} {\alpha }} (-\alpha t + A(x) )^{ -\frac {1} {\alpha } - i \frac {\Im \lambda } {\alpha \Re \lambda } }$, where $A\ge 0$ vanishes exactly on $ K $, which is a solution of the ODE $u'= \lambda | u |^\alpha u$. We refine this ansatz inductively, using ODE techniques. We complete the proof by energy estimates and a compactness argument. This strategy is reminiscent of~[3, 4]. (10.1007/s10884-020-09841-8)
  DOI : 10.1007/s10884-020-09841-8
• Paley-Wiener theorems for a p-adic spherical variety
  
  • Delorme Patrick
  • Harinck Pascale
  • Sakellaridis Yiannis
  Memoirs of the American Mathematical Society, American Mathematical Society, 2021. (10.1090/memo/1312)
  DOI : 10.1090/memo/1312
• Numerical investigation of solitary wave stability for quantum dissipative systems
  
  • Goudon Thierry
  • Vivion Léo
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2021, 62 (1), pp.011509. We consider a simple model describing the interaction of a quantum particle with a vibrational environment, which eventually acts as a friction on the particle. This equation admits soliton-like solutions, and we numerically investigate their stability when subjected to a small initial impulsion. Our findings illustrate the analogies with the behavior of classical particles and the relevance of asymptotic models. (10.1063/5.0021246)
  DOI : 10.1063/5.0021246