Publications

2022

• Stability of steady states for Hartree and Schrödinger equations for infinitely many particles
  
  • Collot Charles
  • de Suzzoni Anne-Sophie
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2022, 5, pp.429-490. We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation for density matrices. We treat dimensions 2 and 3, extending our previous result. We reach a large class of interaction potentials, which includes the nonlinear Schrödinger equation. This result has an incidence in the density matrices framework. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger equation, and on the use of explicit low frequency cancellations as done by Lewin and Sabin. To relate to density matrices, we use Strichartz estimates for orthonormal systems from Frank and Sabin, and Leibniz rules for integral operators. (10.5802/ahl.127)
  DOI : 10.5802/ahl.127
• Optimal transport pseudometrics for quantum and classical densities
  
  • Golse François
  • Paul Thierry
  Journal of Functional Analysis, Elsevier, 2022, 282 (9), pp.109417. This paper proves variants of the triangle inequality for the quantum analogues of the Wasserstein metric of exponent 2 introduced in Golse et al. (2016) to compare two density operators, and in Golse and Paul (2017) to compare a phase space probability measure and a density operator. The argument differs noticeably from the classical proof of the triangle inequality for Wasserstein metrics, which is based on a disintegration theorem for probability measures, and uses in particular an analogue of the Kantorovich duality for the functional defined in Golse and Paul (2017). Finally, this duality theorem is used to define an analogue of the Brenier transport map for the functional defined in Golse and Paul (2017) to compare a phase space probability measure and a density operator. (10.1016/j.jfa.2022.109417)
  DOI : 10.1016/j.jfa.2022.109417
• Soliton Resolution for Critical Co-rotational Wave Maps and Radial Cubic Wave Equation
  
  • Duyckaerts Thomas
  • Kenig Carlos
  • Martel Yvan
  • Merle Frank
  Communications in Mathematical Physics, Springer Verlag, 2022, 391 (2), pp.779-871. (10.1007/s00220-022-04330-z)
  DOI : 10.1007/s00220-022-04330-z
• Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions
  
  • Ben Porath Immanuel
  • Golse François
  , 2022. This paper discusses the mean-field limit for the quantum dynamics of $N$ identical bosons in $mathbf R^3$ interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021-1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials slightly less general than those considered in [T. Kato, Trans. Amer. Math. Soc. 70 (1951), 195-211]. Our second main result is a new operator inequality satisfied by the quantum Klimontovich solution in the case of an interaction potential with Coulomb type singularity. When evaluated on an initial bosonic pure state, this operator inequality reduces to a Gronwall inequality for a functional introduced in [P. Pickl, Lett. Math. Phys. 97 (2011), 151-164], resulting in a convergence rate estimate for the quantum mean-field limit leading to the time-dependent Hartree equation.
• Compactness methods in Lieb's work
  
  • Sabin Julien
  , 2022. We review some compactness methods appearing in the work of Lieb, with an emphasis on the techniques developed around his 1983 article on the optimizers for the Hardy-Littlewood-Sobolev inequality.
• Multiplicity one at full congruence level
  
  • Le Daniel
  • Morra Stefano
  • Schraen Benjamin
  Journal of the Institute of Mathematics of Jussieu, Cambridge University Press, 2022, 21 (2), pp.637-658. Let F be a totally real field in which p is unramified. Let r be a modular representation of G_F on a two dimension mod p vector space satisfying the Taylor--Wiles hypotheses and tamely ramified and generic at a place v above p. Let m be the corresponding Hecke eigensystem. We describe the m-torsion in the mod p cohomology of Shimura curves with full congruence level at v as a GL_2(k_v)-representation. (10.1017/S1474748020000225)
  DOI : 10.1017/S1474748020000225
• Tunneling estimates and approximate controllability for hypoelliptic equations
  
  • Laurent Camille
  • Léautaud Matthieu
  Memoirs of the American Mathematical Society, American Mathematical Society, 2022, 276 (1357). This article is concerned with quantitative unique continuation estimates for equations involving a " sum of squares " operator L on a compact manifold M assuming: (i) the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of L, and (ii) the analyticity of M and the coefficients of L. The first result is the tunneling estimate ϕ L 2 (ω) ≥ Ce −λ k 2 for normalized eigenfunctions ϕ of L from a nonempty open set ω ⊂ M, where k is the hypoellipticity index of L and λ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation (∂ 2 t +L)u = 0: for T > 2 sup x∈M (dist(x, ω)) (here, dist is the sub-Riemannian distance), the observation of the solution on (0, T) × ω determines the data. The constant involved in the estimate is Ce cΛ k where Λ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation (∂t + L)v = 1ωf in any time, with appropriate (exponential) cost, depending on k. In case k = 2 (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary ∂M can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy developed by the authors in [LL15]. (10.1090/memo/1357)
  DOI : 10.1090/memo/1357
• MEAN-FIELD LIMITS IN STATISTICAL DYNAMICS
  
  • Golse François
  , 2022. These lectures notes are aimed at introducing the reader to some recent mathematical tools and results for the mean-field limit in statistical dynamics. As a warm-up, lecture 1 reviews the approach to the mean-field limit in classical mechanics following the ideas of W. Braun, K. Hepp and R.L. Dobrushin, based on the notions of phase space empirical measures, Klimontovich solutions and Monge-Kantorovich-Wasserstein distances between probability measures. Lecture 2 discusses an analogue of the notion of Klimontovich solution in quantum dynamics, and explains how this notion appears in Pickl's method to handle the case of interaction potentials with a Coulomb type singularity at the origin. Finally, lecture 3 explains how the mean-field and the classical limits can be taken jointly on quantum N-particle dynamics, leading to the Vlasov equation. These lectures are based on a series of joint works with C. Mouhot and T. Paul.
• A short proof of a theorem of Cotti, Dubrovin and Guzzetti
  
  • Sabbah Claude
  Portugaliae Mathematica, European Mathematical Society Publishing House, 2022, 79 (1), pp.29-43. (10.4171/PM/2077)
  DOI : 10.4171/PM/2077
• The non-relativistic limit of the Vlasov-Maxwell system with uniform macroscopic bounds
  
  • Brigouleix Nicolas
  • Han-Kwan Daniel
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2022. We study in this paper the non-relativistic limit from Vlasov-Maxwell to Vlasov-Poisson, which corresponds to the regime where the speed of light is large compared to the typical velocities of particles. In contrast with \cite{Asano-Ukai-86-SMA}, \cite{Degond-86-MMAS}, \cite{Schaeffer-86-CMP} which handle the case of classical solutions, we consider measure-valued solutions, whose moments and electromagnetic fields are assumed to satisfy some uniform bounds. To this end, we use a functional inspired by the one introduced by Loeper in his proof of uniqueness for the Vlasov-Poisson system \cite{Loeper-2006}. We also build a special class of measure-valued solutions, that enjoy no higher regularity with respect to the momentum variable, but whose moments and electromagnetic fields satisfy all required conditions to enter our framework. (10.5802/afst.1702)
  DOI : 10.5802/afst.1702
• Multipopulation minimal-time mean field games
  
  • Sadeghi Saeed
  • Mazanti Guilherme
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2022, 60 (4), pp.1942–1969. In this paper, we consider a mean field game model inspired by crowd motion in which several interacting populations evolving in $\mathbb R^d$ aim at reaching given target sets in minimal time. The movement of each agent is described by a control system depending on their position, the distribution of other agents in the same population, and the distribution of agents on other populations. Thus, interactions between agents occur through their dynamics. We consider in this paper the existence of Lagrangian equilibria to this mean field game, their asymptotic behavior, and their characterization as solutions of a mean field game system, under few regularity assumptions on agents' dynamics. In particular, the mean field game system is established without relying on semiconcavity properties of the value function. (10.1137/21M1407306)
  DOI : 10.1137/21M1407306
• Observability for the Schrödinger Equation: an Optimal Transportation Approach
  
  • Golse François
  • Paul Thierry
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2022, 32 (5). We establish an observation inequality for the Schr\"odinger equation on $\bR^d$, uniform in the Planck constant $\hbar\in[0,1]$. The proof is based on the pseudometric introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. \textbf{223} (2017), 57--94]. This inequality involves only effective constants which are computed explicitly in their dependence in $\hbar$ and all parameters involved.
• Weighted Korn and Poincaré-Korn inequalities in the Euclidean space and associated operators
  
  • Carrapatoso Kleber
  • Dolbeault Jean
  • Hérau Frédéric
  • Mischler Stéphane
  • Mouhot Clément
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2022, 343 (3), pp.1565-1596. We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincaré-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, on a bounded domain. (10.1007/s00205-021-01741-5)
  DOI : 10.1007/s00205-021-01741-5
• Intersection theory of nef b-divisor classes
  
  • Dang Nguyen-Bac
  • Favre Charles
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2022. We prove that any nef b-divisor class on a projective variety is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef b-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony in this context. We show that any big and basepoint free curve class is a power of a nef b-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of b-divisors which were defined in our previous work. We further discuss their properties in the case of toric varieties and connect them with classical functional objects in convex integral geometry.
• Large-time behavior of compressible polytropic fluids and nonlinear Schrödinger equation
  
  • Carles Rémi
  • Carrapatoso Kleber
  • Hillairet Matthieu
  Quarterly of Applied Mathematics, American Mathematical Society, 2022, 80, pp.549-574. In this paper we analyze the large-time behavior of weak solutions to polytropic fluid models possibly including quantum and capillary effects. Formal a priori estimates show that the density of solutions to these systems should disperse with time. Scaling appropriately the system, we prove that, under a reasonable assumption on the decay of energy, the density of weak solutions converges in large times to an unknown profile. In contrast with the isothermal case, we also show that there exists a large variety of asymptotic profiles. We complement the study by providing existence of global-in-time weak solutions satisfying the required decay of energy. As a byproduct of our method, we also obtain results concerning the large-time behavior of solutions to nonlinear Schrödinger equation, allowing the presence of a semi-classical parameter as well as long range nonlinearities. (10.1090/qam/1618)
  DOI : 10.1090/qam/1618
• The asymptotics of the optimal holomorphic extensions of holomorphic jets along submanifolds
  
  • Finski Siarhei
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2022. We study the asymptotics of the $L^2$-optimal holomorphic extensions of holomorphic jets associated with high tensor powers of a positive line bundle along submanifolds. More precisely, for a fixed complex submanifold in a complex manifold, we consider the operator which for a given holomorphic jet along the submanifold of a positive line bundle associates the $L^2$-optimal holomorphic extension of it to the ambient manifold. When the tensor power of the line bundle tends to infinity, we give an explicit asymptotic formula for this extension operator. This is done by a careful study of the Schwartz kernels of the extension operator and related Bergman projectors. It extends our previous results, done for holomorphic sections instead of jets. As an application, we prove the asymptotic isometry between two natural norms on the space of holomorphic jets: one induced from the ambient manifold and another from the submanifold. (10.48550/arXiv.2207.02761)
  DOI : 10.48550/arXiv.2207.02761
• Towards Optimal Transport for Quantum Densities
  
  • Caglioti Emanuele
  • Golse François
  • Paul Thierry
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2022. An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\bR^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165--205] for density operators on $L^2(\bR^d)$, and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that, under some boundedness and constraint hypothesis on the Kantorovich potentials, optimal quantum couplings involve a gradient type structure similar in the quantum paradigm to the Brenier transport map. On the contrary, when the two quantum densities have finite rank, the structure involved by the optimal coupling has, in general, no classical counterpart.
• Hybrid convergence of Kähler–Einstein measures
  
  • Pille-Schneider Léonard
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2022, 72 (2), pp.587-615. (10.5802/aif.3455)
  DOI : 10.5802/aif.3455
• Regularization estimates and hydrodynamical limit for the Landau equation
  
  • Carrapatoso Kleber
  • Rachid Mohamad
  • Tristani Isabelle
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2022, 163 (9), pp.334-432. In this paper, we study the Landau equation under the Navier-Stokes scaling in the torus for hard and moderately soft potentials. More precisely, we investigate the Cauchy theory in a perturbative framework and establish some new short time regularization estimates for our rescaled nonlinear Landau equation. These estimates are quantified in time and optimal, indeed, we obtain the instantaneous expected anisotropic gain of regularity (see [53] for the corresponding hypoelliptic estimates on the linearized Landau collision operator). Moreover, the estimates giving the gain of regularity in the velocity variable are uniform in the Knudsen number. Intertwining these new estimates on the Landau equation with estimates on the Navier-Stokes-Fourier system, we are then able to obtain a result of strong convergence towards this fluid system. (10.1016/j.matpur.2022.05.009)
  DOI : 10.1016/j.matpur.2022.05.009
• Global weak solutions for quantum isothermal fluids
  
  • Carles Rémi
  • Carrapatoso Kleber
  • Hillairet Matthieu
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2022, 72 (6), pp.2241-2298. We construct global weak solutions to isothermal quantum Navier-Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent reformulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform. (10.5802/aif.3489)
  DOI : 10.5802/aif.3489
• Partial regularity in time for the space homogeneous Landau equation with Coulomb potential
  
  • Golse François
  • Gualdani Maria Pia
  • Imbert Cyril
  • Vasseur Alexis
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2022, 55 (4ème série) (6), pp.1575-1611. We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has Hausdorff dimension at most 1/2. (10.24033/asens.2524)
  DOI : 10.24033/asens.2524
• MEAN-FIELD AND CLASSICAL LIMIT FOR THE N-BODY QUANTUM DYNAMICS WITH COULOMB INTERACTION
  
  • Golse François
  • Paul Thierry
  Communications on Pure and Applied Mathematics, Wiley, 2022. This paper proves the validity of the joint mean-field and classical limit of the quantum N-body dynamics leading to the pressureless Euler-Poisson system for factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction potential is assumed to be the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a smooth solution that is rapidly decaying at infinity. One key ingredient in the proof is an inequality from [S. Serfaty, with an appendix of M. Duerinckx arXiv:1803.08345v3 [math.AP]]. (10.1002/cpa.21986)
  DOI : 10.1002/cpa.21986
• THE BOLTZMANN-GRAD LIMIT FOR THE LORENTZ GAS WITH A POISSON DISTRIBUTION OF OBSTACLES
  
  • Golse François
  Kinetic and Related Models, AIMS, 2022. In this note, we propose a slightly different proof of Gallavotti's theorem ["Statistical Mechanics: A Short Treatise", Springer, 1999, pp. 48-55] on the derivation of the linear Boltzmann equation for the Lorentz gas with a Poisson distribution of obstacles in the Boltzmann-Grad limit. (10.3934/krm.2022001)
  DOI : 10.3934/krm.2022001
• Large-time behavior of small-data solutions to the Vlasov–Navier–Stokes system on the whole space
  
  • Han-Kwan Daniel
  Probability and Mathematical Physics, MSP, 2022, 3 (1), pp.35-67. (10.2140/pmp.2022.3.35)
  DOI : 10.2140/pmp.2022.3.35
• Quantum and Semiquantum Pseudometrics and applications
  
  • Golse François
  • Paul Thierry
  Journal of Functional Analysis, Elsevier, 2022. We establish a Kantorovich duality for he pseudometric $\cE_\hb$ introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. \textbf{223} (2017), 57--94], obtained from the usual Monge-Kantorovich distance $\MKd$ between classical densities by quantization of one side of the two densities involved. We show several type of inequalities comparing $\MKd$, $\cE_\hb$ and $MK_\hb$, a full quantum analogue of $\MKd$ introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. \textbf{343} (2016), 165--205], including an up to $\hbar$ triangle inequality for $MK_\hb$. Finally, we show that, when nice optimal Kantorovich potentials exist for $\cE_\hb$, optimal couplings induce classical/quantum optimal transports and the potentials are linked by a semiquantum Legendre type transform.