# Laure Saint-RaymondEcole normale supérieure de Lyon | ENS Lyon · Département de Mathématiques

Laure Saint-Raymond

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93

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Introduction

**Skills and Expertise**

## Publications

Publications (93)

In a previous paper by the authors, a cluster expansion method had been developed to study the fluctuations of the hard sphere dynamics around the Boltzmann equation. This method provides a precise control on the exponential moments of the empirical measure, from which the fluctuating Boltzmann equation and large deviation estimates have been deduc...

In [7], a cluster expansion method has been developed to study the fluctuations of the hard sphere dynamics around the Boltzmann equation. This method provides a precise control on the exponential moments of the empirical measure, from which the fluctuating Boltzmann equation and large deviation estimates have been deduced. The cluster expansion in...

The evolution of a gas can be described by different models depending on the observation scale. A natural question, raised by Hilbert in his sixth problem, is whether these models provide consistent predictions. In particular, for rarefied gases, it is expected that continuum laws of kinetic theory can be obtained directly from molecular dynamics g...

We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by consi...

It has been known since Lanford [19] that the dynamics of a hard sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a duality method coupled with a pruning argument to prove that the covar...

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for s...

We present a mathematical theory of dynamical fluctuations for the hard sphere gas in the Boltzmann-Grad limit. We prove that: (1) fluctuations of the empirical measure from the solution of the Boltzmann equation, scaled with the square root of the average number of particles, converge to a Gaussian process driven by the fluctuating Boltzmann equat...

We develop a rigorous theory of hard-sphere dynamics in the kinetic regime, away from thermal equilibrium. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed by the Boltzmann equation. Deviations from this behaviour are described by dynamical correlations, which can be fully characterized for s...

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the sl...

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the sl...

The density stratification in an incompressible fluid is responsible for the propagation of internal waves. In domains with topography, these waves exhibit interesting features. In particular, numerical and lab experiments show that, in two dimensions, for generic forcing frequencies, these waves concentrate on attractors. The goal of this paper is...

We consider the statistical motion of a convex rigid body in a gas of N smaller (spherical) atoms close to thermodynamic equilibrium. Because the rigid body is much bigger and heavier, it undergoes a lot of collisions leading to small deflections. We prove that its velocity is described, in a suitable limit, by an Ornstein-Uhlenbeck process. The st...

These lecture notes present some challenging problems regarding the multiscale analysis of some systems exhibiting singularities at the macroscopic scale. We are interested namely in shocks for the compressible Euler equations in 1D, vortex sheets for the incompressible Euler equations in 2D, and spatial concentrations for the Boltzmann equation. W...

We derive the linear acoustic and Stokes-Fourier equations as the limiting
dynamics of a system of $N$ hard spheres of diameter $\eps$ in two space
dimensions, when~$N\to \infty$, $\eps \to 0$, $N\eps =\alpha \to \infty$, using
the linearized Boltzmann equation as an intermediate step.Our proof is based on
Lanford's strategy \cite{lanford}, and on...

We review various contributions on the fundamental work of Lanford deriving the Boltzmann equation from hard-sphere dynamics in the low density limit. We focus especially on the assumptions made on the initial data and on how they encode irreversibility. The impossibility to reverse time in the Boltzmann equation (expressed for instance by Boltzman...

We treat hydrodynamic limits of the Vlasov-Maxwell-Boltzmann system for one and two species of particles in a viscous incompressible regime.

In this article, we extend the result of Boltzmann on characterisation of
collision invariants from the case of hard disks to a class of two-dimensional
compact, strictly-convex particles.

We derive the Stokes–Fourier equations in dimension 2 as the limiting dynamics of a system of N hard spheres of diameter ε when , , , using the linearized Boltzmann equation as an intermediate step. Our proof is based on the strategy of Lanford [6], and on the pruning procedure developed in [3] to improve the convergence time. The main novelty here...

Cette note montre comment on peut obtenir le mouvement brownien comme limite hydrodynamique d'un système déterministe de sphères dures quand le nombre de particules N tend vers l'infini et que leur diamètre ε tend vers 0, dans la limite de relaxation rapide (avec un choix d'échelles de temps et d'espace convenable). Comme suggéré par Hilbert dans s...

This paper presents in a synthetic way some recent advances on hydrodynamic limits of the Boltzmann equation. It aims at bringing a new light to these results by placing them in the more general framework of asymptotic expansions of Chapman-Enskog type, and by discussing especially the issues of regularity and truncation.

We provide a rigorous derivation of the brownian motion as the hydrodynamic
limit of a deterministic system of hard-spheres as the number of particles $N$
goes to infinity and their diameter $\varepsilon$ simultaneously goes to $0,$
in the fast relaxation limit $N\varepsilon^{d-1}\to \infty $ (with a suitable
scaling of the observation time and len...

Ohm’s law states that the current density j at a given location in a plasma is proportional to the electric field E at that location. We propose here a rigorous derivation of this law (and of some extensions of it) starting from a microscopic model consisting of two species of charged particles interacting both via the self-consistent electromagnet...

Understanding the mechanisms governing the ocean circulation is a challenge for geophysicists, but also for mathematicians who have to develop tools to analyze these complex models (involving a large number of time and space scales).
A particularly important mechanism for the large-scale circulation is the boundary layer phenomenon, which accounts...

Boltzmann brought a fundamental contribution to the understanding of the notion of entropy, by giving a microscopic formulation of the second principle of thermodynamics. His ingenious idea, motivated by the works of his contemporaries on the atomic nature of matter, consists of describing gases as huge systems of identical and indistinguishable el...

We provide a rigorous derivation of the brownian motion as the hydrodynamic
limit of systems of hard-spheres as the number of particles $N$ goes to
infinity and their diameter $\varepsilon$ simultaneously goes to 0, in the fast
relaxation limit $N \varepsilon^{d-1} \to \infty$ (with a suitable scaling of
the observation time and length). As suggest...

We establish the existence of renormalized solutions of the Vlasov-Maxwell-Boltzmann system with a defect measure in the presence of long-range interactions. We also present a control of the defect measure by the entropy dissipation only, which turns out to be crucial in the study of hydrodynamic limits. Nous établissons l'existence de solutions re...

We provide a rigorous derivation of the Boltzmann equation as the mesoscopic
limit of systems of hard spheres, or Newtonian particles interacting via a
short-range potential, as the number of particles $N$ goes to infinity and the
characteristic length of interaction $\e$ simultaneously goes to $0,$ in the
Boltzmann-Grad scaling $N \e^{d-1} \equiv...

We fill in all details in the proof of Lanford's theorem. This provides a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of Newtonian particles interacting via a short-range potential, as the number of particles $N$ goes to infinity and the characteristic length of interaction $\varepsilon$ simultaneously goes to $...

This paper is concerned with a complete asymptotic analysis as $\nu \to 0$ of
the Munk equation $\d_x\psi-\nu \Delta^2 \psi= \tau$ in a domain $\Omega\subset
\mathbf R^2$, supplemented with homogeneous boundary conditions for $\psi $ and
$\partial_n \psi$. This equation is a simple model for the circulation of
currents in closed basins. A crude ana...

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L1 satisfying some appropriate transport relationv⋅∇xfλ=(1−Δx)β2(1−Δv)α2gλ may be i...

In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around an inhomogeneous (nonzonal) stationary profile. This extends the study (Cheverry et al. in Semiclassical and spectral analysis of oceanic waves, Duke Math. J., accepted), where the profile was assumed to be zonal only and w...

This article is concerned with an oceanographic model describing the asymptotic behaviour of a rapidly rotating and incompressible fluid with an inhomogeneous rotation vector; the motion takes place in a thin layer. We first exhibit a stationary solution of the system which consists of an interior part and a boundary layer part. The spatial variati...

We establish improved hypoelliptic estimates on the solutions of kinetic
transport equations, using a suitable decomposition of the phase space. Our
main result shows that the relative compactness in all variables of a bounded
family $f_\lambda(x,v)\in L^p$ satisfying some appropriate transport relation
$$v\cdot\nabla_x f_\lambda =
(1-\Delta_x)^\fr...

In this work we prove that the shallow water flow, subject to strong wind
forcing and linearized around an adequate stationary profile, develops for
large times closed trajectories due to the propagation of Rossby waves, while
Poincar\'e waves are shown to disperse. The methods used in this paper involve
semi-classical analysis and dynamical system...

The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier–Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore comp...

The present paper is devoted to the study of the incompressible Euler limit of the Boltzmann equation via the relative entropy method. It extends the convergence result for well-prepared initial data obtained by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech....

Cette Note est consacrée à la description des effets d'un forçage surfacique, par exemple dû au vent, sur des fluides en rotation rapide dont l'évolution est régie par une équation linéaire. La particularité de l'analyse menée ici réside dans le caractère fortement oscillant en temps du vent, qui peut alors entrer en résonance avec la force de Cori...

La version librement accessible est celle du working paper intitulé "Mathematical study of resonant wind-driven oceanic motions" We are interested here in describing the linear response of a highly rotating fluid to some surface stress tensor, which admits fast time oscillations and may be resonant with the Coriolis force. In addition to the usual...

Waves associated to large scale oceanic motions are gravity waves (Poincar\'e waves which disperse fast) and quasigeostrophic waves (Rossby waves). In this Note, we show by semiclassical arguments, that Rossby waves can be trapped and we characterize the corresponding initial conditions.

The aim of this chapter is to describe the state of the art about the incompressible Euler limit of the Boltzmann equation, which is not so complete as the incompresible Navier-Stokes limit presented in the previous chapter.
Due to the lack of regularity estimates for inviscid incompressible models, the convergence results describing the incompress...

In all existing works on the subject, the general strategy to derive hydrodynamic limits is to proceed by analogy, that is to recognize the structure of the expected limiting hydrodynamic model in the corresponding scaled Boltzmann equation. This explains for instance why all hydrodynamic limits are not equally understood.
The aim of this chapter i...

At the present time, the incompressible Navier-Stokes limit is the only hydrodynamic asymptotics of the Boltzmann equation for which an optimal convergence result is known (and for which we are actually able to implement all the mathematical tools presented in the previous chapter). By “optimal”, we mean here that this convergence result
holds glob...

The last chapter of this survey is devoted to the compressible Euler limit, and is actually a series of remarks and open problems more than a compendium of results.
We will discuss some perspectives regarding the mathematical treatment of this asymptotics. Slight adaptations of the modulated entropy method presented in the previous chapter should...

The kinetic theory, introduced by Boltzmann at the end of the nineteenth century, provides a description of gases at an intermediate level between the hydrodynamic description which does not allow to take into account phenomena far from thermodynamic equilibrium, and the atomistic description which is often too complex. For a detailed presentation...

This paper is devoted to the study of large-scale oceanic motions forced by some wind at the surface and braked by solid friction at the bottom. As the effect of the Coriolis force is dominant, this is a typical singular penalization problem set in a bounded domain, which has been already studied by filtering methods in [N. Masmoudi, Comm. Pure App...

The Boltzmann equation and its formal hydrodynamic limits.- Mathematical tools for the derivation of hydrodynamic limits.- The incompressible Navier-Stokes limit.- The incompressible Euler limit.- The compressible Euler limit.

The incompressible Euler equations are obtained as a weak asymptotics of the Boltzmann equation in the fast relaxation limit (the Knudsen number goes to zero), when both the Mach number (defined as the ratio between the bulk velocity and the speed of sound) and the inverse Reynolds number (which measures the viscosity of the fluid) go to zero.The e...

We are interested here in describing the linear response of the ocean to
some wind forcing, which admits fast time oscillations and may be
resonant with the Coriolis force. In addition to the usual Ekman layer,
we exhibit another - much larger - boundary layer, and some global
vertical profile. That means in particular that the wind effect is no
lo...

This chapter discusses the influence of the Earth's rotation on geophysical flows, both from a physical and a mathematical point of view. It gathers, from the physical literature, the main pieces of information concerning the physical understanding of oceanic and atmospheric flows. For the scales considered—that is, on domains extending over many t...

The sixth problem proposed by Hilbert, in the occasion of the International Congress of Mathematicians held in Paris in 1900,
asks for a global understanding of the gas dynamics. For a perfect gas, the kinetic equation of Boltzmann provides a suitable
model of evolution for the statistical distribution of particles. Hydrodynamic models are obtained...

The discrete time Navier-Stokes equations in whole space are obtained as a fluid limit for the properly scaled discrete time BGK equation by the method developed by Bardos, Golse and Levermore to study hydrodynamic limits of the Boltzmann equation (Bardos, C.; Golse, F.; Levermore, C.D. Fluid Dynamic Limits of Kinetic Equations II: Convergence Proo...

We are interested in a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a physical introduction to the mo...

We are interested in the life span and the asymptotic behavior of the solutions to a system governing the motion of a pressureless gas that is submitted to a strong, inhomogeneous magnetic field ε-1B(x) of variable amplitude but fixed direction; this is a first step in the direction of the study of rotating Euler equations. This leads to the study...

We present here a series of works which aims at describing geophysical flows in the equatorial zone, taking into account the dominating influence of the earth rotation. We actually proceed by successive approximations computing for each model the response of the fluid to the strong Coriolis penalisation. The main diculty is due to the spatial varia...

The mathematical contributions by X.G. Lu (J. Statist. Phys. 98 (5/6) (2000) 1335–1394) and by M. Escobedo et al. (Electronic J. Differential Equations, Monograph 4 (2003)) presented in this Note constitute the first stage in the understanding of the superfluid dynamics, especially of the Bose–Einstein condensation, by means of kinetic models. The...

The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R
3. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L
1 topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This c...

We are interested in the life span and the asymptotic behaviour of the solutions to a system governing the motion of a pressureless gas, submitted to a strong, inhomogeneous magnetic field $ \e^{-1} B(x)$, of variable amplitude but fixed direction -- this is a first step in the direction of the study of rotating Euler equations. This leads to the s...

We prove that the renormalized solutions of the Boltzmann equation considered in a bounded domain with different types of (kinetic) boundary conditions converge to the Stokes-Fourier system with different types of (fluid) boundary conditions when the main free path goes to zero. This extends the work of F. Golse and D. Levermore [9] to the case of...

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray--type solutions towards a vector field which satisfies the usual 2D Navier--Stokes equati...

We give here a complete derivation of the Navier-Stokes-Fourier equations from a model collisional kinetic equation, the BGK model. Though physically unrealistic, this model shares some common features with more classical models such as the Boltzmann equation. Then the program developed by Bardos, Golse and Levermore (Fluid dynamic limits of kineti...

We consider here the problem of deriving rigorously, for well-prepared initial data and without any additional assumption, dissipative or smooth solutions of the incompressible Euler equations from renormalized solutions of the Boltzmann equation. This completes the partial results obtained by Golse [B. Perthame and L. Desvillettes eds., Series in...

Consider the plane motion of a plasma subject to a magnetic field orthogonal to the plane. The equation on the density obtained in the gyrokinetic limit (as |B| tends to infinity), the so-called drift equation, lets appear a defect measure μ corresponding to a possible lack of energy at large velocities [F. Golse, L. Saint-Raymond, The guiding cent...

Resume En utilisant le concept de “solutions dissipatives” introduit par Lions [12], on obtient les equations d'Euler des fluides incompressibles comme limite hydrodynamique de l'equation de BGK convenablement adimensionnee. Ce resultat de stabilite repose sur des estimations fines de l'entropie et de la dissipation d'entropie, et sur des propriete...

A new result of L1-compactness for velocity averages of solutions to the transport equation is stated and proved in this Note. This result, proved by a new interpolation argument, extends to the case of any space dimension Lemma 8 of Golse–Lions–Perthame–Sentis [J. Funct. Anal. 76 (1988) 110–125], proved there in space dimension 1 only. This is a k...

Let E be a topological space, E' a metric space and (S) a system of evolution equations admitting a solution in E' for all initial data in E and stable with respect to initial data on E. We prove that the set of initial data such that (S) admits a unique solution is a Gdelta subset of E. In particular, if the uniqueness property is satisfied on a d...

Appropriately scaled families of DiPerna-Lions renormalized solutions of the Boltzmann equation are shown to have fluctuations whose limit points (in the weak L 1 topology) are governed by a Leray solution of the limiting Navier-Stokes equations. This completes the arguments in C. Bardos, F. Golse and C. Levermore [Commun. Pure Appl. Math. 46, 667–...

Consider a plasma in a strong constant magnetic field with self-consistent electric field. We present here the formal derivation that leads to the so-called guiding center approximation, and justify it in the case of a well-prepared initial density of particles. More precisely, we prove that the motion of the particles can be approximatively decomp...

Consider the plane motion of a plasma subject to a magnetic field orthogonal to the plane. The equation on the density obtained in the gyrokinetic limit (as |B| tends to infinity), the so-called drift equation, lets appear a defect measure μ corresponding to a possible lack of energy at high velocities [7]. In the present Note, it is proved that μ...

: Using the stability results of Bressan & Colombo [BC] for strictly hyperbolic 2 � 2 systems in one space dimension, we prove that the solutions of isentropic and non-isentropic
Euler equations in one space dimension with the respective initial data (ρ0, u
0) and (ρ0, u
0, &\theta;0=ρ0
γ− 1) remain close as soon as the total variation of (ρ0, u...

This paper establishes various asymptotic limits of the Vlasov–Poisson equation with strong external magnetic field, some of which were announced in [14]. The so-called “guiding center approximation” is proved in the 2D case with a constant magnetic field orthogonal to the plane of motion, in various situations (noncollisional or weakly collisional...

Consider the kinetic wave-particle collisions model, obtained by Degond and Peyrard in [4], on a periodic domain. Following Bardos, Golse and Levermore in [2], we prove the stability of Maxwellians with respect to small perturbations. Then, as the system has an infinite number of entropies, we can derive hydrodynamic limits. According to the scalin...

The macroscopic dynamics of a kinetic equation involving a model wave–particle collision operator of plasma physics (see Degond and Peyrard, C. R. Acad. Sci. Paris 323, 1996) is investigated. Using relative entropy estimates about an absolute Maxwellian, it is shown, as in Bardos, Golse and Levermore (Comm. Pure Appl. Math. 46(5), 1993), that any p...

Consider the plane motion of a gas of charged particles subject to the self-consistent electric field and to a constant external magnetic field orthogonal to the plane of motion. As the intensity of the magnetic field tends to infinity, the asymptotic behavior, in the. long time limit, of the average density of particles obeys the vorticity formula...

Consider the plane motion of a gas of charged particles subject to the self-consistent electric field and to a constant external magnetic field orthogonal to the plane of motion. As the intensity of the magnetic field tends to infinity, the asymptotic behavior, in the long time limit, of the average density of particles obeys the vorticity formulat...

Using the stability results of Bressan and Colombo [1]for strictly hyperbolic 2×2 systems in one space dimension, we prove that the solutions of isentropic and nonisentropic Euler equations in one space dimension with the respective initial datas (ρ0,u0) and (ρ0,u0,θ0 = ρ0γ-1) remain close as soon as the total variation of (ρ0,u0) is sufficiently s...

## Projects

Project (1)

Rigorous mathematical study of phenomena of reflection and interaction of internal gravity waves for application in oceanography and in connection with wave turbulence.