Laure Saint-Raymond

Educational background

- Former student of the Ecole Normale Supérieure de Paris (1994-1998)

- Master of Applied Mathematics, University Paris VI (1996)

- Master of Plasma Physics, University Versailles Saint-Quentin (1996)

- PhD in Applied Mathematics, University Paris VII (2000)

- Accreditation to supervise research, University Paris VII (2002)

Professional experience
- Teaching assistant (Agrégée préparatrice), Ecole Normale Supérieure de Paris,
Département de Mathématiques et Applications (1999-2000)

- Research Scientist, Centre National de la Recherche Scientifique,
Laboratoire Jacques-Louis Lions, University Paris VI (2000-2002)

- Full Professor, University Paris VI Pierre et Marie Curie
Laboratoire Jacques-Louis Lions (2002-2007).

- Full Professor, Département de Mathématiques et Applications,
Ecole Normale Supérieure (since September 2007).

- Head of the team « Analysis» (2007-2009)

- Director of Studies (2007-2011)

- Deputy director of the Department of Mathematics (since janvier 2011)

Academic Awards and Honors
- Louis Armand Prize, *French Academy of Sciences *(2003).

- Claude-Antoine Peccot Award, *Collège de France (2004).*

- Pius XI Gold Medal, *Pontificia Academia Scientarium (2004).*

- "Analysis of Partial Differential Equations" Prize with François Golse,
*Society for Industrial and Applied Mathematics (2006).*

- Prize for Young Scientist with Isabelle Gallagher,* Ville de Paris (2006).*

- EMS Prize, *European Mathematical Society (2008).*

- Ruth Lyttle Satter Prize, *American Mathematical Society (2009).*

- Lecture "Un texte, un mathématicien", *Bibliothèque Nationale de France (2010).*

- Invitation to the Abel Symposium, *Norwegian Mathematical Society (2010).*

- Prize of "Annales de l'Institut Henri Poincaré C : analyse non linéaire" (2011).

- Irène Joliot-Curie Prize, *French ministry of Research and Education & French Academy of Sciences (2011).*

- Invitation to the International Congress of Mathematicians, Seoul (2014).

- Invitation to the International Congress of Industrial and Applied Mathematics, Beijing (2015).

Some scientific duties

- PhD advisor of Delphine Salort (2005), Thibaut Allemand (2010), Daniel Han Kwan (2011).

- Member of editorial boards of « Kinetic and Related Models » , « Archive for Rational Mechanics and Analysis », « Bulletin et Mémoires de la Société Mathématique de France »

- Member of the French National Council of Universities, section 26 (since 2011).

List of 12 major publications

[1] The Vlasov-Poisson system with strong magnetic field, with F. Golse
*J. Math. Pures Appl.* 78, 791--817 (1999).

[2] Convergence of solutions to the Boltzmann equation in the incompressible Euler limit,
*Arch. Ration. Mech. Anal. *166, 47--80 (2003).

[3] From the Boltzmann equation to the Stokes-Fourier system in a bounded domain, with N. Masmoudi
*Comm. Pure Appl. Math.* 56, 1263--1293 (2003).

[4] The Navier-Stokes Limit of the Boltzmann Equation for Bounded Collision Kernels, with F. Golse
*Inventiones Math.* 155, 81--161 (2004).

[5] Hydrodynamic limits of the Boltzmann Equation,
*Lecture Notes in Mathematics, Springer *1971, 1--195 (2009).

[6] Mathematical study of the betaplane model, with I. Gallagher
*Mémoires de la SMF* 107, 1--116 (2006).

[7] Mathematical study of rotating fluids with resonant surface stress, with A.-L. Dalibard
*J. Differential Equations* 246 , 2304--2354 (2009).

[8] Trapping Rossby waves by wind forcing, with C. Cheverry, I. Gallagher et T. Paul
*Duke J. Math.* 161, 845-892 (2012).

[9] Compactness in kinetic transport equations and hypoellipticity, with D. Arsenio.
*J. Funct. Anal.* 261, 3044-3098 (2011).

[10] From Newton to Boltzmann : hard spheres and short-range potentials, with I. Gallagher et B. Texier.
*To appear in Zurich Lecture Notes in Advanced Mathematics, EMS Publications* (2013).

[11] The Brownian motion as the limit of a deterministic system of hard spheres, with T. Bodineau and I. Gallagher. *Submitted* (2013).

[12] From the Vlasov-Maxwell-Boltzmann system to the incompressible viscous MHD, with D. Arsenio.
*In preparation* (2013).

Presentation of the research interests

My contributions are mainly concerned with **problems of asymptotic analysis** related to the dynamics of gases, plasmas or fluids. The aim is to show that there are continuous transitions between the different levels of modelling, and that it is therefore relevant to use simplified models in some physical regimes (characterized by dimensionless parameters such as the Knudsen number, the Mach number or the Rossby number…).

In chronological order, I have taken an interest in
(a) *gyrokinetic limits* [1] which describe the behaviour of plasmas in strong magnetic fields (for instance plasmas in tokamaks) ;

(b) *hydrodynamic limits of the Boltzmann equation* [2,3,4,5] which lead to fluid models such as the Euler or Navier-Stokes equations in strongly collisional regimes (fast relaxation towards local thermodynamic equilibrium);

(c) *fast rotating fluids*, especially to describe large-scale oceanic motions for which the Coriolis force has a dominating influence [6] (which may be combined for instance to boundary effects to give rise to boundary layer phenomena [7], or to the convection by macroscopic oceanic currents to account for the formation of stationary oceanic [8]) ;

(d) *magneto-hydrodynamic limits of the Vlasov-Maxwell-Boltzmann system* [12] which lead to a large variety of fluid models for strongly collisional plasmas (and to many mathematical challenges, in particular to subtle problems of functional analysis [9]) ;

(e) *the derivation of the Boltzmann equation* starting from system of particles in the low density limit (Boltzmann-Grad scaling) [10,11], which asks fundamental questions, especially about the propagation of chaos and the appearance of irreversibility.

Problems (b) and (e) are two important ramifications of the sixth problem asked by Hilbert in the occasion of the International Congress of Mathematicians held in Paris in 1900, concerning the axiomatization of physics:
« *Le livre de M. Boltzmann sur les Principes de la Mécanique nous incite à établir et à discuter du point de vue mathématique d'une manière complète et rigoureuse les méthodes basées sur l'idée de passage à la limite, et qui de la conception atomique nous conduisent aux lois du mouvement des continua.* »

I will describe here (briefly) only the contributions related to one of these research directions:
**Concerning hydrodynamic limits of the Boltzmann equation**, my work follows the program proposed by C. Bardos, F. Golse and D. Levermore, which consists in deriving the equations of fluid mechanics from the renormalized solutions to the Boltzmann equation built by DiPerna and Lions. These solutions, which exist globally without any restriction on the size or on the regularity of the initial data (since their stability is controlled only by the energy and entropy bounds), are defined in a very weak sense, which guarantees neither the uniqueness, nor the local conservations of momentum and energy. The fact that they have the expected behaviour in the fast relaxation limit is therefore an important argument to justify their relevance.

The most significant result is probably the one obtained in collaboration with F. Golse in [4]. It establishes the convergence towards the Leray solutions of the incompressible Navier-Stokes equations in the suitable regime. It relies on a moment method with a fine control of conservation defects, together with compactness properties using the hypoelliptic structure of the Boltzmann equation. This result is optimal insofar as it is global in time and does not require any assumption on the initial data except the natural entropy bound giving the scaling. It can be further extended to spatial domains with boundary since one can determine [3,5] the asymptotic form of Maxwell’s reflection condition depending on the size of the accommodation coefficient.

The inviscid regime is a little bit less understood since there is no good theory of global solutions for the Euler equations (at least in 3 space dimensions). The alternative is to consider either strong solutions which are defined only locally, or « dissipative solutions » defined only by some stability inequality (but coincide with the smooth solution as long as it is Lipschitz). The contribution [2] shows that one can obtain rigorous convergence results in both settings, provided that the initial datum is well-prepared, and the reflection on the boundary is purely specular. It uses some relative entropy, together with fine estimates of the flux terms by the entropy dissipation. Refinements of this method actually allow accounting for acoustic waves and initial relaxation layers in the case of ill-prepared initial data [5].

The other part of Hilbert’s program consists in **deriving the Boltzmann equation** from the dynamics of large systems of interacting particles, upstream from the problem of hydrodynamic limits. This represents for me a more recent research direction, partially motivated by works on weak turbulence.

At the present time, the most important result is the theorem by Lanford in the case of hard spheres (and its extension by King to the case of compactly supported potentials of interaction). This result states that the Boltzmann equation provide a good approximation of the microscopic dynamics of an average particle in the Boltzmann-Grad limit, outside from a set of initial configurations of vanishing measure. More precisely, the pseudo-trajectories of the Boltzmann equation almost coincide with the trajectories of particles, as long as there is no recollision. Our joint work [10] with I. Gallagher and B. Texier has consisted in controlling in a precise way the recollision mechanism to give a complete proof of these theorems.

The main drawback of these results about the low density limit is the very short time on which they are valid, which corresponds to less than one collision per particle in average. The main difficulty is to get a priori bounds on the size of collision trees. In a very recent work in collaboration with T. Bodineau and I. Gallagher [11], we have managed to obtain such bounds globally in time by considering some fluctuation regime (around an equilibrium), and to deduce a first result of hydrodynamic limits starting directly from the deterministic system of particles.

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