Laure Saint-Raymond is a mathematician working in partial differential equations, fluid mechanics, and statistical mechanics. She is a professor at l’École Normale Supérieure de Paris and the Université Pierre et Marie Curie (also known as Paris VI). In 2013, she became the youngest member ever elected to the French Academy of Sciences, in the Mechanics and Computer Science section, where several other top-notch mathematicians are members. This feat generated some publicity, in *Le Monde* and from the Université Pierre et Marie Curie, for instance. She was featured in an interview on universcience.tv. More recently, *La Recherche *had an article about Saint-Raymond and her colleagues Thierry Bodineau and Isabelle Gallagher when they won *le prix La Recherche mention mathématiques. * The prize was for their impressive work on describing the Brownian motion as a limit^{*} of a deterministic system of hard-spheres.

You can trace Saint-Raymond’s work in MathSciNet, with its persistent themes, such as the Boltzmann equation and limiting phenomena in statistical mechanics. Her work has inspired several long reviews in MathSciNet, including MR2683475, MR1952079, and MR3157048. In an earlier post, I mentioned that Cédric Villani’s review of her paper with François Golse on the Navier-Stokes limit of the Boltzmann equation “verges on being a short course.” The text of Nader Masmoudi’s long review of her volume in *Lecture Notes in Mathematics* is below.

Meanwhile, IMPA has videos of Saint-Raymond giving a mini-course on some of her work: Class 1, Class 2, and Class 3.

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^{*} Let the number of spheres go to infinity and let their diameters go to zero.

MR2683475

Saint-Raymond, Laure (F-ENS-DAM)

Hydrodynamic limits of the Boltzmann equation.

Lecture Notes in Mathematics, 1971. Springer-Verlag, Berlin, 2009. xii+188 pp. ISBN: 978-3-540-92846-1

From a physical point of view, we expect that a gas can be described by a fluid equation when the mean free path (Knudsen number) goes to zero. In his sixth problem, on the occasion of the International Congress of Mathematicians held in Paris in 1900, Hilbert asked for a full mathematical justification of these derivations. During the last two decades this problem has attracted a lot of interest.

Let us first give some background about this problem (see Chapters 1 and 2 in the book). The molecules of a gas can be modeled by spheres that move according to the laws of classical mechanics. However, due to the enormous number of molecules to be considered, it is hopeless to describe the state of the gas by giving the position and velocity of each individual particle. Hence, we must use some statistics and instead of giving the position and velocity of each particle, we specify the density of particles $F(x,v)$ at each point $x$ and velocity $v$. Under some assumptions (rarefied gas, etc.), it was proved by Boltzmann (and Lanford for a rigorous proof in the hard sphere case) that this density is governed by the Boltzmann equation (B): $$ \partial_t F + v\cdot \nabla_{x} F = B(F,F). $$ To derive fluid equations from the Boltzmann equation, one has to introduce several dimensionless parameters: the Knudsen number ${\rm Kn}$ (which is related to the mean free path), the Mach number ${\rm Ma}$ and the Strouhal number ${\rm St}$ (which is a time rescaling). With these parameters, one can rewrite the Boltzmann equation as $$ {\rm St}\cdot \partial_t F + v\cdot \nabla_{x} F = \frac1{\rm Kn} B(F,F) $$ with $F = M (1 + {\rm Ma}\cdot f)$ where $M$ is a fixed Maxwellian. It is worth noting that the Reynolds number ${\rm Re}$ is completely determined by the relation ${\rm Ma} = {\rm Kn}\cdot {\rm Re}$. Several fluid equations can be derived that depend on these dimensionless parameters: Compressible Euler system, acoustic waves, Incompressible Navier-Stokes-Fourier system, Stokes-Fourier system, Incompressible Euler system, etc. There are several approaches to deal with this problem: the weak compactness method initiated by C. Bardos, F. Golse and C. D. Levermore, asymptotic expansions [see A. De Masi, R. Esposito and J. L. Lebowitz, Comm. Pure Appl. Math. 42 (1989), no. 8, 1189–1214; MR1029125], the energy method [Y. Guo, Comm. Pure Appl. Math. 59 (2006), no. 5, 626–687; MR2172804; erratum, Comm. Pure Appl. Math. 60 (2007), no. 2, 291–293; MR2275331], etc.

This book gives an overview of some of these results and mainly the derivation of the Incompressible Navier-Stokes [F. Golse and L. Saint-Raymond, Invent. Math. 155 (2004), no. 1, 81–161; MR2025302] and Incompressible Euler [L. Saint-Raymond, Arch. Ration. Mech. Anal. 166 (2003), no. 1, 47–80; MR1952079] systems.

After the construction of the renormalized solutions to the Boltzmann equation by R. J. DiPerna and P.-L. Lions [Ann. of Math. (2) 130 (1989), no. 2, 321–366; MR1014927], there was a program initiated by Bardos, Golse and Levermore [J. Statist. Phys. 63 (1991), no. 1-2, 323–344; MR1115587; Comm. Pure Appl. Math. 46 (1993), no. 5, 667–753; MR1213991] to derive incompressible models from the Boltzmann equation. In particular the main objective was to recover the Leray [J. Leray, Acta Math. 63 (1934), no. 1, 193–248; MR1555394; JFM 60.0726.05] global weak solutions of the incompressible Navier-Stokes system starting from the DiPerna-Lions solutions.

There were five main assumptions in their first work:

(1) Because of a problem coming from the rapid time-oscillations of acoustic waves, only the time independent case was considered.

(2) Local conservation laws were assumed even though these are not known to hold for the renormalized solutions.

(3) The lack of high-order moment estimates required the restriction of the discussion to the momentum equation and no heat equation was derived.

(4) A key equi-integrability estimate was assumed on the solutions of the Boltzmann equation. This is due to the fact that the natural space for the Boltzmann equation is $L\log L$ whereas for the Navier-Stokes system the natural space is $L^2$.

(5) Due to a technical estimate for the inverse of the linearized Boltzmann kernel, only very particular collision kernels were considered.

These five assumptions have been removed one by one in the past two decades:

(1) In [P.-L. Lions and N. Masmoudi, Arch. Ration. Mech. Anal. 158 (2001), no. 3, 173–193, 195–211; MR1842343] the time-oscillating acoustic waves were treated using a compensated compactness type argument coming from the compressible-incompressible limit [P.-L. Lions and N. Masmoudi, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 5, 387–392; MR1710123].

(2)–(3) In [P.-L. Lions and N. Masmoudi, op. cit., MR1842343 (pp. 195–211)], the assumption on the local conservation in the momentum equation was removed, and in [Comm. Pure Appl. Math. 55 (2002), no. 3, 336–393; MR1866367], Golse and Levermore were able to derive the Stokes-Fourier system. The main idea is to recover the moment conservation laws at the limit.

(4) The main breakthrough of [F. Golse and L. Saint-Raymond, op. cit.; MR2025302] was a new $L^1$ averaging lemma that allows one to prove the key equi-integrability estimate.

(5) In [F. Golse and L. Saint-Raymond, J. Math. Pures Appl. (9) 91 (2009), no. 5, 508–552; MR2517786] the result was extended to hard cutoff potentials satisfying Grad’s cutoff assumption and in [C. D. Levermore and N. Masmoudi, Arch. Ration. Mech. Anal. 196 (2010), no. 3, 753–809; MR2644440] it was also extended to both hard and soft potentials. Another important extension was done by D. Arsénio [“From Boltzmann’s equation to the incompressible Navier-Stokes-Fourier system with long-range interactions”, Arch. Ration. Mech. Anal., to appear], who treated the non-cutoff case.

We also note that the case where the problem is considered in a bounded domain was treated in [N. Masmoudi and L. Saint-Raymond, Comm. Pure Appl. Math. 56 (2003), no. 9, 1263–1293; MR1980855] where Navier and Dirichlet boundary conditions were derived starting from the Maxwell boundary condition.

Chapter 3 of this book presents the main mathematical tools used in dealing with the hydrodynamic limit. In particular several estimates coming from the entropy, the entropy dissipation and Darrozès-Guiraud information are presented. Also the new $L^1$averaging lemma is proved.

Chapter 4 deals with the incompressible Navier-Stokes limit using the weak compactness method. In particular the author shows how to combine the ideas from [N. Masmoudi and L. Saint-Raymond, op. cit.; MR1980855] to treat the case of a bounded domain with Maxwell boundary conditions.

Chapter 5 deals with the incompressible Euler limit using the relative entropy method [L. Saint-Raymond, op. cit.; MR1952079].

Finally, Chapter 6 gives a survey of the known results about the compressible Euler limit. It is worth noting that if we are interested in starting from the renormalized solutions then none of the methods used in the incompressible case can be adapted. The author gives some open problems and perspectives.

Reviewed by Nader Masmoudi