Agnel Joseph

Garima Agarwal

Swapnil Mahajan

Jean-Christophe Gelly

Lakshmipuram S Swapna

Bernard Offmann

Frédéric Cadet

Aurélie Bornot

Manoj Tyagi

Hélène Valadié

Agnel Praveen

Ric Cadet

Bohdan Schneider

Catherine Etchebest

Narayanaswamy Srinivasan

Alexandre G De Brevern
email: <alexandre.debrevern@univ-paris-diderot.fr

A short survey on

Keywords: protein structures, biochemistry, amino acids, secondary structures, propensities, structural alphabet, structure prediction, structural superimposition, mutation, binding site, Bayes theorem, Support Vector Machines

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Protein structures have been classically described in two regular states ( -helix and -strand) and the remaining unassigned regions as

α β
an irregular state (coil), this state correspond to a large number of diverse conformations. Nonetheless, the use of only three states oversimplifies the description of protein structures. A detailed description for 50 of the residues classified as coils is missed even when % they encompass repeating local structure. Description of local protein structures have hence focused on the elaboration of complete sets of small prototypes or structural alphabets (SAs), that help to approximate every part of the protein backbone ( ).

" "

Offmann , et al., 2007 Designing a structural alphabet requires identification of a set of average recurrent local protein structures that (efficiently) approximates every part of known structures. As each residue is associated to one of these prototypes, the whole 3D protein structure can be translated into a series of prototypes (letters) in 1D, as the sequence of prototypes.

gives an example of encoding of a protein structures with a Structural Alphabet. The N-terminal extremity of Figure 1 Aspergillus acid phosphatase ( ) chain B is shown. To each residue, a local protein structure prototype was associated. niger [START_REF] Kostrewa | Crystal structure of Aspergillus niger pH 2.5 acid phosphatase at 2.4 A resolution[END_REF] Thus, the coil region could be precisely described as a succession of small protein prototypes instead of a succession of identical states.

Protein Blocks

Secondary structure assignments are widely used to analyze protein structures. However, it often gives a coarse description of 3D protein structures, with about half of the residues being assigned to an undefined state ( ). Moreover, the Bornot and de Brevern, 2006 structural diversity observed in -helices and -strands, is hidden. Indeed, -helices are frequently not linear, and are either curved (58 )

α β α %
or kinked (17 ) ( ). The absence of secondary structure assignment for a significant proportion of the residues has led % [START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF] to the development of local protein structure libraries that are able to approximate all (or almost all) of the local protein structures without using classical secondary structures. These libraries yielded prototypes that are representative of local folds found in proteins. The complete set of local structure prototypes defines a structural alphabet (

). [START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF] Ten years ago, Pr. Serge Hazout developed a novel structural alphabet with two specific goals ( ): (i) to obtain a de Brevern, et al., 2000 good local structure approximation and (ii) to predict local structures from sequence. Fragments that are five residues in length were coded in terms of the / dihedral angles. A Root Mean Square Deviation on Angle (RMSDA) score was used to quantify the structural φ ϕ difference among the fragments (

). Using an unsupervised cluster analyser related to Self Organizing Maps [START_REF] Rangwala | Local structural motifs of protein backbones are classified by self-organizing neural networks[END_REF] (SOM ( ; )), a three step training process was carried out. The first step involved learning of structural [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF][START_REF] Kohonen | Self-Organizing Maps[END_REF] difference of fragments in terms of RMSDA and in the second step the transition probability (probability of transition from one fragment to another in a sequence) was also considered along with the RMSDA, ., in a similar way to Markov model ( ). In the i.e [START_REF] Rabiner | A tutorial on hidden Markov models and selected application in speech recognition[END_REF] third step, the constraint based on transition probability was removed. Optimal prototypes were identified by considering both the structural approximation and the prediction rate. A set of 16 prototypes called Protein Blocks (PBs), represented as average dihedral vectors, was obtained at the end of this process ( ). de Brevern, et al., 2000 ). Our earlier work [START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF]de Brevern, et al., 2000de Brevern, et al., 2002Etchebest, et al., 2005 on PBs have shown that PBs are effective in describing and predicting conformations of long fragments ( ;Benros, et al., 2006 Benros, et ;;;;;al., 2009Bornot, et al., 2009[START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF]de Brevern and Hazout, 2001de Brevern and Hazout, 2003 de Brevern, et al., ) and short loops ( ; ; ), analyzing protein contacts ( ), in 2002 Fourrier, et al., 2004Tyagi, et al., 2009Tyagi, et al., 2009Faure, et al., 2008 building a transmembrane protein ( ; ), and in defining a reduced amino acid alphabet to aid de [START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF]de Brevern, et al., 2009 design of mutations (

). This reduced amino acid alphabet was recently proved suitable for predicting protein Etchebest, et al., 2007 families or sub-families and secretory proteins of ( ;

). We have also used protein blocks P. falciparum Zuo andLi, 2009 Zuo andLi, 2009 to superimpose and to compare protein structures ( ; ;

). [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF]Tyagi, et al., 2006Tyagi, et al., 2006 Other laboratories have taken advantage of PBs to reconstruct globular protein structures ( ), design peptides ( Dong, et al., 2007 ) and to define binding site signatures (

). Novel prediction methodologies ( ; [START_REF] Thomas | Prediction of peptide structure: how far are we?[END_REF]Dudev and Lim, 2007Li, et al., 2009 ; ) and fragment-based local statistical potentials ( ) were also [START_REF] Rangwala | Local structural motifs of protein backbones are classified by self-organizing neural networks[END_REF][START_REF] Zimmermann | LOCUSTRA: accurate prediction of local protein structure using a two-layer support vector machine approach[END_REF]Li, et al., 2009 developed. The features of this alphabet have been compared by Karchin et al. ( ) with those of 8 other structural [START_REF] Karchin | Hidden Markov models that use predicted local structure for fold recognition: alphabets of backbone geometry[END_REF] alphabets showing that our PB alphabet is highly informative, with the best predictive ability of those tested. Among the available SAs, it is the most widely used SA today.

Applications Binding site signature

PBs enable the detection of structural similarity between proteins with excellent efficiency. Dudev, Lim and co-workers, (Dudev and ;

), used this concept to locate structural motifs of metal/ligand-binding sites in proteins ( Lim, 2001 Yang, et al., 2008 Dudev andLim, ). They encoded a protein structure databank in terms of PBs and they located PBs encompassing specific metal-binding sites. Then, 2007 a discontinuous PB pattern, similar to a PROSITE pattern, was defined. First, the structural motifs of Cys Zn-finger domains, which are 4 known to adopt a specific structure, have been analyzed. Then, they focused on structural motifs of Mg -binding sites in a set of 2 + non-redundant Mg -binding proteins. Four Mg -structural motifs that showed important relationships between them were identified.

+

2 +

Other features of the proteins were also defined ( ). This strategy can be easily extended to other cases. Recently, Dudev and Lim, 2007 they have extended the approach to DNA and RNA binding sites, highlighting a novel non-specific motif enabling diverse interactions with DNA and RNA as with proteins (

). [START_REF] Wu | A structural-alphabet-based strategy for finding structural motifs across protein families[END_REF]

Definition of a reduced amino acid alphabet

Reduced amino acid alphabet is a popular concept that is explored by many research teams. Indeed, the appropriate selection of an amino acid type in a reliable set is particularly helpful to limit the number of experiments. Most of approaches were mainly based on sequence properties, ., ( ; ;

). i.e Akanuma, et al., 2002Clarke, 1995[START_REF] Kamtekar | Protein design by binary patterning of polar and nonpolar amino acids[END_REF] In this area, PBs not only help in describing protein structures, but are also useful in extraction of sequence structure relationships.

-Based on this relation, we proposed association of amino acids in a limited number of clusters. This approach permits an exchange of amino acids which are equivalent in terms of sequence structure relationship, while maintaining local protein structure conformation ( -). Zuo and Li used this reduced amino acid alphabet to predict different properties through a learning approach ( Etchebest, et al., 2007 Zuo ; ). and Li, 2009 Zuo andLi, 2009 Long structural fragments PBs are 5 residues long fragments. To assess the structural stability of these short fragments, we identified the most frequent series of 5 consecutive PBs which are 9 residues long. Then, we selected 72 most frequent series and named them Structural Words (SWs).

Interestingly, SWs encompass 92 of the residues (nearly 100 of the repetitive structures and 80 of secondary structure coil). By using % % % most of the SWs, it was possible to create a simple network describing most of the transitions between the SWs in proteins (de Brevern, et

). SWs yield a pertinent description of a large part of 3D structures, but as they constitute a sub-set of all five PBs combinations, al., 2002 they do not allow a description for every part of the protein structures. So, we have developed a novel approach named Hybrid Protein Model (HPM (

)). This innovative approach made it possible to create longer prototypes comprising 10 to 13 de Brevern and Hazout, 2000 residues. ( ; ; ; ; ; Benros, et al., 2006Benros, et al., 2003Benros, et al., 2009Benros, et al., 2002 , et al., 2009Bornot, et al., 2009de Brevern and Hazout, 2001 led to the construction of networks of local protein structures (

).

Hazout, 2005

Structural alignment

The structural alphabet allows translating protein three-dimensional structures into a series of letters ( see). Consequently, it Figure 1a is possible to use classical sequence alignment methodology to perform structure-based alignment (see ). The main difficulty lies Figure 1b in obtaining a pertinent substitution matrix, to find the similarity score between PBs for alignments. Using the homologues of known 3-D structure in PALI database (

) encoded in terms of PBs, a PB substitution matrix was computed ( ). A Gowri, et al., 2003Tyagi, et al., 2006 dedicated webserver has been developed ( ) that performs optimal alignments of a query protein http://bioinformatics.univ-reunion.fr/PBE/ structure with entries of 3-D structures in a database, using PBs and the substitution matrix 18 . A recent benchmark has proved that this [ ] method is most efficient in mining PDB and identifying proteins with similar 3-D structure (

). [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF] From this work, new developments have been performed. A first one directly relates to the use of substitution matrix and concerns the characterization of conformational patterns in active and inactive forms of kinases. Comparison of closely related kinases indicates a higher global similarity between the active state kinases compared to inactive states (as reflected from their PB scores) (Agarwal, et al., ). The second axis focuses on the database, which is the basis for generating the substitution matrix, ., PALI database. The 2010 i.e superimposed structures of PALI show regions with correct alignments linked with regions more difficult to align (named variable regions). A novel optimisation of the superposition based on PBs, shows a global improvement in the variable regions. Hence, PBs improve PALI database alignments (Agarwal .,

). The last axis concerns the alignment approach. Even though the et al in preparation recent benchmark has proved the quality of the methodology ( ), some structural alignmentsshow poor consistency. [START_REF] Tyagi | Protein structure mining using a structural alphabet[END_REF] Optimisation of the substitution matrix and a novel alignment methodology improved both the mining and the superimposition of protein structures (Joseph ., ). illustrates a very difficult case of superimposition of the threedimensional structures of et al in preparation Figure 3 acid phosphatase (

) and periplasmic glucose-1-phosphatase ( ). Aspergillus niger [START_REF] Kostrewa | Crystal structure of Aspergillus niger pH 2.5 acid phosphatase at 2.4 A resolution[END_REF]Escherichia coli Lee, et al., 2003 The superimposition obtained with our previous approach ( ) is quite poor. Our novel procedure allows the recognition Tyagi, et al., 2006 of highly similar regions (shown on

) and gives a spectacular improvement in the superimposition.

show that Figure 3e Figures 3c and3d the bottom part of the protein structures can be truly superimposed.

gives the alignment in terms of PBs. Figure 4 Prediction Like secondary structure prediction, it is possible to predict local structures in terms of structural alphabet (see for a summary Table 1 of all prediction approaches). Indeed, concomitant to an accurate local 3D structures description, definition of PBs was driven by prediction capabilities (

). The prediction principle is based on Bayes theorem. First, a set of protein chains used in de Brevern, et al., 2000 ' training were encoded in terms of PBs, using minimal RMSDA criterion. Then, sequence windows of 15 residues length were considered for calculating the propensities associated with each PB. For each PB, the probability of occurrence of an amino acid at each position in the sequence window was calculated and an occurrence matrix was generated, ., 16 for the 16 PBs. Bayes theorem was applied to i.e predict the structure of new sequences. A prediction rate of 34.4 was achieved ( ; ).

% de Brevern, et al., 2004de Brevern, et al., 2000 Nonetheless, only one amino acid occurrence matrix is associated to each PB. Consequently, the sequence information is averaged. A clustering approach related to SOM ( ; ) performed on PBs sequences revealed well-defined sequence [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF][START_REF] Kohonen | Self-Organizing Maps[END_REF] families for some PBs. For each sequence family, an amino acid occurrence matrix was then computed. This strategy increased sequence specificities for some PBs and permitted to achieve an improved prediction rate of 40.7 ( ; % de Brevern, et al., 2004 de Brevern, et al., ). Finally, a simulated annealing approach in the process of sequence family generation, helped to improve the overall prediction to 2000

(

). Importantly, this approach did not bring any biased or unbalanced improvements between the PBs. % Etchebest, et al., 2005 Combining the secondary structure information with the Bayesian prediction did not result in significant improvement of the prediction rate. A website, LocPred ( ), which includes most of the tools developed so far, is http://www.dsimb.inserm.fr/~debrevern/LOCPRED/ available to perform these predictions ( ). Predictions were also performed with the SWs ( ; de Brevern, et al., 2004[START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF] ) and specifically for short loops ( ; ). de Brevern, et al., 2002Fourrier, et al., 2004Tyagi, et al., 2009 A knowledge-based approach for predicting local backbone structure was also developed. In this case, overlapping fragments of 5 residues from a query sequence are extracted and queried against a pentapeptide database. In this database, which was built from SCOP database culled at 95 identity, each pentapeptide is mapped to a Protein Block. In absence of any hit in the database, pentapeptides in % " "

which constraint of identity in the central position (position 3) is relaxed are considered. Overall performance of the approach was about 62 .

%

Recent developments have been made by other teams. Li and co-workers who proposed an innovative approach for PB prediction, taking into account information from secondary structure and solvent accessibilities ( ). Prediction rates were significantly Li, et al., 2009 improved ( ). Interestingly, this approach was found useful for fragment threading, pseudo sequence design, and http://sg.ustc.edu.cn/lssrap/ local structure predictions.

Support Vector Machines methodology coupled with evolutionary information greatly improved the prediction rates. Hence, Zimmermann and Hansmann developed a method for PB prediction using SVMs with a radial basis function kernel, leading to an improvement of the prediction rate to 60 61 (

). This method called Locustra is available online at -% [START_REF] Zimmermann | LOCUSTRA: accurate prediction of local protein structure using a two-layer support vector machine approach[END_REF] . In a very recent work, Rangwala, Kauffman and Karypis have developed a novel http://www.fz-juelich.de/nic/cbb/service/service.php tool named svmPRAT (

). It involves formulating the annotation problem as a classification or regression problem [START_REF] Rangwala | Local structural motifs of protein backbones are classified by self-organizing neural networks[END_REF] using support vector machines (

). The use of such approach allows an impressive increase of http://www.cs.gmu.edu/~mlbio/prosat/ prediction rate of about 69 . PB prediction is part of MONSTER (Minnesota prOteiN Sequence annoTation servER, % ). Thus, in less than a decade the prediction rate of PBs has doubled in a very efficient way. http://bio.dtc.umn.edu/monster/ As emphasized by Li and co-workers ( ), it is often difficult to compare accurately the different studies because of Li, et al., 2009 different definitions considered for local structures, or different dataset and/or different criteria used for evaluating success predictions.

In order to extend the analyses to long structural fragments, HPM strategy was used to construct a new library of local structures. 120 structural clusters (named Local Structure Prototypes, LSPs) were then proposed to describe fragments that are 11-residue long (Benros, et

). An original prediction method based on logistic regressions was first developed for predicting local structures from a single al., 2006 sequence. This method proposed a short list of the best structural candidates among the 120 LSPs of the library. Considering a geometrical assessment, a prediction rate of 51.2 was reached. This result was quite significant, given the fragment length and the high number of %

classes (

). Recently, an improved prediction method based on SVMs and evolutionary information was proposed. A Benros, et al., 2006 global prediction rate of 63.1 was achieved and prediction for 85 of proteins was improved. This method was shown to be among the % % most efficient of cuttingedge local structure prediction strategies ( ). Bornot, et al., 2009

Conclusions and Perspectives

Since 1989, nearly twelve different structural alphabets have been developed, ., ( ; ; e.g Fetrow, et al., 1997 Ku andHu, 2008 Sander, et ; ; ), for dedicated reviews see ( ; ). al., 2006[START_REF] Unger | A 3D building blocks approach to analyzing and predicting structure of proteins[END_REF][START_REF] Unger | The importance of short structural motifs in protein structure analysis[END_REF][START_REF] Joseph | Local Structure Alphabets[END_REF][START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF] However, almost none has been used outside their developing laboratories. PBs alphabet is the only exception. It is mainly due to the ease with which protein 3D structures can be encoded as PBs. It can be considered as the classical standard of Structural Alphabet as DSSP is the classical standard for secondary structure assignment ( ). [START_REF] Kabsch | Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded and geometrical features[END_REF] PBs have been utilized in numerous different applications, ., modelling of a transmembrane protein implicated in malarial infection e.g ( ; ;

). It has also led to the development of an excellent superimposition method de [START_REF] Martin | Protein secondary structure assignment revisited: a detailed analysis of different assignment methods[END_REF]de Brevern, 2009de Brevern, et al., 2009 ( ) and is now used by various research teams ( ). We have also developed confidence indexes Tyagi, et al., 2008 Joseph, et al., 2010 associated to prediction accuracy ( ; ;

). We now link the uncertainties Bornot, et al., 2009de Brevern, et al., 2000Etchebest, et al., 2005 with the prediction of protein flexibility, looking at data from X-ray analysis, Molecular Dynamics (Bornot ., ) and NMR. et al submitted In the same way, superimposition approaches are currently improved. We also examine protein protein interactions in the light of PBs -

(Swapna ., ). et al in preparation

Ackowledgements:

The authors would like to the for their comments that help improve the manuscript.

Figure 2 Figure 3 Figure 4

 234 Figure 1Principe of encoding of protein structures using Structural Alphabet The N terminal extremity of chain B of acid phosphatase ( ) (a) is encoded in terms of a structural Aspergillus nigerKostrewa, et al., 1999 alphabet (b). Each residue is approximated by a specific prototype, here a Protein Block. Hence, the crude description as a coil region (done by any secondary structure assignment method) is replaced by a more precise series of PBs . dcbdfkl

  de Brevern and Hazout, 2001 de Brevern ). This resulted in higher structural variability for the longer fragments through a significant increase in the number ofand Hazout, 2003

	prototypes,	e.g	., 100 to 130 prototypes (	; Benros, et al., 2006 Benros, et al., 2009 de Brevern and Hazout, 2001 de Brevern and Hazout, ; ;
	2003	). These longer fragments were used to perform structural superimposition (	de Brevern and Hazout, 2001	), methodological
	optimisation (	; Benros, et al., 2003 de Brevern and Hazout, 2003	), and analysis of sequence structure relationship ( -	Benros, et al., 2006	;
	Benros			;	;	). A modified version of HPM proposed by Pr. Serge Hazout, has

  These works were supported by grants thank reviewers from the French Ministry of Research, University of Paris Diderot Paris 7, University of Saint-Denis de la R union, French National Institute for Blood Transfusion (INTS), French Institute for Health and Medical Research (INSERM) and Indian Department of Biotechnology. APJ and GA are supported by CEFIPRA number 3903-E and Council of Scientific and Industrial Research, respectively. AB had a grant from the French Ministry of Research, MT has a post-doctoral fellowship from NIH and HV had a post-doctoral fellowship from CEA. NS and AdB acknowledge to CEFIPRA for collaborative grant (number 3903-E). BS and AdB acknowledge to Partenariat Hubert Curien Barrande (2010 2011). New assessment of a structural alphabet . In Silico Biol . 5 : 283 -289 de Brevern AG . 2009 ; New opportunities to fight against infectious diseases and to identify pertinent drug targets with novel methodologies . Infect Disord Drug Targets . 9 : 246 -247 de Brevern AG , Autin L , Colin Y , Bertrand O , Etchebest C . 2009 ; In silico studies on DARC . Infect Disord Drug Targets . 9 : 289 -303 de Brevern AG , Benros C , Gautier R , Valadie H , Hazout S , Etchebest C . 2004 ; Local backbone structure prediction of proteins . In Silico Biol . 4 : 381 -386 de Brevern AG , Etchebest C , Benros C , Hazout S . 2007 ; Pinning strategy : a novel approach for predicting the backbone structure in terms of protein blocks from sequence Benros C , de Brevern AG . 2004 ; Use of a structural alphabet for analysis of short loops connecting repetitive structures . BMC Bioinformatics . 5 : 58 -Gowri VS , Pandit SB , Karthik PS , Srinivasan N , Balaji S . 2003 ; Integration of related sequences with protein three-dimensional structural families in an updated version of PALI database . Nucleic Acids Res . 31 : 486 -488 Hazout S . 2005 ; Une nouvelle m thode d apprentissage: Self-Learning by Information Share-Out (SLISO)

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François Golse

From the Kinetic Theory of Gases to Continuum Mechanics

Keywords: Hydrodynamic limits, Kinetic models, Boltzmann equation, Entropy production, Euler equations, Navier-Stokes equations PACS: 47.45-n, 51.10.+y, 51.20.+d, 47.10.ad

Recent results on the fluid dynamic limits of the Boltzmann equation based on the DiPerna-Lions theory of renormalized solutions are reviewed in this paper, with an emphasis on regimes where the velocity field behaves to leading order like that of an incompressible fluid with constant density.

In memory of Carlo Cercignani Relating the kinetic theory of gases to their description by the equations of continuum mechanics is a question that finds its origins in the work of Maxwell [30]. It was subsequently formulated by Hilbert as a mathematical problem -specifically, as an example of his 6th problem on the axiomatization of physics [START_REF] Hilbert | Internat. Congress of Math[END_REF]. In Hilbert's own words "Boltzmann's work on the principles of mechanics suggests the problem of developing mathematically the limiting processes which lead from the atomistic view to the laws of motion of continua". Hilbert himself studied this problem in [START_REF] Hilbert | [END_REF] as an application of his theory of integral equations. The present paper reviews recent progress on this problem in the past 10 years as a consequence of the DiPerna-Lions global existence and stability theory [12] for solutions of the Boltzmann equation. This Harold Grad lecture is dedicated to the memory of Carlo Cercignani, who gave the first Harold Grad lecture in the 17th Rarefied Gas Dynamics Symposium, in Aachen (1990), in recognition of his outstanding influence on the mathematical analysis of the Boltzmann equation in the past 40 years.

THE BOLTZMANN EQUATION: FORMAL STRUCTURE

In kinetic theory, the state of a monatomic gas at time t and position x is described by its velocity distribution function F ≡ F(t, x, v) ≥ 0. It satisfies the Boltzmann equation

∂ t F + v • ∇ x F = C (F) ,
where C (F)(t, x, v) := C (F(t, x, •))(v) is the Boltzmann collision integral defined for each continuous, rapidly decaying function f ≡ f (v) by

C ( f )(v) := R 3 ×S 2 ( f (v ′ ) f (v ′ * ) -f (v) f (v * )) d 2 2 |(v -v * ) • ω| dv * dω ,
assuming that gas molecules behave as perfectly elastic hard spheres of diameter d. In this formula, we have denoted

v ′ ≡ v ′ (v, v * , ω) := v -(v -v * ) • ωω , v ′ * ≡ v ′ * (v, v * , ω) := v * + (v -v * ) • ωω , |ω| = 1 . (1)
Molecular interactions more general than hard sphere collisions can be considered by replacing

d 2 2 |(v -v * ) • ω| with appropriate collision kernels of the form b(|v -v * |, | v-v * |v-v * | • ω|).
In this paper, we restrict our attention to the case of hard sphere collisions to avoid dealing with more technical conditions on the collision kernel.

Properties of the collision integral

While the collision integral is a fairly intricate mathematical expression, the formulas (1) entail remarkable symmetry properties. As a result, the collision integral satisfies, for each continuous, rapidly decaying f ≡ f (v), the identities 3, and

R 3 C ( f ) dv = 0 , R 3 C ( f )v k dv = 0 , k = 1, 2,
R 3 C ( f )|v| 2 dv = 0 . (2)
The first relation expresses the conservation of mass (or equivalently, of the number of particles) by the collision process, while the second and the third express the conservation of momentum and energy respectively. Perhaps the most important result on the collision integral is Boltzmann's H Theorem: for each continuous, rapidly decaying f ≡ f (v) > 0 such that ln f has polynomial growth as |v| → +∞, R 3 C ( f ) ln f dv ≤ 0 , and

R 3 C ( f ) ln f dv = 0 ⇔ C ( f ) = 0 ⇔ f is a Maxwellian, (3)
i.e. there exists ρ, θ > 0 and u ∈ R 3 such that

f (v) = M (ρ,u,θ ) (v) := ρ (2πθ ) 3/2 exp - |v -u| 2 2θ . (4)

Dimensionless variables

Fluid dynamic limits are obtained as properties of solutions of the Boltzmann equation under appropriate scaling assumptions. We therefore recast the Boltzmann equation in dimensionless variables, so as to identify the dimensionless parameters that control the scalings of the time and space variables, following [4,[START_REF] Sone | Kinetic Theory and Fluid Dynamics[END_REF].

First we choose a macroscopic length scale L (for instance the size of the container where the gas is enclosed, or of an object immersed in the fluid, or the typical length scale on which the variation of macroscopic fluid quantities is observed), as well as a macroscopic observation time scale T o (i.e. the time scale on which the evolution of the fluid quantities is observed.)

We next define reference scales of density ρ and temperature θ by setting

F dx dv = ρL 3 , vF dx dv = 0 , 1 2 |v| 2 F dx dv = 3 2 ρθ .
The collision time scale T c is defined in turn by

d 2 2 M (ρ,0,θ) (v)M (ρ,0,θ ) (v * )|(v -v * ) • ω| dv dv * dω = ρ
C = 1 4π 2 |v -v * |e -(|v| 2 +|v * | 2 )/2 dv dv * ,
we see that the Boltzmann equation in dimensionless variables takes the form

Sh ∂ t F + v • ∇ x F = 1 Kn Ĉ ( F) , (5)
where the dimensionless collision integral is

Ĉ ( F)(t, x, v) = R 3 ×S 2 ( F(t, x, v′ ) F(t, x, v′ * ) -F(t, x, v) F(t, x, v * ))|( v -v * ) • ω| d v * dω . (6)
Obviously, the dimensionless collision integral Ĉ ( F) satisfies exactly the same properties as the original expression C (F), i.e. the conservation laws of mass, momentum and energy (2) and Boltzmann's H Theorem (3). Henceforth, we always consider the Boltzmann equation ( 5) in dimensionless variables, dropping all hats for notational simplicity. Thus, the conservation properties of the collision operator (2) imply that rapidly decaying (in v) solutions of the dimensionless Boltzmann equation ( 5) satisfy the following local conservation laws:

                   Sh ∂ t R 3 F dv + div x R 3 vF dv = 0 (conservation of mass), Sh ∂ t R 3 vF dv + div x R 3 v ⊗ vF dv = 0 (conservation of momentum), Sh ∂ t R 3 1 2 |v| 2 F dv + div x R 3 v 1 2 |v| 2 F dv = 0 (conservation of energy). (7)
Likewise, Boltzmann's H Theorem implies that solutions F > 0 of the Boltzmann equation that are rapidly decaying while ln F has polynomial growth as |v| → +∞ satisfy the differential inequality

Sh ∂ t R 3 F ln F dv + div x R 3 vF ln F dv = 1 Kn R 3 C ( f ) ln f dv ≤ 0 . ( 8
)

THE COMPRESSIBLE EULER LIMIT AND HILBERT'S EXPANSION

Whenever a gas evolves in a fluid dynamic regime (at the length scale L), the average time between successive collisions involving a typical gas molecule is much smaller than the time necessary for an acoustic wave to travel a distance L. In other words, fluid dynamic regimes are characterized by the condition T c ≪ T a , or equivalently by he condition Kn ≪ 1.

In [START_REF] Hilbert | [END_REF], Hilbert studied the Boltzmann equation ( 5) in the asymptotic regime defined by Kn = ε ≪ 1 and Sh = 1. His idea was to seek the solution

F ε of ∂ t F ε + v • ∇ x F ε = 1 ε C (F ε ) (9)
as a formal power series in ε with smooth coefficients -known as Hilbert's expansion:

F ε (t, x, v) = ∑ n≥0 ε n f n (t, x, v) , with f n smooth in t, x, v , for each n ≥ 0 . (10)
He found that the leading order term in that expansion is of the form

f 0 (t, x, v) = M (ρ,u,θ )(t,x) (v) ,
where (ρ, u, θ ) is a solution of the compressible Euler system

         ∂ t ρ + div x (ρu) = 0 , ρ(∂ t u + u • ∇ x u) + ∇ x (ρθ ) = 0 , ∂ t θ + u • ∇ x θ + 2 3 θ div x u = 0 . (11)
Caflisch [START_REF] Caflisch | [END_REF] succeeded in turning Hilbert's formal result into a rigorous statement bearing on solutions of the Boltzmann equation, by using a truncated variant of the Hilbert expansion above. Specifically, given a smooth solution (ρ, u, θ ) of the compressible Euler system on some finite time interval [0, T ), he constructs a family of solutions of the Boltzmann equation that converges to M (ρ,u,θ ) uniformly in t ∈ [0, T ) as ε → 0. Before Caflisch's result, Nishida [31] had proposed another proof of the compressible Euler limit of the Boltzmann equation under more stringent regularity assumptions, viz. analyticity, using some abstract variant of the Cauchy-Kowalewski theorem. One striking advantage of the Hilbert expansion is its versatility, abundantly illustrated by the great diversity of physically meaningful applications to be found in the work of Sone [START_REF] Sone | Kinetic Theory and Fluid Dynamics[END_REF][START_REF] Sone | Molecular Gas Dynamics, Theory, Techniques, and Applications[END_REF]. However, there are some serious difficulties with the Hilbert expansion, some of which can be treated with adequate mathematical techniques. First, the radius of convergence of the Hilbert power series is 0 in general, so that essentially all mathematical arguments based on Hilbert's expansion use a truncated variant thereof. In general, truncated Hilbert expansions are not everywhere nonnegative, and are not exact solutions of the Boltzmann equation. One obtains exact solutions of the Boltzmann equation by adding to the truncated Hilbert expansion some appropriate remainder term, satisfying a variant of the Boltzmann equation that becomes weakly nonlinear for small enough ε (see for instance [START_REF] Caflisch | [END_REF]11,2].) The truncated Hilbert expansion with the remainder term so constructed is a rigorous, pointwise asymptotic expansion (meaning that ε

-n |F ε -( f 0 + ε f 1 + . . . + ε n f n )| → 0 pointwise in (t, x, v))
of the solution F ε of (9) as ε → 0. Another difficulty in working with Hilbert's expansion, even truncated at some finite order, is that

f n = O(|∇ n t,
x f 0 |) for each n ≥ 0. Since generic solutions of the compressible Euler system lose regularity in finite time [33], truncated Hilbert expansions make sense on finite time intervals only. For instance, if a solution (ρ, u, θ ) of the compressible Euler system involves a shock wave, only the 0-th order term in the associated Hilbert expansion, i.e. f 0 (t, x, v) = M (ρ,u,θ )(t,x) (v) is well defined. In general, if the geometric structure and the position of the singularities in the solution of the hydrodynamic equations are known precisely, one can bypass this difficulty by adding to the truncated Hilbert expansion appropriate boundary layer terms. If the structure of these singularities is unknown, or one does not even know whether the hydrodynamic solution is smooth, one cannot use the Hilbert expansion.

GLOBAL EXISTENCE THEORY FOR THE BOLTZMANN EQUATION

To avoid the various shortcomings of the Hilbert expansion method, one needs a theory of global solutions for the Boltzmann equation based on the only estimates that are uniform in Kn as Kn → 0. These estimates are those deduced from the conservation laws (2) and Boltzmann's H Theorem (3), or from their differential formulations ( 7)- (8).

Henceforth, we are concerned with solutions of the Boltzmann equation for a gas filling the Euclidian space R 3 and at equilibrium at infinity. By Galilean invariance and with a convenient choice of units, we can assume without loss of generality, that this equilibrium state at infinity is the Maxwellian M (1,0,1) , denoted by M in the sequel. In other words, we seek the solution of

             Sh ∂ t F + v • ∇ x F = 1 Kn C (F) , (x, v) ∈ R 3 × R 3 , t > 0 , F(t, x, v) → M as |x| → +∞ , F t=0 = F in . (12)
A convenient quantity measuring the distance between two distribution functions in the context of the Boltzmann equation is the relative entropy:

for F ≡ F(x, v) ≥ 0 and G ≡ G(x, v) > 0 a.e. in (x, v) ∈ R 3 × R 3 , H(F|G) := R 3 ×R 3 (F ln(F/G) -F + G)(x, v) dx dv . ( 13
)
Notice that a ln(a/b)a + b ≥ 0 for each a ≥ 0 and b > 0, with equality if and only if a = b. Hence the integrand is a nonnegative measurable function and H(F|G) = 0 if and only if F = G a.e. on R 3 × R 3 .

Since ln M = -3 2 ln(2π) -1 2 |v| 2 , a formal argument based on the local conservation laws (7) and the differential inequality (8) shows that any classical solution F of ( 12) with appropriate decay as |v| → +∞ satisfies

Sh ∂ t R 3 (F ln(F/M) -F + M) dv + div x R 3 v(F ln(F/M) -F + M) dv ≤ 0 .
Integrating in x both sides of this inequality and assuming that F → M fast enough as |x| → +∞, we conclude that

sup t≥0 H(F(t, •, •)|M) ≤ H(F in |M) , and ∞ 0 R 3 ×R 3 -C (F) ln F ds dx dv ≤ Sh H(F in |M) . (14)
Observe that the collision integral C (F) acts as a nonlocal integral operator analogous to a convolution in the v variable and as a pointwise product in the x variable. The fact that C (F) is quadratic in F while H(F|M) is "essentially homogeneous of degree 1 as F ≫ 1" suggests that C (F) may not be defined for all nonnegative measurable functions F satisfying the entropy bound [START_REF] Golse | [END_REF] above. Yet, for each measurable

F > 0 on R 3 × R 3 , one has |x|+|v|≤r |C (F)| √ 1 + F dx dv ≤ C |x|≤r (-C (F) ln F + (1 + |v| 2 )F) dx dv so that C (F)/ √ 1 + F ∈ L 1 loc (R + × R 3 × R 3 ), i.e
. is locally integrable in (t, x, v). This suggests dividing both sides of the Boltzmann equation by √ 1 + F, thereby leading to the notion of renormalized solution. Definition. (DiPerna-Lions [12]) A renormalized solution relative to M of the Boltzmann equation is a nonnegative function

F ∈ C(R + , L 1 loc (R 3 × R 3 )) satisfying H(F(t)|M) < +∞ for each t ≥ 0 and M(Sh ∂ t + v • ∇ x )Γ(F/M) = 1 Kn C (F)Γ ′ (F/M) in the sense of distributions on R * + × R 3 × R 3 , for each Γ ∈ C 1 (R + ) satisfying Γ ′ (Z) ≤ C/ √ 1 + Z.
With this notion of solution, one can prove the global existence and weak stability of solutions of the Cauchy problem for the Boltzmann equation, with initial data that are not necessarily small perturbations of either the vacuum state or of a Maxwellian equilibrium. Theorem. (DiPerna-Lions-Masmoudi [12,27,29]) For each measurable F in ≥ 0 a.e. on R 3 × R 3 satisfying the condition H(F in |M) < +∞, there exists a renormalized solution of the Boltzmann equation ( 12) with initial data

F in . This solution F satisfies      Sh ∂ t R 3 F dv + div x R 3 vF dv = 0 , Sh ∂ t R 3 vF dv + div x R 3 v ⊗ vF dv + div x m = 0 , (15)
where m = m T ≥ 0 is a matrix-valued Radon measure, and the entropy inequality

Sh H(F(t, •, •)|M) + Sh R 3
Tr(m(t)) -

t 0 R 3 ×R 3 C (F) ln F(s, x, v) ds dx dv ≤ Sh H(F in |M) , t > 0 . (16)
A classical solution of the Boltzmann equation with appropriate decay as |v| → +∞ would satisfy all these properties with m = 0; besides the entropy inequality is a weakened variant of Boltzmann's H Theorem -which would imply that this inequality is in fact an equality.

The main advantage of the notion of renormalized solutions is that a) such solutions always exist for each initial data with finite relative entropy with respect to M, and b) such solutions are weakly stable, in the sense that if a sequence (F n ) n≥0 of renormalized solutions of the Boltzmann equation converges to F in the sense of distributions and satisfies H(F n t=0 |M) ≤ C for all n ≥ 0, where C is some positive constant, then F is also a renormalized solution of the Boltzmann equation, satisfying (15) and (16). Unfortunately, there is no uniqueness theorem for this notion of solution, so that a renormalized solution of the Boltzmann equation is not completely determined by its initial data. But if the Cauchy problem for the Boltzmann equation has a classical solution F, each renormalized solution of the Boltzmann equation with the same initial data as F coincides with F a.e. in (t, x, v) (see [26].)

FLUID DYNAMIC LIMITS OF THE BOLTZMANN EQUATION

As explained above, all fluid dynamic limits of the Boltzmann equation are characterized by the scaling condition Kn ≪ 1: hence we set Kn = ε throughout the present section.

Besides, all the fluid dynamic limits considered in this paper correspond with weakly nonlinear regimes at the kinetic level -which does not imply that the nonlinearities are weak at the macroscopic level. Such regimes have been systematically explored by Sone at the formal level (see [START_REF] Sone | Kinetic Theory and Fluid Dynamics[END_REF] and the references therein), by using the Hilbert expansion method. In other words, the distribution functions F considered are small perturbations of the Maxwellian state M at infinity. Henceforth, we denote by δ ε ≪ 1 the order of magnitude of the difference F -M. A typical example of such a distribution function is

F(t, x, v) = M (1,δ ε u(t,x),1) (v), since M (1,δ ε u(t,x),1) (v) = M(v)(1 + δ ε u(t, x) • v + O(δ 2 ε )).
In this example, the distribution function F defines a velocity field u F and a temperature field θ F by the formulas

u F := R 3 vF dv R 3 F dv = δ ε u and θ F := R 3 |v -u F | 2 F dv 3 R 3 F dv = 1 .
Introducing the speed of sound c F := 5θ F /3, we see that the Mach number Ma := u F /c F = δ ε u, so that the scaling parameter δ ε can be thought of as the (order of magnitude of the) Mach number.

The acoustic limit

The acoustic limit is the linearized variant of the compressible Euler limit considered by Hilbert himself. Theorem. (Golse, Jiang, Levermore, Masmoudi [START_REF] Golse | [END_REF]23]) Let Kn = ε, Ma = δ ε = O( √ ε) and Sh = 1. For each ρ in , u in , θ in ∈ L 2 (R 3 ), let F ε be a family of renormalized solutions of the Boltzmann equation (12) with initial data

F in ε = M (1+δ ε ρ in ,δ ε u in ,1+δ ε θ in ) .
Then, in the limit as ε → 0,

1 δ ε R 3 (F ε (t, x, v) -M(v))(1, v, 1 3 |v| 2 -1) dv → (ρ, u, θ )(t, x) in L 1 loc (R * + × R 3 )
, where (ρ, u, θ ) is the solution of the acoustic system

   ∂ t ρ + div x u = 0 , ρ t=0 = ρ in , ∂ t u + ∇ x (ρ + θ ) = 0 , u t=0 = u in , ∂ t θ + 2 3 div x u = 0 , θ t=0 = θ in .
While the result in [START_REF] Golse | [END_REF] holds for the most general class of molecular interactions satisfying some angular cutoff assumption in the sense of Grad [START_REF] Grad | Rarefied Gas Dynamics[END_REF] (in fact, a much weaker version of Grad's assumption [23,25]), an earlier contribution of the same authors with Bardos [5] introduced a key new idea in the derivation of hydrodynamic limits of renormalized solutions of the Boltzmann equation and treated the case of bounded collision kernels (e.g. cutoff Maxwell molecules).

The incompressible Euler limit

It is a well-known fact that, in the low Mach number limit, the flow of an inviscid fluid can be approximately decomposed into its acoustic and vortical modes, whose interaction vanishes with the Mach number. The result below explores the counterpart for vortical modes of the acoustic limit of the Boltzmann equation. Because of the low Mach number scaling, vortical modes evolve on a longer time scale than acoustic modes, consistently with the fact that the conditions ∇ x (ρ + θ ) = 0 and div x u = 0 characterize the equilibrium points of the acoustic system. Theorem. (Saint-Raymond [32]) Let Kn = ε, and Sh = Ma = δ ε = ε α with 0 < α < 1. Let u in ∈ H 3 (R 3 )1 satisfy divu in = 0, and let u ∈ C([0, T ]; H 3 (R 3 )) be a solution of the incompressible Euler equations

∂ t u + u • ∇ x u + ∇ x p = 0 , div x u = 0 , u t=0 = u in .
Let F ε be a family of renormalized solutions of the Boltzmann equation (12) with initial data

F in ε = M (1,δ ε u in ,1) .
Then, in the limit as ε → 0, one has

1 δ ε R 3 vF ε (t, x, v) dv → u(t, x) in L ∞ ([0, T ]; L 1 loc (R 3 )) .
The proof of this result is based on the relative entropy method, described in the next section. Actually, there had been precursors of this theorem due to the author [START_REF] Bouchut | Kinetic Equations and Asymptotic Theory[END_REF] and to Lions-Masmoudi [29], where the relative entropy method was introduced for this type of problem. Unfortunately, the statements in [START_REF] Bouchut | Kinetic Equations and Asymptotic Theory[END_REF]29] rested on extra assumptions on the family of solutions of the Boltzmann equation that remain unverified. These assumptions were removed by some clever argument in [32], which therefore contains the first complete proof of the theorem above.

The Stokes limit

We continue our exploration of vortical modes with the Stokes limit of the Boltzmann equation. The scaling is weakly nonlinear at the macroscopic level of description, and the time scale is chosen so as to keep track of entropy production in the fluid dynamic limit. Theorem. (Golse, Levermore, Masmoudi [START_REF] Golse | [END_REF]25]) Let Kn = Sh = ε, and Ma = δ ε = o(ε). For each (u in , θ in ) ∈ L 2 × L ∞ (R 3 ) such that div x u in = 0 and each ε ∈ (0, u in L ∞ ), let F ε be a family of renormalized solutions of the Boltzmann equation (12) with initial data

F in ε = M (1-δ ε θ in ,δ ε u in ,1+δ ε θ in ) .
Then, in the limit as ε → 0, one has

1 δ ε R 3 (F ε (t, x, v) -M(v))(v, 1 3 |v| 2 -1) dv → (u, θ )(t, x) in L 1 loc (R + × R 3 × R 3 ) ,
where (u, θ ) is a solution of the Stokes-Fourier system

∂ t u + ∇ x p = ν∆ x u , div x u = 0 , u t=0 = u in , ∂ t θ = 2 5 κ∆ x θ , θ t=0 = θ in .
The viscosity and heat conductivity in this theorem are given by the formulas (equivalent to the usual ones in [START_REF] Sone | Kinetic Theory and Fluid Dynamics[END_REF]):

ν = 1 5 D * (v ⊗ v -1 3 |v| 2 I) , κ = 2 3 D * ( 1 2 (|v| 2 -5)v) , (17)
where D * denotes the Legendre dual of the Dirichlet form D of the collision operator linearized about M, i.e.

D(Φ)

:= 1 2 R 3 ×R 3 ×S 2 |Φ + Φ * -Φ ′ -Φ ′ * | 2 |(v -v * ) • ω|MM * dv dv * dω .
The fluid dynamic model obtained in the statement above is the Stokes-Fourier system; notice that the motion and temperature equations are decoupled in the absence of an external force field deriving from a potential. Previously Lions and Masmoudi [29] arrived at the particular case of the statement above corresponding to an initial data for which θ in = 0, leading to the motion equation only, i.e. the evolution Stokes equation. For want of a better control of the high speed tails of the distribution function, their argument cannot be generalized to obtain the Stokes-Fourier system presented above. The proof in [START_REF] Golse | [END_REF] uses a different idea originating from [5].

The incompressible Navier-Stokes limit

Finally, we remove the weakly nonlinear scaling assumption at the macroscopic level of description, while keeping entropy production effects at leading order, and obtain the incompressible Navier-Stokes equations as a fluid dynamic limit of the Boltzmann equation. Theorem. (Golse, Saint-Raymond [18,19]

) Let Kn = Sh = Ma = δ ε = ε. For each (u in , θ in ) ∈ L 2 × L ∞ (R 3
) such that div x u in = 0, let F ε be a family of renormalized solutions of the Boltzmann equation (12) with initial data

F in ε = M (1-εθ in ,εu in ,1+εθ in ) ,
for each ε ∈ (0, u in L ∞ ). There exists at least one subsequence ε n → 0 such that

1 ε n R 3 (F ε n (t, x, v)-M(v)) v, 1 3 |v| 2 -1 dv → (u, θ )(t, x) in weak-L 1 loc (R + × R 3 × R 3 ) ,
where (u, θ ) is a "Leray solution" of the Navier-Stokes-Fourier system with viscosity ν and heat conductivity κ given by formula (17):

∂ t u + div x (u ⊗ u) + ∇ x p = ν∆ x u , div x u = 0 , u t=0 = u in , ∂ t θ + div x (uθ ) = 2 5 κ∆ x θ , θ t=0 = θ in .

Boltzmann equation Kn

= ε ≪ 1 Ma Sh Fluid dynamic limit δ ε ≪ √ ε 1 Acoustic system δ ε ≪ ε ε Stokes-Fourier system δ ε = ε α , 0 < α < 1 δ ε Incompressible Euler equations ε ε Incompressible Navier-Stokes equations
Let us briefly recall the notion of Leray solution of the Navier-Stokes-Fourier system. In [24] (arguably one of the most important papers in the modern theory of partial differential equations), Leray defined a convenient notion of weak solution of the Navier-Stokes equations, and proved that, in space dimension 3, any initial velocity field with finite kinetic energy launches at least one such solution defined for all times. Leray solutions are not known to be uniquely defined by their initial data; however, if an initial data launches a smooth solution, all Leray solutions with the same initial data must coincide with that smooth solution. At the time of this writing, it is yet unknown (and a major open problem in the analysis of partial differential equations) whether Leray solutions launched by any smooth initial data remain smooth for all times. Thus, we do not know whether different subsequences ε n → 0 in the theorem above lead to the same Leray solution (u, θ ) of the Navier-Stokes-Fourier system in general.

A Leray solution of the Navier-Stokes-Fourier system above is a pair (u, θ ) consisting of a velocity field u and a temperature field θ , both continuous on R + with values in L 2 (R 3 ) equipped with its weak topology, that solves the Navier-Stokes-Fourier system in the sense of distributions, satisfies the initial condition, and verifies the Leray inequality:

1 2 R 3 (|u| 2 + 5 2 |θ | 2 )(t, x) dx + t 0 R 3 (ν|∇ x u| 2 + κ|∇ x θ | 2 )(s, x) dx ds ≤ 1 2 R 3 (|u in | 2 + 5 2 |θ in | 2 )(t, x) dx . (18)
The Leray inequality is an equality for classical solutions of the Navier-Stokes equations, exactly as the DiPerna-Lions entropy inequality ( 16) is an equality for classical solutions of the Boltzmann equation. This indicates that the Leray existence theory for the Navier-Stokes equations and the DiPerna-Lions existence theory for the Boltzmann equation are parallel theories. The theorem above explains how these theories are related in the hydrodynamic limit. Partial results on this theorem have been obtained by Lions-Masmoudi [28]. While the reference [18] treated the case of bounded collision kernels, the theorem above was later extended to all hard cutoff potentials in the sense of Grad -which includes the case of hard spheres considered in this paper -in [19]. The arguments in [18,19] have been recently refined by Levermore and Masmoudi [25] to encompass both soft as well as hard potentials, under a cutoff assumption more general than that proposed by Grad in [START_REF] Grad | Rarefied Gas Dynamics[END_REF].

While these results bear on the most general case of renormalized solutions without restrictions on the size of initial data in space dimension 3, the Navier-Stokes limit of the Boltzmann equation had previously been obtained in the case of global smooth solutions for small initial data by Bardos and Ukai [7]. The Navier-Stokes limit of the Boltzmann equation had also been established on finite time intervals by adapting the Caflisch method based on Hilbert truncated expansions, by DeMasi, Esposito and Lebowitz [11].

The fluid dynamic limits discussed in this section can therefore be summarized as in table 1. Notice that these limits have been established for molecular interactions more general than hard sphere collisions; see the references listed in the statements of the various theorems above for the conditions on the collision kernel b(vv * , ω). All these results assume some angular cutoff on the collision kernel as proposed by Grad [START_REF] Grad | Rarefied Gas Dynamics[END_REF] -or slightly more general, as in [25].

More importantly, some of the conditions bearing on the parameters Kn, Ma and Sh may be not optimal. Formal arguments suggest that the acoustic limit should hold whenever δ ε ≪ 1 instead of δ ε ≪ √ ε, while the incompressible Euler limit should hold under the weaker condition δ ε ≫ ε instead of δ ε = ε α with 0 < α < 1.

Let us conclude this section with an important remark on the physical meaning of the "incompressible" fluid dynamic limits of the Boltzmann equation. What is proved in the last three theorems is that, to leading order, the velocity field u satisfies the same equations as the velocity field of an incompressible fluid with constant density. This does not mean that the gas is incompressible in that regime. Also, in the case of an incompressible fluid with the same heat capacity and heat conductivity as the gas, the diffusion term in the equation for the temperature field would be multiplied by 5/3. This difference comes from the work of the pressure: see the discussion in footnotes 6 on p. 93 in [START_REF] Sone | Kinetic Theory and Fluid Dynamics[END_REF] and 43 on p. 107 of [START_REF] Sone | Molecular Gas Dynamics, Theory, Techniques, and Applications[END_REF], together with section 3.7.2 in [START_REF] Sone | Molecular Gas Dynamics, Theory, Techniques, and Applications[END_REF].

Likewise, the inequality ( 18) was written by Leray in [24] with θ ≡ 0. For an incompressible fluid with constant density ρ, the quantity 1 2 ρ|u(t, x)|2 dx is the kinetic energy of the fluid at time t, and the Leray inequality is interpreted as a statement on the dissipation of energy in the fluid. The meaning of (18) with θ ≡ 0 is obviously different, since the quantity 1 2 (|u(t, x)| 2 + 5 2 θ (t, x) 2 ) dx is not the total energy of the gas at time t.

MATHEMATICAL TOOLS FOR THE HYDRODYNAMIC LIMIT

The linearized collision integral

In all the fluid dynamic limits considered in the previous section, the solution F ε of the Boltzmann equation ( 12) is a small perturbation of the uniform Maxwellian equilibrium state M. Therefore, the linearization about M of the Boltzmann collision integral plays an important role in these limits. Thus, we consider this linearized collision integral intertwined with the multiplication by M, and set

L M φ = -M -1 DC (M) • (Mφ ), or equivalently L M φ (v) := R 3 ×S 2 (φ (v) + φ (v * ) -φ (v ′ ) -φ (v ′ * ))|(v -v * ) • ω|M(v * ) dv * dω . (19)
Hilbert [START_REF] Hilbert | [END_REF] proved that L M is an unbounded, Fredholm, self-adjoint nonnegative operator on

L 2 (R 3 ; M dv) 2 , with domain L 2 (R 3 ; (1 + |v| 2 )M dv) and nullspace KerL M = Span{1, v 1 , v 2 , v 3 , |v| 2 }.
Hilbert's argument, written for the hard sphere case, was later extended by Grad [START_REF] Grad | Rarefied Gas Dynamics[END_REF], who defined some appropriate class of collision kernels b(vv * , ω) for which the linearized collision integral satisfies the Fredholm alternative. Grad's idea was that grazing collisions between neutral gas molecules are rare events that can be somehow neglected, at variance with the case of plasmas or ionized gases. Henceforth, we denote

φ = R 3 φ (v)M(v) dv for each φ ∈ L 1 (R 3 ; M dv) .
With this notation, the Fredholm alternative for the integral equation L M f = S with unknown f and source term S ∈ L 2 (R 3 ; M dv) can be stated as follows: a) either S = Sv 1 = Sv 2 = Sv 3 = S|v| 2 = 0, in which case the integral equation has a unique solution f

satisfying f ∈ L 2 (R 3 ; (1 + |v| 2 )M dv) and f = f v 1 = f v 2 = f v 3 = f |v| 2 = 0 , henceforth denoted f = L -1
M S, or b) there exists φ ∈ KerL M such that Sφ = 0, in which case the integral equation

L M f = S does not have any solution in L 2 (R 3 ; (1 + |v| 2 )M dv).
The moment method for the Navier-Stokes limit: formal argument Define g ε by the formula

F ε = M(1 + δ ε g ε ). If F ε satisfies (12) with Kn = Sh = δ ε = ε, the relative fluctuation of distribution function g ε satisfies ε∂ t g ε + v • ∇ x g ε + 1 ε L M g ε = Q M (g ε , g ε ) , (20)
where Q M is the symmetric bilinear operator defined by

Q M (φ , φ ) = M -1 C (Mφ ).
Multiplying each side of (20) by ε and letting ε → 0 shows that, if g ε n → g for some subsequence ε n → 0, the limiting fluctuation g is a "local Maxwellian state", i.e. is of the form

g(t, x, v) = ρ(t, x) + u(t, x) • v + θ (t, x) 1 2 (|v| 2 -3) . (21)
Multiplying each side of ( 20) by M and vM, and integrating in v ∈ R 3 shows that

ε∂ t g ε + div x vg ε = 0 , ε∂ t vg ε + div x v ⊗ vg ε = 0
in view of (2). Passing to the limit as ε n → 0, and taking into account the local Maxwellian form (21) of g leads to

div x u = div x vg = 0 , ∇ x (ρ + θ ) = div x v ⊗ vg = 0 . (22)
The first equality is the solenoidal condition for the velocity field u, while the second implies that ρ + θ = 0, assuming that ρ, θ → 0 as |x| → +∞.

Next we multiply each side of ( 20) by 1 ε vM and integrate in v ∈ R 3 to obtain

∂ t vg ε + div x 1 ε Ag ε = -∇ x 1 ε |v| 2 3 g ε ,
where

A(v) := v ⊗ v -1 3 |v| 2 I. One has A kl = A kl v 1 = A kl v 2 = A kl v 3 = A kl |v| 2 = 0 for each k, l = 1, 2, 3, so that Âkl := L -1 M A kl ∈ L 2 (R 3 ; (1 + |v| 2 )M dv) is well-defined. Since L M is self-adjoint on L 2 (R 3 ; M dv), one has 1 ε Ag ε = 1 ε (L M Â)g ε = Â 1 ε L M g ε = Â(Q M (g ε , g ε ) -Â(ε∂ t g ε + v • ∇ x g ε ) → Â(Q M (g, g) -Âv • ∇ x g . (23)
By ( 21) and the solenoidal condition in [START_REF] Hilbert | [END_REF], the second term takes the form

Âv • ∇ x g = Â ⊗ v ⊗ v : ∇ x u = ν(∇ x u + (∇ x u) T ) . (24)
Indeed Âi j A kl = ν(δ ik δ jl + δ il δ jk -2 3 δ i j δ kl ), which can be recast as ν = 1 10 Â : 3 and each R ∈ O 3 (R). This formula for ν is equivalent to the first relation in (17).

A since A(Rv) = RA(v)R T for each v ∈ R
As for the first term, since g ∈ KerL M according to [START_REF] Hilbert | Internat. Congress of Math[END_REF], one has Q M (g, g) = 1 2 L M (g 2 ) (see [3], fla. (60) on p. 338.) Hence

Â(Q M (g, g) = 1 2 ÂL M (g 2 ) = 1 2 (L M Â)g 2 = 1 2 Ag 2 = 1 2 A ⊗ v ⊗ v : u ⊗ u = u ⊗ u -1 3 |u| 2 I , (25)
in view of the elementary identity A i j A kl = (δ ik δ jl + δ il δ jk -2 3 δ i j δ kl ). Let ξ ≡ ξ (x) ∈ C ∞ c (R 3 ) be a divergence-free test vector field. Substituting (24) and ( 25) in (23) shows that

0 = ∂ t ξ • vg ε dx -∇ξ : 1 ε Ag ε dx → ∂ t ξ • u dx -∇ξ : (u ⊗ u -1 3 |u| 2 I) dx + ν ∇ξ : (∇ x u + (∇ x u) T ) dx . Since ξ is divergence-free ∇ξ : 1 3 |u| 2 I dx = 1 3 |u| 2 div x ξ dx = 0 , while ∇ξ : (∇ x u) T dx = -∇(divξ ) • u dx = 0 . Therefore ∂ t ξ • u dx -∇ξ : u ⊗ u dx + ν ∇ξ : ∇ x u dx = 0 ( 26
)
for each divergence-free test vector field ξ

≡ ξ (x) ∈ C ∞ c (R 3 ). Now, if T ∈ D ′ (R 3
) is a vector-valued distribution satisfying T, ξ = 0 for each divergence-free test vector field ξ ≡ ξ (x) ∈ C ∞ c (R 3 ), there exists a scalar distribution π ∈ D ′ (R 3 ) such that T = ∇π. In other words, (26) means precisely that u is a weak solution of the motion equation in the Navier-Stokes system.

Compactness tools

An important ingredient in the proof of all fluid dynamic limits of the Boltzmann equation considered above is the fact that the relative fluctuation of distribution function g ε := (F ε -M)/δ ε M converges in some sense, possibly after extracting some subsequence ε n → 0. The key argument is the following inequality resulting from (16):

R 3 h(δ ε g ε ) (t) dx = H(F ε (t, •, •)|M) ≤ H(F in ε |M) = O(δ 2 ε ) for each t ≥ 0 ,
for the initial data considered in the four theorems stated in the previous section, where h(z) := (1 + z) ln(1 + z)z.

Since h(z) ≃ z 2 /2 as z → 0, this control is as good as a bound in L ∞ (R + ; L 2 (R 3 × R 3 ; M dx dv)) for the values of

g ε not exceeding O(1/δ ε ). Thus (1 + |v| 2 )g ε is relatively compact in weak-L 1 ([0, T ] × [-R, R] 3 × R 3 ;
M dt dx dv) for each R, T > 0, and all its limit points as ε → 0 belong to L ∞ (R + ; L 2 (R 3 × R 3 ; M dx dv)). In the case of the acoustic or Stokes-Fourier limit, the uniqueness of the solution of the limiting fluid equations implies that the whole family (1 + |v| 2 )g ε converges weakly.

Since the leading order term in ( 20) is 1 ε L M g ε and L M is a linear operator, the weak compactness of the family (1 + |v| 2 )g ε is enough to conclude that any limit point g of that family as ε → 0 must satisfy L M g = 0, and therefore is an infinitesimal Maxwellian, i.e. is of the form [START_REF] Hilbert | Internat. Congress of Math[END_REF].

In addition, for the Navier-Stokes-Fourier limit, the compactness of the family g ε in L 1 loc for the strong topology (implying the a.e. pointwise convergence of a subsequence) is needed to pass to the limit in nonlinear terms. We use repeatedly some compactness results for moments of the distribution function in the velocity variable v based on bounds on the streaming operator -see [16,15]. These compactness results are referred to as compactness by velocity averaging. A typical example of a velocity averaging theorem used in the Navier-Stokes limit of the Boltzmann equation is as follows. We state it in the steady case for the sake of simplicity. Theorem. (Golse, Saint-Raymond [17]

) Let f n ≡ f n (x, v) be a bounded sequence in L 1 (R N x × R N v ) such that the sequence v • ∇ x f n is bounded in L 1 (R N x × R N v ), while f n itself is bounded in L 1 (R N x ; L p (R N v )) for some p > 1. Then a) f n is weakly relatively compact in L 1 loc (R N x × R N v ); and b) for each φ ∈ C c (R N ), the sequence of velocity averages R N f n (x, v)φ (v) dv is strongly relatively compact in L 1 loc (R N ).

The conservation laws

The formal argument presented above in the case of the Navier-Stokes limit shows the importance of the local conservation laws of mass, momentum and energy in the derivation of fluid dynamic models from the Boltzmann equation. Unfortunately, renormalized solutions of the Boltzmann equation are not known to satisfy the local conservation laws of momentum and energy in (7). They satisfy instead the approximate conservation laws

Sh ∂ t R 3 Γ F ε M v 1 2 |v| 2 M dv + div x R 3 Γ F ε M v ⊗ v 1 2 |v| 2 v M dv = 1 ε R 3 Γ ′ F ε M C (F ε ) v 1 2 |v| 2 dv . (27)
Therefore, one must show that the conservation defects

1 εδ ε Sh R 3 Γ ′ F ε M C (F ε ) v |v| 2 dv → 0 in L 1 loc (R + × R 3
) as ε → 0, and identify the limits as ε → 0 of the terms

1 δ ε R 3 Γ F ε M -Γ(1) v 1 2 |v| 2 M dv and 1 δ ε Sh R 3 Γ F ε M -Γ(1)   v ⊗ v -1 3 |v| 2 I (|v| 2 -5)v   M dv .
This raises an important question regarding the tail of the distribution functions F ε as |v| → +∞. That the family

(1 + |v| 2 )g ε is relatively compact in weak-L 1 ([0, T ] × [-R, R] 3 × R 3 ;
M dt dx dv) for each R, T > 0 is in general not enough -for instance, in the acoustic limit, Sh = 1 and one needs to identify the limit of the energy flux

1 δ ε R 3 Γ F ε M -Γ(1) 1 2 |v| 2 vM dv , which is a 3rd order moment in v of 1 δ ε (Γ(1 + δ ε g ε ) -Γ(1)) ∼ Γ ′ (1)g ε .
Controlling the high speed tail of (fluctuations of) the distribution function is an essential step in the derivation of fluid dynamic limits of the Boltzmann equation, and involves rather technical estimates based on the entropy and entropy production estimates (16) together with the dispersion effects of the streaming operator Sh ∂ t + v • ∇ x (see [4,18,19]).

The relative entropy method

In inviscid hydrodynamic limits, i.e. the compressible or incompressible Euler limits, entropy production does not balance streaming. Therefore the velocity averaging method fails for such limits. The idea is to use the regularity of the solution of the target equations, together with relaxation towards local equilibrium in order to obtain some compactness on fluctuations of the distribution function.

Pick for instance u, a smooth solution of the target equations -e.g. the incompressible Euler equations -and study the evolution of the quantity

Z ε (t) = 1 δ 2 ε H(F ε |M (1,δ ε u(t,x),1) )
where F ε is a renormalized solution of ( 12) with Sh = Ma = δ ε = ε α and 0 < α < 1. This is the leading order of the relative entropy of the Boltzmann solution with respect to the local Maxwellian state defined by u, in the incompressible Euler scaling. At the formal level, it is found that

dZ ε dt (t) = - 1 δ 2 ε R 3 ∇ x u : R 3 (v -δ ε u) ⊗2 F ε dv dx + 1 δ ε R 3 ∇ x p • R 3 (v -δ ε u)F ε dv dx .
The second term in the r.h.s. of the equality above vanishes with ε since 1 δ ε R 3 vF ε (t, x, v) dv → divergence-free vector field.

The key idea is to estimate the first term in the r.h.s. as follows

1 δ 2 ε R 3 ×R 3 ∇ x u : (v -δ ε u) ⊗2 F ε dv dx ≤ CZ ε (t) + o(1)
where C = O( ∇ x u L ∞ ). Then, one concludes with Gronwall's inequality. The relative entropy method stems from an idea of H.T. Yau (for Ginzburg-Landau lattice models, see [39]); it was later adapted to the Boltzmann equation by the author [START_REF] Bouchut | Kinetic Equations and Asymptotic Theory[END_REF] and Lions-Masmoudi [29]. It is especially designed to handle sequences of weak solutions of the Boltzmann equation converging to a classical solution of the fluid equation.

CONCLUSIONS

The DiPerna-Lions theory of renormalized solutions of the Boltzmann equation allows one to obtain derivations of fluid dynamic regimes from the kinetic theory of gases without unphysical assumptions on the size or regularity of the data. Following the program outlined in [4], these derivations are based on a) relative entropy and entropy production estimates, together with b) functional analytic methods in Lebesgue (L p ) spaces.

At present, the program in [4] leaves aside the compressible Euler limit of the Boltzmann equation, or the asymptotic regime leading to the compressible Navier-Stokes equations. Little progress has been made on these issues since the work of Nishida [31] and Caflisch [START_REF] Caflisch | [END_REF].

The problem of deriving fluid dynamic limits from the Boltzmann equation in the steady regime is also of considerable importance for practical applications. Formal results are of course well understood with the classical Hilbert or Chapman-Enskog expansion techniques -see the book of Sone [START_REF] Sone | Kinetic Theory and Fluid Dynamics[END_REF]. Unfortunately, the theory of the steady Boltzmann equation with large data is not as mature as its counterpart for the Cauchy problem, in spite of interesting contributions by Arkeryd and Nouri [1], and there is no analogue of the DiPerna-Lions theory for the steady case yet.

But even for evolution problems in regimes that are weakly nonlinear at the kinetic level, the relative entropy is not the solution to all difficulties. In several asymptotic regimes of the Boltzmann equation, the leading order and next to leading order fluctuations of the distribution function may interact to produce highly nontrivial macroscopic effects in the fluid dynamic limit. Examples of such asymptotic regimes are a) ghost effects, introduced by Sone, Aoki, Takata, Sugimoto and Bobylev in [34], reported in Sone's Harold Grad Lecture [START_REF] Sone | Rarefied Gas Dynamics[END_REF] and [START_REF] Sone | Kinetic Theory and Fluid Dynamics[END_REF][START_REF] Sone | Molecular Gas Dynamics, Theory, Techniques, and Applications[END_REF], and systematically studied by Sone, Aoki and the Kyoto school, b) Navier-Stokes limits recovering viscous heating terms, due to Bobylev [8] and Bardos-Levermore-Ukai-Yang [6] -see also the discussion in [START_REF] Sone | [END_REF], and c) hydrodynamic limits for thin layers of fluid -see [START_REF] Golse | From the Boltzmann equation to fluid dynamics in thin layers[END_REF].

  while the acoustic time scale is defined by T a = L/ √ θ . The dimensionless variables are t = t/T o , x = x/L, and v = v/ √ θ , while the dimensionless distribution function is F = θ 3/2 F/ρ. Introducing two dimensionless parameters, the Strouhal number Sh = T a /T o and the Knudsen number Kn = CT c /T a = 2/d 2 Lρ with

TABLE 1 .

 1 Fluid dynamic limits of the Boltzmann equation, depending on the dimensionless parameters Kn, Ma and Sh.

The notation H m (R n ) designates the Sobolev space of square integrable functions on R n whose partial derivatives of order ≤ m in the sense of distributions are square integrable functions on R n . A vector field is said to belong to H m (R n ) if all its components belong to H m (R n ).

The notation L p (R N ; f dv) (where p ≥ 1 and f > 0 is a measurable function defined a.e. on Ω) designates the set of measurable functions φ defined a.e. on R N that satisfyΩ |φ (v)| p f (v) dv < +∞ .

Acknowledgements. The author thanks Profs. Aoki, Levermore and Sone for their generous scientific advice during the preparation of this paper.