## Abstract

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 elementary particles. The state of a gas can therefore be described in a statistical way. The evolution, which introduces couplings, loses part of the information, which is expressed by the decay of the so-called mathematical entropy (the opposite of physical entropy!).

## 1. The Boltzmann equation and the entropy inequality

More precisely, the state of a monatomic gas is described by some distribution function *f*=*f*(*t*,*x*,*v*), which measures the density of particles having position and velocity at time *t*. The Boltzmann equation predicts its evolution under the combined effect of transport and collisions.

### (a) The Boltzmann equation

In the absence of collisions and external forces, according to Newton’s principle, particles move freely, with uniform rectilinear motion. Changing our minds from the Lagrangian point of view,
to the Eulerian point of view, we obtain the free transport equation
If the interactions are governed by a very short-range potential, then one can assume, at first sight, that collisions are local in both *x* and *t* and consider only binary collisions (the occurrence of multiple collisions being extremely rare). Because of the conservation of momentum and energy (elastic collisions), post- and pre-collisional velocities (*v*,*v*_{1}) and (*v*′,*v*′_{1}) have to satisfy the following relations:
for some *n*∈*S*^{2}. The statistical distribution of deflection angles *n* is then predicted by a function *b*=*b*(|*v*−*v*_{1}|,(*v*−*v*_{1})⋅*n*), referred to as the collision cross section, which depends only on the microscopic interaction potential.

A counting argument leads then to the Boltzmann equation [1],
the Mach number *Ma* and Knudsen number *Kn* being physical parameters which will be discussed later on. The collision term, which acts only on the *v*-variable, comprises a gain term, corresponding to the creation of particles of velocity *v* by collision between particles of velocities *v*′ and *v*′_{1}, and a loss term, owing to the disappearance of particles of velocity *v* because of collision with particles of velocity *v*_{1}.

Note that the joint probability of having particles of velocity (*v*′,*v*′_{1}) (respectively of velocities (*v*,*v*_{1})) before the collision is assumed to be equal to *f*(*t*,*x*,*v*′)*f*(*t*,*x*,*v*′_{1}) (respectively to *f*(*t*,*x*,*v*)*f*(*t*,*x*,*v*_{1})), meaning that there is independence.

### Remark 1.1

This assumption is crucial to the appearance of irreversibility; some comments are in order:

— the direction of time is fixed by an arbitrary choice, one of the date at which complete information is given (in the sense that all marginals of the distribution function at

*N*particles are supposed to be known); the amount of information which is lost depends on the (non-negative) distance to this particular date;— chaos (which is the usual terminology for independence, that is, for the factorization of marginals) is propagated only asymptotically in the thermodynamic limit ; and

— the Boltzmann equation predicts the most probable evolution: this does not exclude the existence of pathological behaviours but indicates that their probability vanishes asymptotically as .

It seems then quite obvious that the irreversibility encoded in the Boltzmann equation is related to probability. We will give a quantitative statement in the next section.

The main properties of the Boltzmann equation come actually (at least at a formal level) from the following symmetry argument: for all and all ,
which lies on the structure of the collision cross-section *b* and on the change of variables (*v*,*v*_{1})↦(*v*′,*v*′_{1}) with Jacobian 1.

By the definition of *v*′ and *v*′_{1}, the functions 1, *v* and |*v*|^{2} are obviously invariants of collisions, namely
In particular, integrating the Boltzmann equation against these functions, we retrieve the local conservations of mass, momentum and energy
which is not surprising because the elementary collision processes conserve all these quantities.

Starting from the same identity, and using both the additivity and the monotonicity of the logarithm, we obtain
with equality if and only if *f* is a Gaussian
also called a Maxwellian, because it was the equilibrium distribution predicted by Maxwell. This means, in particular, that for the solutions to the Boltzmann equation one has
The functional is therefore a Lyapunov functional for the Boltzmann dynamics. Unless the distribution *f* is a global Maxwellian (with parameters *ρ*,*u* and *θ* not depending on *t* and *x*), we therefore expect the dynamics not to be reversible.

### (b) The entropy inequality

Consider a gas in the infinite space , assumed to be at rest with the distribution *M* at infinity: without loss of generality, we can suppose that
Combining both the mass and energy conservations and the entropy inequality, we obtain the following inequality:
We then define two convex non-negative functionals; namely,

— the

*relative entropy*1.1— and the

*entropy dissipation*1.2

so that the relative entropy inequality can be rewritten in much shorter form 1.3

### Remark 1.2

This inequality is the only (*a priori*) estimate which corresponds to physical principles. By the combination of energy and entropy, we indeed obtain the free energy. We then expect that it provides a suitable mathematical framework for the study of the Boltzmann equation.

— The bound coming from the control on the relative entropy is not sufficient in order that the collision term

*Q*(*f*,*f*) makes sense (at least in the non-homogeneous case). The collision operator, which acts more or less like a convolution with respect to the*v*variable, is indeed only a multiplication with respect to*t*and*x*.— Provided that we define a very weak notion of solution (renormalized solution in the sense of DiPerna and Lions), it is however the only framework in which the existence of global solutions without restriction on the size of the initial data is proved [2]. The counterpart is that the uniqueness is not known, nor is the local conservation of momentum and energy, nor even the fact that the kinetic equation is satisfied in the sense of distributions.

— Note that, in order to derive the Boltzmann equation from the principles of Hamiltonian mechanics [3], we need in particular to assume that the macroscopic density of the gas is small enough, because concentrations lead to pathological trajectories (with multiple collisions or recollisions breaking chaos). Under such an assumption ( bound on the moments of

*f*), renormalized solutions are nothing but weak solutions. The point is that it is not clear at all that such a bound could be propagated.— It is then natural to ask whether renormalized solutions are physically relevant. A partial answer to this question is given by the study of hydrodynamic limits.

### (c) Hydrodynamic regimes

The problem of hydrodynamic limits concerns the connection between the Boltzmann equation and fluid models (which correspond to a rougher level of description). The question is to know whether these various descriptions are consistent, at least in some physical regimes. We can also hope to have a refined understanding of entropy and irreversibility at the macroscopic level, because these notions seem to have been introduced heuristically at the origin.

The idea, formulated by Hilbert to solve in his sixth problem [4], is the following: if elementary particles in a gas undergo many collisions, then the local thermodynamic equilibrium should be reached almost instantaneously. In the first approximation, the solution *f* to the Boltzmann equation should therefore be a Maxwellian, completely characterized by its density, bulk velocity and temperature. In other words, we expect that the gas dynamics reduces to a system of hydrodynamic equations.

Hydrodynamic approximations of the Boltzmann equation are thus obtained in the fast relaxation limit, i.e. when the Knudsen number

The relevant fluid model depends then on the physical characteristics of the flow to be considered; in particular, on the other non-dimensional parameter arising in the Boltzmann equation, namely the Mach number This parameter measures the compressibility of the gas: we indeed expect the acoustic waves, the speed of which is determined by the temperature, to compensate for the fluctuations in density. The higher the thermal speed, the weaker the compressibility.

Note that the Knudsen and Mach numbers together completely characterize the dynamics for perfect gases (i.e. those gases for which the total volume occupied by microscopic elementary particles is negligible compared with the macroscopic volume of the gas). For instance, the Reynolds number *Re* measuring the inverse viscosity satisfies the von Karman relation
The compressible Euler limit is the first-order approximation obtained in the fast relaxation limit, it can be derived easily at a formal level using, for instance, Grad’s moment method, namely writing the local conservation laws together with the constraint that the gas has to be locally at thermodynamic equilibrium. For small Mach number, the incompressible Navier–Stokes equation is a good approximation of the compressible system [5], which is used, for instance, in applied computations.

Note that not all fluid models can be recovered from the Boltzmann equation. For instance, the compressible Navier–Stokes equations are not an admissible asymptotics because we should have *Ma*=*O*(1), *Re*=*O*(1) and *Kn*→0. This is related to the fact that the Boltzmann equation is derived under a rarefaction assumption, which implies in particular that there is no excluded volume in the state relation of the gas. However, the compressible Navier–Stokes equations can be viewed as a second-order correction, with respect to the compressible Euler equation (see [6–8]).

### Remark 1.3

In all existing mathematical works aimed at deriving hydrodynamic limits, the general strategy 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.

An important part of the literature is devoted to the study of hydrodynamic limits in the framework of smooth solutions. Let us mention, for instance, the pioneering works of Nishida [9] and Caflisch [10] on the compressible Euler limit, and of Bardos & Ukai [11] on the incompressible Navier–Stokes limit (see also [12]). The main drawback of these results is that they require regularity and smallness assumptions on the initial data, which do not seem to be physically relevant.

We focus here on the study of hydrodynamic limits in the framework of solutions defined by the physical energy and entropy bounds, both at the kinetic level and at the fluid level whenever possible. The aim of this paper is precisely to detail the analogies between these kinetic and fluid models, focusing our attention on the use of the entropy inequality and the implications for functional analysis. Convergence results are stated in theorems 2.3 and 3.3, and we refer to [13] for the details of the proofs.

## 2. Incompressible viscous regimes: the entropy inequality as a uniform *a priori* estimate

The hydrodynamic regime which is the best understood at the present time is the one leading to the incompressible Navier–Stokes equations. It is indeed the only asymptotics of the Boltzmann equation for which an optimal convergence result is known. By ‘optimal’, we mean here that this convergence result

— holds globally in time;

— does not require any assumption on the initial velocity profile;

— does not assume any constraint on the initial thermodynamic fields; and

— takes into account boundary conditions and describes their limiting form.

Here, we do not present the convergence proof in detail (in particular, we will consider only the case ). We rather focus on the use of the scaled relative entropy inequality
2.1
obtained for *Ma*=*Kn*=*ε*.

It is indeed the central tool of the asymptotic analysis, exactly as its limiting form, the Leray energy inequality, in the theory of weak solutions to the incompressible Navier–Stokes equations.

### (a) Control of the fluctuation

In order that the scaling assumption is consistent, we consider initial data which are a perturbation of order *ε* around *M*, for instance
so that
We indeed check that, for such distributions, the thermal speed is typically of order 1, whereas the bulk velocity is of order *ε*, which guarantees that the Mach number is effectively of order *ε*.

The first use of the scaled relative entropy inequality is to ensure that the solution to the Boltzmann equation remains a perturbation of order *ε* around *M* for all times, that is, to give some *uniform a priori estimate on the fluctuation* *g*_{ε} defined by
The relative entropy can indeed be recast in the following form:
Because *h*(*z*)∼*z*^{2}/2 as *z*→0, we thus expect the scaled relative entropy to control the *L*^{2} norm of the fluctuation *g*_{ε}

Nevertheless, for fixed *ε*, we have only an *L*^{1}-bound. It comes, for instance, from Young’s inequality
which provides
2.2

Refined estimates can be obtained, introducing the *renormalized fluctuation*
2.3
A simple functional inequality indeed shows that
so that
2.4
Note that the identity allows us, in particular, to retrieve the previous *L*^{1}-bound on the fluctuation.

### (b) Control on the relaxation

As we consider fluctuations around *M*, it is natural to introduce the linearized collision operator
which is a self-adjoint Fredholm operator with the kernel spanned by 1,*v*,|*v*|^{2} (see [14,15] for the proof) and satisfies the coercivity estimate
where *Π* denotes the orthogonal projection on .

Rewriting the kinetic equation in terms of the fluctuation, we obtain
2.5
from which we deduce formally that *g*_{ε} should be close to its orthogonal projection on , which is the expected relaxation towards an infinitesimal Maxwellian.

The second use of the scaled relative entropy inequality is to *control that relaxation process*. The previous argument is indeed not correct in the framework of renormalized solutions, because

— the kinetic equation is not known to be satisfied (even in the sense of distributions) and

— the fluctuation

*g*_{ε}is not in*L*^{2}, so that the Hilbertian theory of cannot be applied.

Our starting point here is therefore the identity for the renormalized fluctuation
2.6
based on the bilinearity of *Q*.

The estimate on the renormalized fluctuation obtained in the previous paragraph, together with easy continuity properties of *Q* (possibly after suitable truncations of the collision cross section *b*), provides a control on the first term in (2.6),
The second term in the right-hand side of (2.6), which is a kind of renormalized collision term, is controlled by the *entropy dissipation*
using the same kind of functional inequality as for (2.4).

We therefore obtain the relaxation estimate
2.7
denoting by *ρ*_{ε}, *u*_{ε} and *θ*_{ε} the density, bulk velocity and temperature associated with the fluctuation , respectively.

### Remark 2.1

Of course this step is specific to the study of the hydrodynamic limits of the Boltzmann equation and has no analogue at the level of the incompressible Navier–Stokes equations: it corresponds indeed to the control of the purely kinetic part of the solution, the one which is precisely neglected in the fluid approximation.

This estimate is, however, crucial in order to understand the dissipation process, which also appears at the macroscopic level and will be studied in the next paragraph.

Starting from the approximate conservation of mass, momentum and energy and using identity (2.6) to decompose the flux terms in the convection and diffusion terms, we obtain which gives the limiting constraint equations as well as the consistency of the Navier–Stokes–Fourier approximation and where denotes the Leray projection onto divergence-free vector fields. The major difficulty is then to establish the stability.

### (c) Limiting form of the entropy inequality in viscous regimes

The specificity of the viscous incompressible regime is that the effects of transport and collisions are comparable in the sense that we have some limiting form of the kinetic equation. Using the regularizing properties of the transport, we therefore expect to control the spatial regularity of the macroscopic fields *ρ*_{ε}, *u*_{ε}, *θ*_{ε} defining the hydrodynamic projection . Note that this control of the spatial regularity by the dissipation is reminiscent of the Leray energy inequality [16].

Start from the following (admissible) square-root renormalization of the scaled Boltzmann equation:
Easy computations based on the estimates of the two previous paragraphs show that, up to a remainder which is controlled in some negative Sobolev space, we then have the identity
2.8
Denote by *g* and *q* some joint limit points of and . We finally obtain the *limiting kinetic equation*
which expresses the fact that diffusion is related to the collision process.

From (2.8), we first deduce—using an averaging lemma (see [17,18,19] for precise statements)—that
Actually, it is even possible to obtain a more stringent result, decomposing the moment into two terms, the first one having a certain amount of regularity with respect to *x* and the second one being small, typically of order *ε*.

### Remark 2.2

This strong compactness is of course a key ingredient [20,21] to study the asymptotics of the motion and temperature equations, insofar as they involve convection terms (which are quadratic functions of the bulk velocity and temperature).

The point is that we have no similar result to control the dependence with respect to time: this is because of the presence of fast oscillating waves, referred to as acoustic waves, which account for the weak compressibility of the gas and which have to be filtered to obtain some strong convergence.

Filtering fast temporal oscillations and using compensated compactness arguments, we obtain the following convergence statement:

### Theorem 2.3

*Let (g*_{ε,0}*) be a family of initial fluctuations satisfying the uniform entropy bound
*
*For any ε, consider a (renormalized) solution g*_{ε} *to (*2.5*) with initial data g*_{ε,0}.

*Then, (g*_{ε}*) is weakly compact in* *and any of its limit point is of the form
*
*where (ρ,u,θ) is a solution to the Navier–Stokes–Fourier system.*

From (2.8), we also obtain the *limiting entropy inequality*,
In general, the inequality is strict because part of the entropy is dissipated in the relaxation layer and part of the entropy remains trapped in acoustic waves.

### Remark 2.4

In an incompressible viscous regime, the entropy inequality is therefore the exact counterpart of the Leray energy inequality for the Navier–Stokes equations, The strategy to study the hydrodynamic limit is thus very close to the argument of Leray, proving the weak stability of solutions to the incompressible Navier–Stokes equations. Most of the technical difficulties actually come from the fact that functional spaces associated with the entropy inequality are only asymptotically Hilbert spaces, so that renormalization techniques have to be used repeatedly.

## 3. Incompressible inviscid regimes: entropy as a suitable metric

Owing to the lack of regularity estimates for inviscid incompressible models, the convergence results describing the incompressible Euler asymptotics of the Boltzmann equation require

— additional regularity assumptions on the solution to the target equations;

— very restricting conditions on the initial data, both on the profile and on the moments; and

— transparent boundary conditions, namely specular reflection.

The strategy of proof is indeed completely different. We start here from the Boltzmann equation with the scaling *Ma*=*ε* and *Kn*=*ε*^{q} for *q*>1
3.1
so that the scaled relative entropy inequality (2.1) is replaced by
3.2
A rapid inspection shows then that the uniform fluctuation and relaxation estimates obtained in the previous section are still relevant, whereas the regularity estimate fails. This corresponds to the fact that there is no viscous dissipation controlling the gradient of the bulk velocity in the limiting model.

The idea is therefore to use, instead of weak stability arguments, some kinetic counterpart of the strong–weak stability principle [22]. A natural way to build the metrics to measure this stability is to modulate the entropy, as explained in the next paragraph.

### (a) The modulated entropy

The idea of using the notion of relative entropy for this kind of problem (first developed by Golse [23], then by Lions & Masmoudi [24]) comes, on the one hand, from the notion of entropic convergence developed by Bardos *et al.* [25] and, on the other hand, from Yau’s [26] elegant derivation of the hydrodynamic limit of the Ginzburg–Landau lattice model.

Precisely, the *modulated entropy* is defined for each (non-negative) test distribution by
3.3
Because the integrand is non-negative and vanishes if and only if , we expect the modulated entropy to measure in some sense the distance between *f*_{ε} and . More precisely, the fluctuation estimates (2.2) and (2.4) can be adapted to obtain some *L*^{1} control on the scaled difference (and even some *L*^{2} estimate on a suitable renormalization of that difference).

For the sake of simplicity, we will consider here only the case when is a local Maxwellian of unit density, unit temperature and bulk velocity *εw* where is a divergence-free vector field.

We indeed assume that the initial data are well prepared in the sense that
This implies in particular that
so that the initial fluctuation is nothing else but *g*^{in}=*u*^{in}⋅*v*.

### Remark 3.1

Such an assumption can be relaxed, provided that we are able to describe precisely both the relaxation layer and the acoustic waves, which are the main obstacles to entropic convergence. The point is that, in such a situation, all the moment equations (density, bulk velocity and temperature) are coupled via the acoustic waves: we have then to consider solutions to the Boltzmann equation having better integrability properties than renormalized solutions in order that the energy flux is even defined [27].

Because of the convexity of the entropy functional, the *entropic convergence* implies the strong convergence of moments.

Simple computations indeed show that
denoting by *M*_{fε} the local Maxwellian having the same moments (1+*ερ*_{ε},*εu*_{ε},1+*εθ*_{ε}) as *f*_{ε}. We also have the identity
3.4
with .

Defining *u* as the weak limit of (*u*_{ε}), we therefore have
3.5

### (b) The stability inequality

Combining the entropy inequality and the conservation laws, we expect to control the modulated entropy for all times, provided that the vector field *w* satisfies suitable conditions. More precisely, we will prove that the modulated entropy converges towards 0 if

—

*w*coincides initially with the data*w*^{in}=*u*^{in}, so that—

*w*is a solution to the incompressible Euler equation

The functional inequality (3.5) then shows that the limiting bulk velocity *u* is the solution to the incompressible Euler equations with initial data *u*^{in}.

The first step of the proof is therefore to differentiate the modulated entropy with respect to time. Of course, such a computation is formal and only the integrated form of the inequality is known to be satisfied for renormalized solutions in the sense of DiPerna and Lions. More precisely, we have the following *modulated entropy inequality*: for any smooth divergence-free vector field *w*
3.6
where *m*_{ε} is some defect measure owing to the non-conservation of energy for renormalized solutions (which can be traced back to entropy inequality (3.2)).

### Remark 3.2

If, instead of modulating only the bulk velocity, we also modulate the temperature, then the same formal computation involves the local conservation of energy For renormalized solutions, such an identity is not known to hold, even introducing defect measures: there is indeed no control on the third moment.

An alternative should consist in modulating some functional, which is controlled by the entropy but has a slower growth at infinity, and in using renormalized conservation laws. Such a renormalized entropy method is introduced in [28] to study fast relaxation limits leading to viscous magneto-hydrodynamics. The point here (in an inviscid regime) is that we have no control on conservation defects.

The right-hand side of (3.6) comprises three terms:

— The initial modulated entropy, which tends to zero provided that

*w*^{in}=*u*^{in}.— The so-called acceleration term, which depends linearly on the bulk velocity

*u*_{ε}, and the limit of which is As*u*is divergence-free (the incompressibility constraint being a simple consequence of the local conservation of mass), it can be rewritten so that its contribution is zero if*w*is a solution to the incompressible Euler equations.— The flux term which has to be controlled in terms of both the modulated entropy and the entropy dissipation in order to obtain the expected stability inequality.

The flux term is indeed decomposed using the collision operator linearized around , denoted , and the identity
which is the counterpart of (2.6) in the case when the reference state is the local Maxwellian instead of the global Maxwellian *M*.

We thus identify two terms, a convection term of the type to be estimated by the scaled modulated entropy (using some functional inequality such as (3.5)) and a diffusion term controlled by the entropy dissipation.

### (c) Limiting form of the stability inequality in inviscid regimes

By Gronwall’s lemma, we then have
Taking limits as *ε*→0, we thus obtain
3.7
for any divergence-free vector field .

Stability inequality (3.7) provides the expected convergence *under some smoothness condition* on the solution to the limiting equations (Lipschitz regularity). Note that we also have the convergence of the scaled entropy dissipation to 0, which is consistent with the fact that the initial data are well prepared.

### Theorem 3.3

*Let* *be some divergence-free vector field. For any fixed ε, consider a (renormalized) solution g*_{ε} *to (*3.1*) with initial data u*_{0}*⋅v.*

*Then, (g*_{ε}*) is weakly compact in* *and converges entropically to
*
*where u is the Lipschitz solution to the incompressible Euler equation, as long as the latter does exist.*

### Remark 3.4

Relaxing the regularity assumption should require new ideas: the stability in energy and entropy methods is indeed controlled by the Lipschitz norm of the limiting field.

In three dimensions, the incompressible Euler equations are not known to have weak solutions, so that we do not expect to extend our convergence result for distributions with lower regularity. In return, in two dimensions, the mathematical theory of the incompressible Euler equations is much better understood and singular solutions, for instance vortex patches, are known to exist globally in time: it should then be relevant to study the hydrodynamic limit of the Boltzmann equation in this setting.

Improvements of the modulated entropy method to obtain stability around non-smooth solutions have been obtained at the present time only for scalar conservation laws.

Stability inequality (3.7) is the exact counterpart of the *Dafermos weak–strong uniqueness principle* for the incompressible Euler equations [29]
3.8

### Remark 3.5

Starting from (3.8), Lions [30] has defined dissipative solutions to the incompressible Euler equations with initial data *u*^{in} to be functions satisfying ∇⋅*u*=0 and *u*_{|t=0}=*u*^{in} in the sense of distributions and such that
for all and all divergence-free test functions .

Such solutions always exist globally in time; they are not known to be weak solutions of the incompressible Euler equations in conservative form, but they coincide with the unique smooth solution with same initial data as long as the latter do exist.

Note that the notion of dissipative solutions has been introduced especially to investigate the inviscid limit of the incompressible Navier–Stokes equations, and then used to study various asymptotics, such as the gyrokinetic or quasi-neutral limits of the Vlasov–Poisson equation.

## 4. What about the compressible regime?

The study of the compressible (inviscid) limit of the Boltzmann equation is at the present time rather at the stage of project than of result, even of partial result. We can however draw the main lines of the possible strategy by analogy with what is known on the limit model.

Some of the difficulties have already been mentioned in the incompressible case (inviscid regime):

— lack of regularity or weak regularity of the solution to the limit system and

— control of large velocities, especially of the third moment in

*v*.

Other difficulties are specific to the compressible case:

— the entropy space is no longer asymptotically Hilbertian and

— the relaxation far from equilibrium is poorly understood.

In the medium term, the hope is to obtain results in one space dimension (that is, assuming translation invariance with respect to the two other directions) 4.1 because the hyperbolic system of conservation laws obtained as the formal limit then has a relatively well-understood Cauchy theory for weak solutions (in the sense of Glimm).

### (a) *A priori* estimates coming from the entropy inequality

The modulated entropy and modulated entropy dissipation define some kind of *non-Hilbertian distances*.

Computations (3.4) performed in the previous section show, for instance, that the relative entropy controls the following macroscopic functionals:
denoting by *M*_{f} the local Maxwellian having the same moments (*R*,*U*,*Θ*) as *f*. This means that *R* belongs to some Orlicz space and that (away from vacuum) the bulk velocity is typically in *L*^{2}, whereas the temperature is only in *L*^{1}.

As in the case of the inviscid incompressible limit and even though the entropy dissipation is expected to converge in some sense towards a measure carried on shocks, we are not able to obtain uniform regularity estimates on the moments. Indeed, the scaling of the entropy dissipation does not allow us to control the transport
whatever the renormalization *γ*.

### Remark 4.1

Note that the control obtained from the entropy bound is much weaker than the *a priori* estimates that are known for Glimm’s solutions to the one-dimensional compressible Euler equations [31], typically and *BV* bounds.

The counterpart of Glimm’s functional at the kinetic level, the so-called potential for interaction of the one-dimensional Boltzmann equation (introduced by Bony and used by Cercignani [32] to build weak distributional solutions), is not known to provide *any control on the regularity of moments*.

### (b) Entropy–entropy dissipation inequalities and relaxation

In order to derive rigorously compressible hydrodynamic limits of the Boltzmann equation, we need to understand the effects of the nonlinear relaxation process, especially to establish quantitative variants of the mechanism for decreasing the entropy. More precisely, one would like to prove an entropy–entropy dissipation inequality: this is a functional inequality of the type
where *H*↦*Θ*(*H*) is some continuous function, strictly positive when *H*>0. Such an inequality coupled with Boltzmann’s *H* theorem would indeed imply that any solution to the spatially homogeneous Boltzmann equation satisfies *H*( *f*|*M*_{f})(*t*)→0 as , and one would be further able to compute an explicit rate of convergence.

An old conjecture of Cercignani, formulated at the beginning of the 1980s, was that the spatially homogeneous Boltzmann equation would satisfy a linear entropy–entropy dissipation inequality
4.2
for some *λ*( *f*) depending on *f* only via some estimates of moments, Sobolev regularity, lower bound. Bobylev & Cercignani [33] have actually disproved this conjecture, by considering distributions close to the equilibrium *M*, with a very tiny bump at large velocities.

However, Mouhot [34] has established that the *exponential trend to equilibrium* which should be implied by (4.2), namely
4.3
holds true in the particular case of hard spheres (and for more general hard potentials with a cut-off, under a very strong decay condition). The idea of the proof is to combine linear and nonlinear techniques: quantitative estimates of exponential decay on the evolution semigroup associated with the linearized collision operator are used to estimate the rate of convergence when the solution is close to equilibrium (where the linear part of the collision operator is dominant), whereas the existing nonlinear entropy method, combined with some *L*^{1} *a priori* estimates, is used to estimate the rate of convergence for solutions far from equilibrium.

Such a method should open new perspectives in the field of compressible hydrodynamic limits. Of course, this would suppose that we could obtain pointwise estimates on the moments of the solution to the spatially inhomogeneous Boltzmann equation, which remains an outstanding problem.

### Remark 4.2

The results we have mentioned here actually concern the homogeneous Boltzmann equation. The relaxation mechanism is much less understood in the inhomogeneous case.

However, in the limit *Kn*→0, we expect the relaxation mechanism to be essentially local so that the homogeneous theory should apply. A rigorous argument, based on the modulation of the entropy dissipation (which is a convex functional), has been developed for instance to describe relaxation layers in incompressible hydrodynamic limits (see [27]).

### (c) Which strategy?

What we would like to prove first is the following one-dimensional statement.

### Conjecture 4.3

Let (*R*_{0},*U*_{0},*Θ*_{0}) be some small *BV* fluctuation of the equilibrium state (1,0,1). For any fixed *ε*, consider a (renormalized) solution *f*_{ε} to (4.1) with initial data *M*_{R0,U0,Θ0}.

Then, ( *f*_{ε}) is weakly compact in and any of its limit point is some local thermodynamic equilibrium
such that (*R*,*U*,*Θ*) is a weak solution to the compressible Euler equation.

The previous discussions show that the strategies developed in the framework of incompressible hydrodynamic limits are expected to fail in the compressible regime.

In the absence of regularity estimates (with respect to space and time) on the moments,

*Grad’s moment method*should require at least— either some compensated compactness to take limits in nonlinear convection terms

— or some lower continuity of the entropy dissipation with respect to weak topology to take limits in the constraint equation

At the present time, both statements seem out of reach.

The

*modulated entropy method*is expected to bypass the difficulties coming from the lack of uniform regularity estimates on the moments, using the stability of the limiting system. However,— even though unique, the weak entropic solutions built by Bressan & Bianchini [35] are not known to be strongly stable, because they are not sufficiently smooth (no Lipschitz regularity);

— even restricting our attention to smooth solutions to the compressible Euler equations, we have no stability inequality at the kinetic level because of the lack of control on large velocities.

Note that large velocities and large spatial frequencies seem to play very similar roles, which is consistent with the form of the free transport operator.

An alternative to both methods is to define solutions of the compressible Euler equations by some exact (and not only asymptotic) kinetic formulation and to prove the *weak stability of this kinetic formulation*. We could then hope to prove the convergence of scaled Boltzmann equation (4.1), in the fast relaxation limit *ε*→0, directly at the kinetic level. This is the purpose of a work in progress with Diogo Arsenio.

## Footnotes

One contribution of 11 to a Theme Issue ‘Entropy and convexity for nonlinear partial differential equations’.

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