PERSISTENCE OF BOLTZMANN ENTROPY IN FLUID MODELS

Higher order entropies are 
kinetic entropy estimators 
suggested by Enskog expansion of Boltzmann entropy. 
These quantities are 
quadratic in the 
density $\rho$, velocity $v$ 
and temperature $T$ renormalized derivatives. 
We investigate asymptotic expansions of 
higher order entropies for 
compressible flows 
in terms of the Knudsen 
$\epsilon_k$ and Mach $\epsilon_m$ numbers 
in the natural situation where 
the volume viscosity, the shear viscosity, and the 
thermal conductivity 
depend on temperature, 
essentially in the form $T^x$. 
Entropic inequalities 
are obtained when 
||$\log \rho$||BMO,$\quad$ 
$\epsilon_m$||$v/\sqrt{T}$|| L ∞ ,$\quad$ 
||$\log T$||$BMO$,$\quad$ 
$\epsilon_k$||$h\partial_{x} \rho$/$\rho$|| L ∞ , 
 
$\epsilon_k$$\epsilon_m$||$h\partial_{x} v$/$\sqrt{T}$ || L ∞ , 
 
$\epsilon_k$||$h\partial_{x}T$/$T$|| L ∞ , 
 
and 
$\epsilon_k^2$||$h^2\partial^2_x T$/$T$|| L ∞ 
 
are small enough, 
where 
$h = 1/(\rho T^{(1/2) -x)}$ 
is a weight associated with the dependence on 
density and temperature of the mean free path.


Introduction
The notion of entropy has been shown to be of fundamental importance in fluid modeling from both a physical and mathematical point of view [3,4,5,6,7,8,10,15,16,19,23,28].We have introduced in previous work [11,12,13,14] a notion of kinetic entropy estimators for fluid models, suggested by Enskog expansion of Boltzmann kinetic entropy.Conditional higher order entropic inequalities have been established in the situations of incompressible flows [11,12,13] as well as compressible flows [14] spanning the whole space.
In this paper, we investigate asymptotic expansions of higher order entropies for compressible fluids and study the corresponding conditional entropic inequalities when the Mach number ǫ m and the Knudsen number ǫ k are small.In contrast, although higher order entropies are suggested by Enskog expansion, only incompressible fluids [11,12,13] or compressible fluid equations with coefficients of order unity were considered in previous work [14].
In Section 2 we first summarize the mathematical and physical motivations for higher order entropies which are kinetic entropy estimators for fluid models.The corresponding balance equations may also be seen as a thermodynamic generalization of Bernstein equation to systems of partial differential equations associated with renormalized variables [12].
We introduce in Section 3 the natural rescaled variables and small parameters associated with fluid models, notably the Mach number ǫ m and the Knudsen number ǫ k .Thanks to the rescaled variables, the governing equations and the higher order entropies are rewriten in terms of ǫ m and ǫ k .We also introduce the molecular coordinates-associated with the particle collision time and the mean free path-which are such that all small parameters are eliminated from the corresponding system of partial differential equations.The volume viscosity, the shear viscosity and the thermal conductivity are assumed to depend on temperature as given by the kinetic theory, that is, essentially in the form of a power law of temperature T κ with a common exponent κ.
In Section 4 we summarize weighted inequalities in Sobolev and Lebesgue spaces [12,14].These inequalities are required for renormalized variables with powers of density or temperature as weights as well as for fluid models with temperature dependent thermal conductivity and viscosities.We further specify how the various inequalities are transformed by a change of scale in the coordinate system.
In Section 5 we investigate Boltzmann kinetic entropy estimators taking into account the natural small parameters of fluid models.We derive parameter dependent balance equations for higher order entropic correctors as well as for extra correctors associated with density which is a hyperbolic variable.We then study entropic estimates by combining the correctors balance equations with weighted inequalities.Entropic estimates are obtained when the quantity is small enough, where h = 1/ρT 1 2 −κ is a weight associated with the dependence on temperature and density of the mean free path.
Note that the quantity χ is small when the Mach number is small since we formally have χ = O(ǫ m ).Assuming that the Mach number is small is equivalent to the underlying assumption of a small Knudsen number since ǫ m = Re ǫ k where Re is the Reynolds number.In addition, χ is scaling invariant for the changes of scales naturally associated with the solutions of the compressible Navier-Stokes equations.Finally, kinetic entropy estimators are shown to be closely associated with Sobolev norms of the fluid entropy in molecular coordinates.

Higher order entropies
In this section we briefly motivate the introduction of higher order entropies by discussing Bernstein equations and inspecting Enskog expansion of Boltzmann kinetic entropy [11,12,14]

A thermodynamic interpretation of Bernstein equations
For parabolic or elliptic scalar equations, a priori gradient estimates can be obtained by using Bernstein method [1,22].Considering the heat equation 2 , Bernstein equation for the k th derivatives can be written in the form The structure of (2.1) appears to be formally similar to that of an entropy balance, where |∂ k u| 2 , k ≥ 1, play the rôle of generalized entropies, even though there also exist zeroth order entropies like u 2 .In the next section, we introduce a kinetic framework supporting this entropic interpretation.

Enskog expansion of Boltzmann kinetic entropy
In a semi-quantum framework, the state of a polyatomic gas is described by a particle distribution function f (t, x, c, i)-governed by Boltzmann equation-where t denotes time, x the n-dimensional spatial coordinate, c the particle velocity, i the index of the particle quantum state, and I is the corresponding indexing set [4,6,8,10].Approximate solutions of Boltzmann's equation can be obtained from a first order Enskog expansion f = f (0) 1 + εφ (1) + O(ε 2 ) where f (0) is the local Maxwellian distribution, φ (1) the perturbation associated with the Navier-Stokes regime and ε the usual Enskog formal expansion parameter.The compressible Navier-Stokes equations can then be obtained upon taking moments of Boltzmann's equation [5,8,10].
The kinetic entropy where k B denotes Boltzmann constant, satisfies the H theorem, that is, the second principle of thermodynamics.Enskog expansion f /f (0) = 1 + εφ (1) ) then induces expansions for S kin in the form where S (0) is the zeroth order fluid entropy evaluated from f (0) and where S (j) is a sum of terms in the form k B i∈I R n 1≤i≤j φ (i) νi f (0) dc with nonnegative integers ν i ≥ 0, 1 ≤ i ≤ j, such that j = 1≤i≤j iν i .For compressible polyatomic gases, using a single term in orthogonal polynomial expansions of perturbed distribution functions, one can establish that where T denotes the absolute temperature, ρ the density, v the gas velocity, specific heat per unit mass, c v the constant volume specific heat per unit mass, r g the gas constant per unit mass, c int the internal specific heat per unit mass, λ the thermal conductivity, η the shear viscosity, κ the volume viscosity, and the actual values of the numerical factors are evaluated here for n = 3.
From the general expression of φ (i) in the absence of external forces acting on the particles, one can further establish that for any j ≥ 2 where and where the coefficients c ν are smooth scalar functions of log T of order unity.In the even case j = 2k, after integrations by parts in R n S (2k) dx, in order to eliminate spatial derivatives of order strictly greater than k, and by using interpolation inequalities, one obtains that | R n S (2k) dx| is essentially controled by the integral of or equivalently of and in the odd case 1] dx.This suggests quantities in the form γ [k] or γ [k] as (2k) th order kinetic entropy correctors-or kinetic entropy deviation estimators [11,12,14].We are thus investigating majorizing entropic correctors that we are free to modify for convenience, e.g., by multiplying the temperature derivatives by the factor c v /r g .These correctors may also be rescaled by mutiplicative constants depending on k and their temperature dependence may be simplified in accordance with that of transport coefficients.Finally, a similar analysis can also be conducted for the Fisher information and suggests the same quantities γ [k] or γ [k]  as higher order kinetic information correctors.

Persistence of Boltzmann entropy
Denoting by γ [0] a nonnegative quantity associated with the zeroth order entropy S (0) , we will investigate entropicity properties of the kinetic entropy estimators γ for the solutions of a second order system of partial differential equations modeling a compressible fluid.For this fluid system, the zeroth order entropy S (0) is already of fundamental importance as imposed by its hyperbolic-parabolic structure and the corresponding symmetrizing properties [10,15,19,20].We thus only consider the quantities γ , 0 ≤ k ≤ l, as a family of mathematical entropy estimators-of kinetic origin-and we will establish that they indeed satisfy conditional entropic principles for solutions of Navier-Stokes type equations, so that, in some sense, there is a persistence of Boltzmann entropy at the fluid level.This point of view differs from that of thermodynamic theories that have already considered entropies differing from that of zeroth order, that is, entropies depending on transport fluxes or macroscopic variable gradients.These generalized entropies have been associated notably with Burnett type equations or extended thermodynamics.In both situations, new macroscopic equations are correspondingly obtained, that is, 'extended fluid models', which are systems of partial differential equations of higher orders than Navier-Stokes type equations.

Nondimensionalization
We introduce in this section the rescaled fluid variables, the rescaled fluid equations, and the natural small parameters needed to investigate asymptotic expansions of higher order entropies.We only consider compressible flows spanning the whole space that are 'constant at infinity'.

Rescaled variables
In order to investigate asymptotic expansions of higher order entropies, we need to specify the order of magnitude of the various terms appearing in fluid governing equations.To this purpose, for each quantity φ, we introduce a typical order of magnitude denoted by < φ >.In particular, we introduce a characteristic length < x >, velocity < v >, density < ρ >, viscosity < η >, and pressure < p >.The order of magnitude of the sound velocity c is then < c > 2 = < p >/< ρ > and from the state law we also have < c > 2 = < r g >< T > where r g denotes the gas constant per unit mass and T the absolute temperature, and the Reynolds number is given by An important aerodynamic length upon consideration is the dissipation length < x > dis defined such that the corresponding Reynolds number is unity < x > dis = < η >/< ρ >< v > and we can then write Re = < x >/< x > dis .We define the characteristic time from the characteristic length < x > and the characteristic velocity < v > by letting < t > = < x >/< v >.
We also introduce a typical mean free path < l > and from the kinetic theory of gases we have < η > = < ρ >< c >< l > [5,8].Denoting by ǫ k the Knudsen number < l >/< x > and ǫ m the Mach number < v >/< c > we then have the Von Karman relation which relates ǫ k and ǫ m .We will assume in the following that the Knudsen number ǫ k is small and the Mach number ǫ m will also be small, since we are especially interested in flows where the characteristic length < x > is the dissipative length < x > dis and the Reynolds number is then unity.Upon defining the reduced quantity φ = φ/< φ > associated with each quantity φ of the fluid model, we can now estimate the order of magnitude of each term in the governing partial differential equations and in the definition of higher order entropies.We will assume, for the sake of simplicity, that we have r g = 1 so that c 2 = T in particular.

Rescaled governing equations
The equations governing compressible flows can be written in the form [10,23] where t is time, x the n-dimensional spatial coordinate, ρ the mass density, v the velocity vector, p the pressure, Π the viscous tensor, e the internal energy per unit mass, and Q the heat flux.The viscous tensor and the heat flux are given by where κ is the volume viscosity, I the unit tensor, η the shear viscosity, λ the thermal conductivity, and we denote by vI the deviatoric part of the strain rate tensor.For the sake of simplicity the internal energy per unit mass e is taken in the form e = c v T where c v is a constant.
Upon using the general notation of Section 3.1 the reduced equations can then be written Note that, in contrast with the rescaled system (3.8)-(3.10), the compressible flow model previously considered in [14] did not contain any small parameter.The asymptotic analysis of higher order entropies for small ǫ k and ǫ m numbers has only been investigated in the simpler situation of incompressibles flows [13].
Remark 3.1 Theoretical calculations and experimental measurements have shown that the viscosity ratio κ/η is of order unity for polyatomic gases [2,5,8].Volume viscosity also arise in dense gases and in liquids so that its absence in monatomic dilute gases is an exception rather than a rule [2,8].

Remark 3.2
The dimension n appearing in the coefficient 2/n of the viscous tensor is normally the full spatial dimension, that is, the dimension n ′ of the velocity phase space of the associated kinetic model.We may still assume that the spatial dimension of the model has been reduced, that is, the equations are considered in R n with n < n ′ .The full size viscous tensor Π ′ is then a matrix of order n ′ , and the corresponding coefficient must be 2/n ′ .However, if we denote by Π the upper left block of size n of Π ′ , that is, the useful part of Π ′ , we may rewrite Π in the form where I is the unit tensor in n dimensions.Therefore, using a smaller dimension n instead of the full dimension n ′ in the coefficient of the viscous tensor is equivalent to increasing the volume viscosity by the amount 2η (n ′ − n)/nn ′ .

Temperature dependence of transport coefficients
Previous analyses have shown that it is necessary to take into account the temperature dependence of transport coefficients [11,12,14].The kinetic theory of polyatomic gases lead to the following simplified assumptions concerning the temperature dependence of the thermal conductivity λ(T ), the volume viscosity κ(T ), and the shear viscosity η(T ), away from small temperatures.We assume that λ, κ, and η are C ∞ (0, ∞) and such that there exist κ, a > 0, a > 0, and a σ > 0 for any σ ≥ 1, with ) Kinetic theory suggests that 1/2 ≤ κ ≤ 1 but the situations where 0 ≤ κ < 1/2 or κ > 1 are still interesting to investigate from a mathematical point of view.

Rescaled higher order entropies
Letting < γ [k] > = < ρr > = < ρ >< r > we deduce after some algebra that the rescaled higher order entropy correctors γ [k] = γ [k] /< ρ >< r > are given by and we have recovered that the (2k) th order entropy correctors are of order O(ǫ 2k k ) as was expected from their formal construction in Section 2.
The mathematical fluid entropy −S (0) can be shown to be a strictly convex function of the conservative variables U = (ρ, ρv, E tot ) where the total energy per unit volume is where ψ [0] is the modified zeroth order entropy [18,19] and C 0 a positive constant to be determined later.The rescaled zeroth order term γ [0] is easily rewritten in the form

Molecular coordinates
The proper framework required to investigate asymptotic expansions of higher order entropies involve the rescaled coordinates ( t, x), the rescaled unknowns ( ρ, v, T ), and the rescaled governing equations (3.8)-(3.10)depending on the small parameters ǫ k and ǫ m .In this framework, parameter dependent a priori estimates and entropic inequalities can directly be obtained from the rescaled fluid governing equations.
On the other hand, it is also possible to completely eliminate the parameters ǫ k and ǫ m from the governing equations and from higher order entropy expansions by using a set of coordinates associated with the molecular properties of the fluid.More specifically, let us introduce the new coordinates associated with the characteristic length ǫ k < x > = < l > and the characteristic time ǫ k ǫ m < t > = < l >/< c > so that x is measured in units of the mean free path < l > and t in units of the particle collision time < l >/< c >.We then have for any multiindex α ∈ N n and any σ ∈ N. By using these molecular coordinates ( t, x), and upon defining the velocity v by we obtain after a little algebra the governing equations where In these equations, with a slight abuse of notation, we have denoted by the same letter the corresponding functions expressed in physical x or molecular x coordinates.It is then remarkable that all small parameters have been eliminated from the system (3.18)-(3.20)so that we may use any result obtained in previous work [14].A second method to investigate asymptotic expansions is therefore to use molecular coordinates and to map back these results to the unknowns (ρ, v, T ) written in terms of the macroscopic coordinates ( t, x).These two methods are of course equivalent but working in physical coordinates is usually more instructive and more flexible.

Weighted inequalities
Higher order entropies naturally introduce weight factors in the form of powers of temperature or density when estimating Lebesgue norms of the flow variables derivatives.We restate here weighted estimates of derivatives [9,12,14,25,26] and further specify the scaling dependence of the corresponding estimating constants.We work with dimensionless quantities and omit hat accents for the sake of notational simplicity.We denote by BM O the space of functions with bounded mean oscillations [17,23,25,26], by C 0 0 the space of continuous functions that vanish at infinity and by H k the usual Sobolev spaces [22,23,25].

Weighted products of derivatives
We investigate weighted products of derivatives of the rescaled unknowns r, τ and w, that will be taken to be r = log ρ, w = v/ √ T , and τ = log T in our applications.In the following Theorem, since in our applications w and τ are parabolic variables, the total number of derivations k can be left unchanged.
There exist scale invariant positive constants δ(n, p, θ) and c(k, n, p), only depending on (n, p, θ) and (k, n, p), respectively, such that if r BMO + τ BMO < δ, then for any a, b with |a| + |b| ≤ θ, any integer l ≥ 1, and any multiindices α j , 1 ≤ j ≤ l, with |α j | ≥ 1, 1 ≤ j ≤ l, and 1≤j≤l |α j | = k, whenever e aτ +br ∂ k τ ∈ L p (R n ), the following inequality holds where we have defined for any smooth scalar function φ where we have defined ∂ k w p = 1≤i≤n ∂ k w i p and where, in the left hand member of (4.3), with a slight abuse of notation, we have denoted by w any of its components w 1 , . . ., w n .
Since in our applications r = log ρ will be a hyperbolic variable, the total number of derivations appearing in the estimates need to be decreased by using a weighted L ∞ norms of the gradients [14].We denote by by C 1 0 the space of continuously differentiable functions that vanish at infinity with their gradients.
where we have defined w = (r, w, τ ), for any m ∈ N, and where In particular, in the situation 2 ≤ k ≤ 3, the second term in the right hand side of in (4.4) is absent.

Weighted products of renormalized derivatives
We now estimate products of derivatives of density, temperature and velocity components rescaled by the proper renormalizing factors.Theorems 4.3 and 4.4 are essentially consequences of Theorems 4.1 and 4.2 and of differential identities [12,14].
for some positive constant T ∞ and ρ be positive such that r = log ρ ∈ BM O.There exist scale invariant positive constants δ(n, p, θ) and c(k, n, p), only depending on (n, p, θ) and (k, n, p), respectively, such that if log ρ BMO + log T BMO < δ, then for any real a and b such that |a| + |b| ≤ θ, any integer l ≥ 1, and any multiindices α j , 1 ≤ j ≤ l, with |α j | ≥ 1, 1 ≤ j ≤ l, and where, in the left hand member, with a slight abuse of notation, we have denoted by v any of its components v 1 , . . ., v n .
Theorem 4.4 Let k ≥ 2 be an integer, θ > 0 be positive , 1 < p < ∞, ρ, v, T , be such that ρ ≥ ρ min , T ≥ T min , and There exist scale invariant positive constants δ(k, n, p, θ) and c(k, n, p, θ), only depending on (k, n, p, θ), where, in the left hand member, with a slight abuse of notation, we have denoted by v any of its components v 1 , . . ., v n and where we have denoted w = (r, w, τ ) and In particular, in the situation where 2 ≤ k ≤ 3, the second term in the right hand side of (4.7) is absent.Note that there is a L ∞ norm for the renormalized velocity v/ √ T term in w ′ BMO .

Asymptotics of Higher order entropy estimates
We investigate in this section parameter dependent higher order entropic estimates for compressible flows.We investigate solutions of the compressible equations in reduced form (3.8)-(3.10)rewritten by suppressing hat accents where ρ is the density, v the velocity, p the pressure, κ(T ) the volume viscosity, •vI the deviatoric part of the strain rate tensor, η(T ) the viscosity, c v the constant volume heat capacity, T the absolute temperature and λ(T ) the thermal conductivity.All these quantities are dimensionless and we have assumed for the sake of simplicity that the internal energy is proportional to temperature e = c v T where c v is a constant.
The relevant assumptions on the thermal conductivity λ, the volume viscosity κ and the shear viscosity η are derived from the kinetic theory of gases as discussed in Section 3.3 [8,10,12].
We consider the case of functions defined on R n with n ≥ 2, that are 'constant at infinity', and we only consider smooth solutions such that where l is an integer such that l ≥ [n/2] + 3, that is, l > n/2 + 2, t is some positive time, ρ ∞ > 0 a fixed positive density and T ∞ > 0 a fixed positive temperature.We also assume that ρ and T are such that ρ ≥ ρ min and T ≥ T min where ρ min > 0 and T min > 0 are fixed positive constants.Such smooth solutions are known to exist either locally in time or globally when the initial state is close to the constant state (ρ ∞ , 0, T ∞ ) [10,18,19,21,23,24,27].

Higher order entropies
Following the physical ansatz (2.5)(3.14)we define the (2k) th order rescaled kinetic entropy corrector γ [k] by where h = 1/ρT 2 for any smooth scalar function φ like ρ, T , v i , 1 ≤ i ≤ n, and where k!/α! are the multinomial coefficients, keeping in mind that hat accents are now omitted.
This choice of γ [k] , with the coefficients c v in front of temperature derivatives, yields more convenient higher order entropic estimates.It eliminates various quadratic terms associated with hyperbolic variables thanks to symmetry properties.This choice can also be associated with symmetrized forms of the system of partial differential equations.Let us denote u = ρ, ρv, ρ(e + 1 2 |v| 2 ) the conservative variable, v = −(∂ U S (0) ) t the entropic variable, and z = ρ, v, T t the natural variable, which is also a normal variable [15,20].Defining the matrix , which is associated with normal forms [15,20], one can then write the higher order entropy correctors in the form γ where h is the weight associated with the dependence of the mean free path on density and temperature h = l/< l >.This choice can also be associated with a 'spatial gradient' Fisher information with for instance γ The weight h is such that the spatial derivative operator h∂ x is invariant by the changes of scales (5.6) naturally associated with the Navier-Stokes equations.
Thanks to the fact that v and T are parabolic variables, we can expect source terms in the form to appear in the governing equation for γ [k] -up to weight factors.However, since ρ is a hyperbolic variable, there will be no such corresponding source term |∂ k+1 x ρ/ρ| 2 for density.A priori estimates for density derivatives and more generally of hyperbolic variables derivatives indeed require to introduce extra entropic corrector terms.These extra corrector terms will yield source terms in the form |∂ k x ρ/ρ| 2 .These terms are similar in spirit to the perturbed quadratic terms introduced by Kawashima in order to obtain hyperbolic variable derivatives estimates for linearized equations around equilibrium states and decay estimates [19].They are used here with renormalized variables as well as with powers of h as weights factors in order to obtain higher order entropic inequalities.We define the quantity γ [k− 1  2 ] by where we have set for convenience and we will see that in the γ [k− 1 2 ] governing equation there is a source term in the form |∂ k x ρ/ρ| 2 -up to weight factors.From a physical point of view, we also note that γ [k− 1  2 ] is of the general form (2.4) for S (2k−1) .Finally, we define the (2k) th order kinetic entropy estimator by (5.10) Note that the quantities γ [i− 1 2 ] , 1 ≤ i ≤ k, are multiplied by the small factor a in (5.10) so as to not modify the majorizing properties of the correctors γ [i] , 0 ≤ i ≤ k.

Balance equations
We present in this section the parameter dependent balance equations for γ [k] and γ [k− 1  2 ] .Similar equations can be derived for γ[k] and γ[k− 1  2 ] but are omitted.
Proposition 5.2 Let (ρ, v, T ) be a smooth solution of the compressible Navier-Stokes equations (5.1)-( 5.3) with regularity (5.4)(5.5)and let 1 ≤ k ≤ l.Then the following balance equation holds in where ϕ γ is given by where g = ρT γ only contains the temperature and velocity (k +1) th derivatives squared as expected from the hyperbolic-parabolic structure of system of partial differential equations.The term where c σνµφ are constants and the sums are over and Π (k+1) µ are defined by where v denotes-with a slight abuse of notation-any of its components v 1 , . . ., v n , and ν must be such that so that there is a total number of k + 1 derivations and there are no derivative of order k + 1 of density.Moreover, there is at most one derivative of order k + 1 of temperature or velocity components in the product and one of the terms is always split between several derivative factors.Furthermore the term ω γ is given by where c νµ are constants and we use similar notation for Π ν as for Π (k+1) µ and the summation extends over so that in particular |α|=k+1 (µ α + µ ′ α + µ ′′ α ) = 0 and there are always at least two factors in the product Π (k+1) µ .
Proof.The proof is lenghty and similar to the proof given in [14] for unscaled equations.
⊓ ⊔ Remark 5.3 Note that the velocity v = ǫ m v naturally appears in the multilinear products (5.17).
Proposition 5.4 Let (ρ, v, T ) be a smooth solution of the compressible Navier-Stokes equations (5.1)-(5.3)with regularity (5.4)(5.5)and let 1 ≤ k ≤ l.Then the following balance equation holds in where ϕ where g = ρT will help to complete the missing gradient terms in where the sums are over are defined as in the governing equation for γ [k] and Π (k+1) µ is always split between several derivative factors.Furthermore the term and one of the products µ is always split between several derivative factors.

A priori estimates
Integrating the balance equation (5.14) for γ [k] and taking into account the smoothness properties with 1 ≤ k ≤ l, we obtain that so that we have to estimate the integrals Similarly, we obtain by integrating the balance equation (5.19) and we also have to estimate the integrals R n |Σ | dx associated with the modified correctors but are omitted for brevity.We denote by χ the quantity and we will establish entropic type inequalities when χ is small enough.We could as well use the quantity but χ ≤ χ(1 + χ) and χ ≤ χ(1 + χ) so that χ and χ are asymptotically equivalent in the neiborghood of zero.These quantities χ and χ are invariant under the change of scales (5.6) described in Remark 5.1.They can also be interpreted as involving the natural variables log ρ, log T , and v/ r g T appearing in Maxwellian distributions [4] and the natural scale h associated with the mean free path.Since we have formally v/ r g T = O(ǫ m ), log(T /T ∞ ) = O(ǫ m ), and log(ρ/ρ ∞ ) = O(ǫ m ), the constraint that χ and χ remain small may also be interpreted as a small Mach number constraint, which is consistent with Enskog expansion [16].In the following, all constants associated with a priori estimates and entropic inequalities may depend on the system parameters a, a, a σ , σ ≥ 1, κ, and c v .However, these dependencies are made implicit in order to avoid notational complexities and only the dependence on k and n is made explicit. (5.28) Proof.In the expression (5.16) for γ , the last term is directly majorized as thanks to the properties of transport coefficients.On the other hand, since the quantities T σ−κ ∂ σ T λ, T σ−κ ∂ σ T κ and T σ−κ ∂ σ T η are uniformly bounded from assumptions (3.12) we only have to estimate the L 2 norm of the products Π (k+1) ν and Π (k+1) µ in order to majorize the terms in the sum of (5.16).When Π (k+1) ν only contains derivatives of v and T -in particular if there is a derivative of order k + 1-we obtain from Theorem 4.3 applied to (v, T ) with k replaced by k + 1, that when χ is small enough , where . However, if the product Π (k+1) ν is split-in particular if there is a derivative of density-we obtain from Theorem 4.4 applied to (ρ, v, T ) with k replaced by k + 1, that when χ is small enough keeping the notation of Theorem 4.4 for w ′ BMO , h∂ x w ′ L ∞ and gh m ∂ m w ′ L 2 .Therefore, we obtain that Combining these estimates, we obtain for χ small enough

,
(5.30) where c 0 and c ′ 0 are constants independent of k, n, ǫ k and ǫ m .
Proof.All split terms of Σ [k− 1  2 ] γ or ω [k− 1  2 ] γ are estimated as in the proof of the proposition 5.5 whereas the special terms are directly estimated in terms of π with constants independent of k, n, ǫ k and ǫ m .⊓ ⊔

Zeroth order estimates
The balance equation for (5.31) can be written-after some algebra-in the form (5.32) Proposition 5.7 Let γ [0] be given by (5.31).Then γ [0] ≥ 0 and there exists positive constants C 0 and δ 0 > 0-independent of ǫ k and ǫ m -such that for χ < δ 0 small enough where we define from (5.15) Proof.The proof is similar to that of the unscaled case [14] and 5.33 is a direct consequence of 5.32.⊓ ⊔

Higher order estimates
We have defined the (2k) th order Boltzmann kinetic entropy estimators by Γ ) such that for C 0 ≥ B 0 , 0 < a ≤ 1, and 0 Moreover, assuming that T ≥ T min and ρ ≤ ρ max , there exists exists Proof.Using the Cauchy-Schwartz inequality, it is straightforward to check that for any 1 Therefore, half of the density part of γ [i] and half of the velocity part of γ The same method also applies for the modified estimators γ[i− 1  2 ] , 1 ≤ i ≤ k, and this yields Inequalities (5.36) and (5.37) upon summing over 1 ≤ i ≤ k.Inequality (5.38) is a consequence of valid for τ min ≤ τ , where τ min = log T min , τ ∞ = log T ∞ and T min ≤ T ∞ , and of We can now combine the estimates of Propositions 5.5, 5.6, and 5.7, and the differential inequalities (5.23) and (5.24) in order to obtain entropic estimates.Theorem 5.9 Let (ρ, v, T ) be a smooth solution of the compressible Navier-Stokes equations (5.1)-(5.3)with regularity (5.4)(5.5)and let 1 ≤ k ≤ l.There exists positive constants b, ā, and δ(k, n) such that for the fixed value a = ā and χ < δ we have the estimates with similar results for the modified higher order entropies (5.40) Proof.The inequality (5.39) is a consequence of valid for a small enough and χ/a small enough.This inequality (5.41) is established from the estimates of Propositions 5.5, and 5.7.The proof is similar to that of the unscaled case [14] since the estimating constants have been shown to be independent of ǫ k and ǫ m , and the proof of (5.40) is similar.

⊓ ⊔
By integrating these differential inequalities, be obtain in particular that where the subscript 0 indicates that the functionals are estimated at the initial time t = 0, with similar results for the modified entropies. (5.43) One can also obtain the following exponential estimates of higher order entropy estimators.Proof.This results from the differential inequality and from Inequality 5.36 upon time integration.The proof for the modified entropy estimators is similar.⊓ ⊔ Theorem 5.9 shows that the (2k) th order kinetic entropy estimators Γ [k] and Γ [k] effectively obey entropic principles.Upon integrating inequalities (5.39) or (5.40), a priori estimates are obtained for the solutions of the compressible Navier-Stokes equations.These entropic inequalities and the related a priori estimates are invariant-up to a multiplicative factor-by the change of scales (5.6) described in Remark 5.1 and naturally associated to the Navier-Stokes equations.Since we have formally v/ r g T = O(ǫ m ), log(T /T ∞ ) = O(ǫ m ), and log(ρ/ρ ∞ ) = O(ǫ m ), the constraint that χ or χ remain small may be interpreted as a small Mach number constraint, which is consistent with Enskog expansion [16].These estimates also provide a thermodynamic interpretation of the corresponding weighted Sobolev norms involving renormalized variables as well the dependence on density and temperature of the local mean free path through the factor h which ensures that the operator h∂ x is scale invariant.Remark 5.11 In the special situation κ = 1/2 the weight h does not depend anymore on temperature and consequently the control over second order derivatives of temperature is not neede in χ or χ.A value κ = 1/2 corresponds to an infinite interaction potential at small interparticle distances.

Sobolev norms in molecular coordinates
Higher order entropic inequalities yield a priori estimates as soon as the quantity χ is small enough.It is possible to rewrite χ in terms of the molecular coordinates ( t, x) introduced in Section 3.5 in such a way that the reduced velocity v, the variations of log ρ and log T , and their derivatives in molecular coordinates have to be small.Higher order entropic inequalities (5.40) directly yield estimates for r, w = ǫ m v/ √ T , and τ = log T , through the integrals Remark 5.12 The weight h can be minorized independently of the maximum tempererature only when κ ≥ 1/2 as given by the kinetic theory of gases.Remark 5.13 It is also possible to investigate a priori estimates of entropic correctors integrals R n γ [k] dx in terms of powers of the knudsen numbers O(ǫ 2k+2 k ).

Proposition 5 . 6
two or more derivative factors so that N ν + N µ − 2 ≥ 1.The same type of estimates can be obtained for the convective contributions ω [k] γ since the corresponding products Π (k+1) µ are always split between several derivative factors.⊓ ⊔ Let (ρ, v, T ) be a smooth solution of the compressible Navier-Stokes equations (5.1)-(5.3)with regularity (5.4)(5.5)and let 1 ≤ k ≤ l.There exist positive constants δ(k, n) and c k = c(k, n)-independent of ǫ k and ǫ m -such that for χ < δ we have .35) and we have to establish that these kinetic entropy estimators obey entropic principles for the solutions of the compressible fluid model (5.1)-(5.3).Lemma 5.8 Let (ρ, v, T ) be a smooth solution of the compressible Navier-Stokes equations (5.1)-(5.3)with regularity (5.4)(5.5),assume that T ≥ T min .There exists B 0 ( Tmin T ∞