This paper compares three popular notions of admissibility for weak solutions of the compressible isentropic Euler equations of gas dynamics: (i) the viscosity criterion, (ii) the entropy inequality (the thermodynamically admissible isentropic solutions), and (iii) the viscosity–capillarity criterion. An exact summation of the Chapman–Enskog expansion for Grad’s moment system suggests that it is the third criterion that is representing the kinetic theory of gases. This, in turn, may shed some light on the ability to recover weak solutions of the Euler equations via a hydrodynamic limit.
In a recent and noteworthy paper, de Lellis & Szekelyhidi  have produced an infinite number of weak solutions to the initial value problem for the isentropic Euler equations of gas dynamics in dimension n≥2. Furthermore, these solutions satisfy an ‘entropy’ inequality (termed ‘the thermodynamically admissible’ inequality in the monograph of Dafermos ). As physical reality would suggest uniqueness, we can logically suppose (at least) the following possibilities:
— there are no isentropic gases, and non-uniqueness is due to an error in this basic (but unrealistic) model in continuum mechanics and
— the admissibility criteria used in  are inadequate and do not reflect physical reality.
As the mathematical community would be hard put to abandon one of its favourite sets of equations (compressible Euler or p-system), it seems to me useful, in this paper, to review three popular admissibility criteria: (i) the viscosity criterion, (ii) the above-mentioned ‘thermodynamically admissible’ solutions satisfying an ‘entropy’ (in fact, energy) inequality, and (iii) a generalization of (i) obtained by including Korteweg’s theory of capillarity . In particular, because (i) and (ii) are built upon the compressible Navier–Stokes equations which themselves are claimed to have derived from the Chapman–Enskog expansion of Boltzmann’s kinetic theory, it seems a valuable exercise to review the basis for the Chapman–Enskog expansion. Specifically, I note that the Chapman–Enskog expansion for the linearized Grad moment approximation to the Boltzmann equation when exactly summed (following Gorban & Karlin [4–6]) does not yield the compressible Navier–Stokes equations but Korteweg theory. Korteweg theory by definition introduces capillarity energy, i.e. gradients of the density. Hence, while highly oscillatory initial data will be physically excluded on this basis, a new contribution is introduced into the ‘energy’ balance law which may not necessarily vanish in a weak limit of a sequence of solutions.
The paper is divided into the following sections after this introduction:
Admissibility criteria for the compressible isentropic Euler equations, n≥2.
A comparison of admissibility criteria.
The Chapman–Enskog expansion for the Boltzmann equation.
Grad’s 13 moments.
The Chapman–Enskog expansion for the Grad 10 moment system.
The dispersion relation, hydrodynamics and the entropy equality.
Spatial dimension, moment truncation and linearization.
Implications for non-uniqueness of Euler equations.
2. Balance laws
Here, we recall the balance laws of compressible gas dynamics. We denote by For the fluid, position in space is given by and time is t>0. We consider, for the sake of simplicity, only mechanical theory, because the balance laws of primary interest are the isentropic Euler equations. In this case, the relevant balance laws are conservation of mass and linear momentum which in the absence of body forces are 2.1 and 2.2 where the summation convention is used.
A classical elastic fluid is given by the constitutive relation when δij is the Kronecker delta and ρ2ψ′(ρ)=p(ρ) is the pressure. This of course includes the special case of isentropic and isothermal gas dynamics, where p(ρ)=ργ, γ>1,γ=1, respectively. The viscous stress tensor of Cauchy and Poisson is given by where and λ,μ are viscosity coefficients. For the sake of simplicity, we make the usual choice The Dutch physicist Korteweg  proposed modelling capillarity effects via the capillarity stress tensor for which we take the special form (see the paper by Dunn & Serrin  for a discussion of the general form of Korteweg’s theory), where the quantity αρ is the surface tension coefficient and α>0 is a constant.
We then call the cases In each of the above cases, the balance laws of mass and momentum imply an additional balance law of mechanical energy (an ‘entropy’ equality). Because the Korteweg fluid’s ‘entropy’ equality includes the others as special cases, we record only its balance of mechanical energy: 2.3 where we recall the choice yields .
3. Admissibility criteria for the compressible isentropic Euler equations, n≥2
The balance laws (2.1) and (2.2) with T=TE are the Euler equations of compressible gas dynamics. Local (in time) smooth solutions for the case p′(ρ)>0 are known to exist and be unique . For the case n≥2, little is known about weak solutions to the initial value problem. One approach to admissibility of weak solutions is to use our hierarchy of continuum models and make the following definitions.
The viscosity (respectively viscosity–capillarity) admissibility criterion admits those weak solutions which are limits of smooth solutions of the viscous elastic (respectively Korteweg) fluid which are obtained when (respectively ).
An immediate consequence of the viscosity admissibility criterion is that weak solutions satisfying the viscosity admissibility criterion must satisfy the inequality 3.1 in the sense of distributions (see Dafermos , section 3.3). The ‘entropy’ inequality, more exactly an energy dissipation inequality, can itself be used as admissibility criterion which again following Dafermos we will call a thermodynamically admissible solution.
The viscosity–capillarity criterion unlike the viscosity criterion will not yield the inequality (3.1), because any weak limiting process will generally provide defect measures from the weak limits of the terms 3.2 as . Furthermore, even to make sense of initial data for an approximating sequence of solutions required by the viscosity–capillarity criterion, the initial data would have to satisfy 3.3
This would clearly penalize high oscillatory initial data in ρ. Furthermore, the only way that solutions of a Korteweg fluid will satisfy the ‘entropy’ inequality (3.1) is if all the terms in (3.2) will approach zero in the sense of distributions.
4. A comparison of admissibility criteria
In §3, three admissibility criteria for the higher dimensional Euler equations are presented. Two of them, the viscosity and viscosity–capillarity criteria, require the relevant weak solution to be constructed by very restrictive approximation schemes. On the other hand, the thermodynamic admissibility criterion of inequality (3.1) (in the spirit of Lax ) has the distinct advantage of being defined independently of the method of construction of the weak solution. Unfortunately, this generality is also its main disadvantage. Specifically, the beautiful result of de Lellis & Szekelyhidi  (see in addition the recent article of Chiodaroli ) tells us:
In n≥2 space dimensions and for any given p(ρ),p′(ρ)>0, there exist bounded initial data (ρ0,u0) with ρ0>c>0 for which there are infinitely many bounded thermodynamically admissible solutions (ρ,u) of the compressible isentropic Euler equations with ρ>c>0.
Theorem 4.1 seems to reject (at least in its present form) the ‘thermodynamic admissibility criterion’ given by inequality (3.1). What can be said for the other two? We know in one space dimension they have been remarkably successful in ruling out unphysical solutions to gas dynamics and even providing existence of solutions as well (see Chen & Perepelitsa  for a recent contribution). Furthermore, the viscosity–capillarity criterion allowed for a consideration of the case when p′(ρ) may change signs as in the materials exhibiting change of phases (see Fan et al.  for a survey and an extensive list of references). One standard argument for preferring the viscosity criterion is based on the Chapman–Enskog expansion for the Boltzmann equation (e.g. Saint-Raymond , section 2.2.2). Hence, it seems reasonable to re-examine the argument based on the Chapman–Enskog expansion and consider the implications.
5. The Chapman–Enskog expansion for the Boltzmann equation
The starting point for the discussion is of course the Boltzmann equation 5.1 Here, f(x,t,ξ) denotes the probability of finding a particle of gas at point , at time t, moving with velocity is the collision operator and ε>0 denotes the Knudson number. As , we expect fast decay to a slow invariant manifold which will be governed by the five macroscopic hydrodynamic variables M:ρ (density) , u (velocity) and Θ (temperature) . The Chapman–Enskog expansion is a formal method for computing f on the invariant manifold as a power series in ε, i.e. 5.2 From (5.1), we see 5.3 so that f(0)(M) is the equilibrium Maxwellian distribution. Historically, truncating the Chapman–Enskog expansion at order 1, ε, ε2 and ε3 is called the Euler, Navier–Stokes–Fourier, Burnett and super-Burnett approximations, respectively. The apparent ability to derive the Navier–Stokes–Fourier theory from the kinetic theory of gases was of course a strong motivation to continue the expansion to higher orders of ε. However, as noted by Bobylev [13,14] truncation at Burnett order yields instability of fluid equilibrium, a decidedly unphysical result (see Bobylev  and Struchtrup  for recent discussions). As had been noted by Rosenau [16–18] and more recently by Bobylev [14,19], the problem with the Chapman–Enskog expansion is not the expansion itself but its truncation. Hence, the expansion must be exactly or approximately summed to obtain an accurate description of the desired invariant hydrodynamic manifold. To validate the Fourier–Navier–Stokes theory based on truncation at the order ε is questionable mathematics at best. (The Latin phrase vaticinium ex eventu readily comes to mind, meaning a pseudo-prophecy that was written after the event (see Kugel , p. 145).)
The Chapman–Enskog procedure becomes more and more computationally tedious as we proceed to higher and higher order terms in ε. Hence, to obtain a quantitative picture of the process, Gorban & Karlin [4,5,6] applied the technique to the linearized (about the rest state) Grad 13 moment approximation to the Boltzmann equation. It is the remarkable observation of Gorban and Karlin that in this special case the Chapman–Enskog expansion can be exactly summed.
6. Grad’s 13 moments
The linearized Grad 13 moment equations are obtained by
— computing the first 13 moment equations from the Boltzmann equation,
— invoking Grad’s closure rule for the distribution function f, and
— linearizing about the rest state of constant density ρo>0, constant temperature Θ0>0 and velocity u=0.
In appropriate non-dimensional form, the linearized Grad 13 moment equations are 6.1 6.2 6.3 6.4 6.5 where the pressure p=ρ+Θ in linear theory, σ is the extra stress, is the heat flux.
If we rescale space and time, x=x′/ε,t=t′/ε, and drop the primes, we introduce the Knudsen number ε into the system, i.e. 6.13 6.14 6.15
Of course, we could solve for σ in (6.15) and obtain ‘visco-elastic dynamics’ of Maxwell type . This again will reflect the rapid decay to the invariant hydrodynamic manifold but not provide a computation of the invariant manifold.
7. The Chapman–Enskog expansion for the Grad 10 moment system
Here, we recall the results of Gorban & Karlin [4–6] for the exact summation of the Chapman–Enskog expansion for the Grad 10 moment system in one space dimension. In fact, Gorban and Karlin presented an exact summation for the full 13 moments in three space dimensions but for the sake of simplicity only their more restricted theory is presented here. More details may be found in their original articles and a review by Slemrod .
For the Grad 10 moment system, we write the expansion 7.1 where σ(n) depends on the current values of p, u and their space derivatives. Substitute (7.1) into (6.13), (6.14), balance orders of ε, and use the equations themselves to eliminate time derivatives ∂tu,∂tp in favour of space derivatives. This yields the form of σCE: 7.2
Super-Burnett Hence, Re w±(k)≤0 for the Navier–Stokes and Burnett truncations, but for wavenumber the super-Burnett truncation yields a Bobylev instability.
In fact, Gorban and Karlin have shown that (7.2) is indeed representative of the entire expansion, i.e. 7.3 These expansions are in primed variables (x,t) and if we rescale back to the unprimed variables, then we have 7.4 The coefficients an, bn satisfy the recursion relations 7.5
Fortunately, the relations (7.5) are in convolution form which just as in continuous Fourier theory makes their (discrete) transform elementary. First, let us agree on the definition of the Fourier transform with inverse transform Then, from (7.4), we see 7.6 and if we define 7.7 7.8 we can write 7.9 Hence, if we know A(k2), B(k2), then we know and the Chapman–Enskog expansion has been summed.
As noted earlier, the key to the computation of A and B is the convolution form of (7.5). Multiply both equations in (7.5) by (−k2)n+1: then sum from n=0 to changing the order of summation in the terms on the right-hand sides. This yields 7.10 and 7.11 where so as to agree with the known first two terms in σCE. Solve for A in (7.11) to obtain 7.12 and substitute in (7.10) to obtain a cubic equation for B. If we set C=k2B, then this cubic equation is 7.13 and an elementary analysis will yield that (7.13) possesses one non-positive real root C(k2), monotone decreasing in k2, C(0)=0, as . We can now recover the exact sum of the Chapman–Enskog expansion σCE via inverse Fourier transform: 7.14 where Thus, the extra stress is represented as a Fourier integral operator.
8. The dispersion relation, hydrodynamics and the entropy equality
In Fourier space, the hydrodynamic equations (6.13) and (6.14) with σ=σCE become 8.1 Set so that 8.2 and then set the determinant of the coefficient matrix to zero. This yields the dispersion relation 8.3 where we have used the fact the C satisfies the cubic (7.13). Again, the fact that C<0 and C satisfies (7.13) implies 5C2−16C+20>0 and hence Re ϖ<0 for k≠0, as . Hence, the rest state is stable.
Furthermore, multiplication of (8.1)1 by , (8.1)2 by (where the overbar denotes complex conjugation) yields Now, use the relation to write the above equality as 8.4 This is the entropy equality in Fourier space. Note the ‘entropy’ and the ‘dissipation’ are respectively positive and negative, k≠0, because A,B are both negative for k≠0. Integration of (8.4) in k, , and application of Parseval’s identity yields 8.5 The term represents a non-local version of the capillarity energy, whereas is a non-local version of the viscous dissipation. Thus, the exact sum of the Chapman–Enskog expansion for the linearized Grad 10 moment equations yields a non-local version of Korteweg’s theory and not Navier–Stokes theory. Furthermore, the viscosity and capillarity coefficients A and B are inseparable because they are linked by equation (7.12).
9. Spatial dimension, moment truncation and linearization
One may reasonably ask if using the special 10 moment theory in one space dimension plays any significant role. The answer is no. If we were to go through the full 13 moment, three-dimensional analysis of Karlin & Gorban , section 2.2, there would be no qualitative difference because they follow the same exact summation procedure as in the 10 moment, one-dimensional case, and we would still have an addition dispersive contribution to stress tensor.
The issue of linearization is of course important as well. Historically, our understanding of continuum mechanics is based on first grasping the essence of linear theory about a rest state and then studying subtleties of nonlinearity. I think the same can be said for the ideas presented here: it would be wonderful if someone could sum the full Chapman–Enskog expansion for the nonlinear Boltzmann equation. But as I expect the chance of this to be accomplished as negligible I have tried here to gain insights into what one might expect from such a summation via recourse to linear theory. So recourse to linear theory is nothing more than a reflection on the limitations of mathematical analysis.
10. Implications for non-uniqueness of Euler equations
As we have seen, exact summation of the Chapman–Enskog expansion in an albeit special linearized example leads to a linearized version of Kortweg’s theory of capillarity which forms the basis for the viscosity–capillarity admissibility criteria. Capillarity will penalize initial oscillations in density, viscosity will penalize oscillations in the multi-dimensional contact discontinuities which are the heart of the non-uniqueness example of Chiodaroli . But there is a caveat: viscosity must be strong enough to dominate capillarity. This is seen in the paper of Schonbek  where a simpler model problem (Korteweg–de Vries–Burgers) was addressed. Thus, if we are to take the viscosity–capillarity criterion as a useful admissibility criterion, then this domination must be built into the relations between viscosity and capillarity coefficients. Finally, one may ask does the sum of the Chapman–Enskog expansion given by Karlin & Gorban  provide this domination. Alas, there the answer seems to be negative. Slemrod  showed that capillarity appears to dominate viscosity hence generating oscillations. Hence, while the exact sum of the Chapman–Enskog expansion provides a strong motivation for using the viscosity–capillarity admissibility criterion, the domination of viscosity over capillarity will need to be introduced a priori into the theory.
The moral of the story, I believe, is as follows. If the Grad truncation is reflecting the qualitative features of the Boltzmann equation, then the Chapman–Enskog expansion for the Grad system should be reflecting qualitative features of the Chapman–Enskog expansion for the Boltzmann equation. In this scenario, we see that it is Korteweg theory and not Navier–Stokes theory that should be the basis for admissibility criteria. Moreover, because Korteweg theory predicts a capillarity energy term of the form αρ|∇ρ|2 in the energy balance equation it seems that any hydrodynamic limit theory for the Boltzmann equation that attempts to provide the classical Euler equations will have to force this capillarity term to vanish. Indeed, this is precisely the state of the art for the incompressible Euler limit as given by Saint-Raymond [12,24], where assumptions must be made on both the data and the desired limit. Otherwise, the weak limit will yield an additional and unavoidable defect measure. Of course this competition between viscous and capillarity terms is not a new observation and has been studied by Lax & Levermore [25–27] and Schonbek  (among others). That the same competition appears within the context of hydrodynamic limits I think has not been noted until now and gives us a further appreciation of the pioneering work of Lax, Levermore and Schonbek.
One contribution of 11 to a Theme Issue ‘Entropy and convexity for nonlinear partial differential equations’.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.