Boltzmann equation and hydrodynamics beyond Navier–Stokes

Abstract

We consider in this paper the problem of derivation and regularization of higher (in Knudsen number) equations of hydrodynamics. The author’s approach based on successive changes of hydrodynamic variables is presented in more detail for the Burnett level. The complete theory is briefly discussed for the linearized Boltzmann equation. It is shown that the best results in this case can be obtained by using the ‘diagonal’ equations of hydrodynamics. Rigorous estimates of accuracy of the Navier–Stokes and Burnett approximations are also presented.

This article is part of the theme issue ‘Hilbert’s sixth problem’.

1. Introduction

The problem of derivation of hydrodynamic equations from the Boltzmann equation can be considered as an important part of the sixth Hilbert problem [1]. It has a long history, which begins with the famous work of Hilbert [2] and subsequent construction of the Chapman–Enskog method [3,4]. One should also mention two of Burnett’s papers [5,6] and the book by Chapman & Cowling [7] first published in 1939. The next important step was made by Grad [8]. Concerning stationary flows with low Mach numbers, we mention works of Kogan et al. [9] and Sone (see [10] and references therein). However, there are still many unclear questions on the status of ‘higher’ equations of hydrodynamics for non-stationary flows with high or moderate Mach numbers. The well-known difficulty is the instability of the Burnett equations with respect to short-wave perturbations discovered in 1982 [11]. Since then many authors have tried to overcome this obstacle. There are several different approaches; we cite only a small number of related papers (see [12–22] and references therein). Most of the publications deal with linear problems. Comprehensive reviews of results and discussion of their relation to the sixth Hilbert problem can be found in recent papers [17,23].

For brevity, we mention only two most developed approaches to nonlinear problems: (a) the author’s approach, based on transition to new coordinates in the space of hydrodynamic variables (see [12–16] and §2 below) and (b) the approach by Struchtrup and Torrilhon (see [21,22] and references therein) based on generalization of the moment Grad method and direct expansion of moment equations in power series in the Knudsen number.

Roughly speaking, the main difference between the two methods is similar to the difference between the classical methods of Chapman–Enskog and Grad, respectively. Both methods have advantages and disadvantages. The present paper is based on the author’s approach [12–16]. We discuss the problem of non-uniqueness of the truncation of the Chapman–Enskog expansion (§2). Is it possible to find a unique ‘optimal’ replacement for classical (ill-posed) Burnett equations? We show that a positive answer to this question can be given in the case of the linearized Boltzmann equation (§3). Our method leads at the Burnett level to a ‘diagonal’ version of hydrodynamic equations (proposition 3.1): two Korteweg–de Vries–Burgers (viscous) equations and one heat equation in the simplest case of axially symmetric flows. A connection of our equations with asymptotic expansion of solutions to the linearized Boltzmann equation from [24] is discussed in detail in §4. It is proved in §5 (proposition 5.1) that the solutions of the diagonal Burnett equations approximate the hydrodynamic quantities with quadratic (in Knudsen number) accuracy. The structure of the Chapman–Enskog expansion and its natural connection with our approach to the truncation problem are also discussed in §5. A short summary of results is given in §6. Unfortunately, the restricted length of the paper does not allow us to discuss in more detail an interesting question of non-uniqueness of the classical Navier–Stokes equations; we hope to do that in a subsequent publication.

2. Boltzmann equation and truncation of Chapman–Enskog series

We consider the Boltzmann equation [25]

2.1

for the distribution function f(x,v,t;ε) where the variables , , t∈R₊ correspond, respectively, to position, velocity and time. The small parameter ε>0 denotes the Knudsen number. The Boltzmann collision operator Q( f,f) reads

2.2

where

is the differential cross-section at the scattering angle θ∈[0,π] (irrelevant arguments x, t and ε of the function f are omitted in (2.2) and below). We denote for brevity that

2.3

and introduce hydrodynamic variables (the density ρ, bulk velocity u and absolute temperature T)

2.4

where c denotes the thermal velocity. By using conservation laws 〈Ψ,Q( f,f)〉=0 for Ψ=1,v,|v|², we obtain from Boltzmann equation (2.2) the ‘exact’ equations of hydrodynamics:

2.5

where p=ρT denotes the pressure. These equations are obviously unclosed, since they include unknown terms ( fluxes)

2.6

The problem of closure of these equations for small values of ε is equivalent to the question ‘How to express fluxes π( f) and q( f) in terms of macroscopic variables (ρ,u,T)?’ The famous work of Hilbert [2] can be considered as the very first step in solving this problem. The formal limit of equations (2.5) at ε=0 coincides with the famous Euler equations of gas dynamics. The second (after Hilbert’s work) important step was made by Chapman [3] and Enskog [4]. Their classical method (see also two of Burnett’s papers [5,6] and the book [7] first published in 1939) allows one to obtain formally closed ‘equations of hydrodynamics’ in the form of symbolic power series

2.7

where all are nonlinear operators applied to ‘hydrodynamic vector’ (ρ,u,T)^tr. The first two operators and correspond to Euler and Navier–Stokes equations. These are the two most important for gas dynamics sets of equations. However, if we add the third operator we are in trouble, since Burnett equations are ill-posed. This was first noted in 1982 [11], when the standard linear stability analysis for equilibrium solutions (constant ρ₀>0, T₀>0 and u₀=0) of the Burnett equations was performed. It showed a ‘catastrophic’ short-wave instability: small perturbations proportional to grow in time like , for large |k|. The connection of ill-posedness of the Burnett equations with their lack of hyperbolicity was later discussed in [12,13]. Several interesting general approaches to deal with this problem were proposed. We confine ourselves in this paper to an alternative procedure of truncation of the Chapman–Enskog series (2.7). This approach was proposed in [12,13], its idea being very simple. We consider an abstract evolution equation for a vector y from some Banach space Y (or simply a vector ordinary differential equation (ODE) in the case of finite-dimensional space Y) having the form similar to equation (2.7),

2.8

where y∈Y , and A(y),B(y),… are differentiable nonlinear operators acting in Y . In other words, we assume the existence of Frechet derivatives (linear operators) A′(y) such that A(y+h)−A(y)=A′(y)h+O(∥h∥²) with standard notation for a norm in Y . Let us assume that the usual way of truncation of equations (2.8) based on the rule ‘neglect all terms of order O(εⁿ⁺¹)’ does not work for n=2 since the corresponding approximate equation,

2.9

does not have any reasonable solutions. What then? An alternative way of truncation is to use a change of variables (all operators are assumed to be time-independent)

2.10

Then y≃z−ε²R(z) and simple calculations yield

where [R,A]=R′A−A′R (do not confuse this with the standard notation for the commutator of two operators when the operators A and R are nonlinear). In fact this notation becomes the usual one [A,R]=AR−RA in the case of linear operators A and R, since A′=A for linear operator A. The result can be expressed in the following form:

Proposition 2.1. —

Any truncated equation

2.11

is formally equivalent to a family of equations

2.12

where

with any differentiable operator R.

Remark 2.2. —

Note that even classical Navier–Stokes equations are not defined uniquely. Only Euler equations are unique.

Thus, we can apply this scheme to Chapman–Enskog series (2.7) truncated for n=2 (classical Burnett equations) and obtain a family of Burnett-type equations which depend on an arbitrary operator R. It was shown in the first publication [13] of this approach that it is possible to choose such R that the corresponding Burnett-type equations become well-posed. However, the choice of R is obviously not unique. Then the next question is to choose the regularizing operator R in some ‘optimal’ way. An attempt to do so in papers [12,14] resulted in derivation of generalized Burnett equations (GBEs), which were applied to some physical problems: shock waves with moderate Mach numbers [16], half-space problems [26] and others. Yet it is difficult to prove that GBEs represent the best possible variant of hydrodynamic equations at the Burnett level. It is even more difficult to give a mathematically rigorous proof that these equations do improve the Navier–Stokes results for small Knudsen numbers. On the other hand, all these problems can be solved if we assume that the physical system under consideration (rarefied gas) is close to equilibrium. Then we can formally replace the Boltzmann equation (2.1) by its linearized version and apply the above approach. This will be done in the next section.

3. Linearized problem

Note that the ill-posedness of the Burnett equations was first found in [11] for their linearized version. It is also clear that the above discussed approach can be applied without changes to the Chapman–Enskog expansion for the linearized Boltzmann equation. We consider equation (2.1) in dimensionless form, make a substitution

3.1

and formally neglect the nonlinear term. Then we obtain the linearized Boltzmann equation and consider the corresponding initial value problem for g(x,v,t) for t>0:

3.2

We introduce the standard notation for scalar product and norm in the (complex) Hilbert space , adding a natural restriction on the initial data,

3.3

where the star denotes the complex conjugate value. This problem is very well studied in the literature (see in particular [24,27]). We shall see below a connection of its solution with our approach based on proposition 2.1 applied to corresponding Chapman–Enskog series. For the sake of simplicity, we consider below only one-dimensional flows, assuming that instead of in (3.2) and (3.3). We also consider only axially symmetric flows such that g depends on only through v_x and |v|. Note that the null-space of L is given by

3.4

Here and below we ignore irrelevant ‘perpendicular’ modes and consider only functions of v_x and |v|. Then we introduce hydrodynamic variables

3.5

which are in agreement with initial hydrodynamic variables (2.4) and equation (3.1) such that , , . Tildes are omitted below. It is easy to check that in the limit ε=0 we obtain the linearized Euler equations

3.6

The standard Chapman–Enskog expansion for equation (3.2) yields the formal linear partial differential equation for the hydrodynamic vector z=(ρ,u,T)^tr. We obtain

3.7

where M_i,i=0, 1,2,…, are constant real (3×3) matrices, defined uniquely by the Chapman–Enskog method. The matrix M₀ corresponds to Euler equations (3.6); M₀ is independent of intermolecular forces and has the following entries :

Note that matrices M_i with i=1,2,… depend on intermolecular forces. If we replace dots in equation (3.7) by zero, then the result coincides with the classical (ill-posed) Burnett equations. Nevertheless, the following statement explains how we can use these ‘bad’ equations.

Proposition 3.1. —

A transformation

3.8

leads (formally) to diagonal Burnett equations

3.9

with some positive numbers α,β,γ.

Proof. —

We perform the Fourier transform of truncated equations (3.7) and obtain the following vector ODE

3.10

where

3.11

Note that the matrix M₀ has three distinct eigenvalues

3.12

We introduce new variables

3.13

and obtain the equation for vector w=(w₊,w₋,s)^tr:

3.14

with some new constant real matrices . To complete the proof of proposition 3.1, we use the following statement:

Lemma 3.2. —

If n≥2, A is any (n×n) matrix and satisfies the equation

then there is a unique (n×n) matrix B such that (a) diag B=0 and (b) the substitution w=y+εBy leads to the equation

Proof. —

Note that y=(1+εB)⁻¹ and therefore y_t+Λy=ε(A+[B,Λ])y+O(ε²).

Let A={a_ij}, B={b_ij}, then [B,Λ]_ij=(BΛ−ΛB)_ij=b_ij(λ_j−λ_i). Hence, we obtain

and this completes the proof. ▪

Now we can easily complete the proof of proposition 3.1 by applying the same lemma to the equation with a new diagonal operator Λ₁=Λ−ε diag A. Then we can ‘kill’ all non-diagonal terms of order O(ε²). For brevity, we omit straightforward calculations which show that the resulting diagonal equations coincide precisely with (3.9) (see also §5 below). The non-diagonal part of modified equations (3.14) will have the third order in ε and therefore can be neglected. This completes the proof of the Fourier-transformed version of proposition 3.1 for sufficiently small values of ε|k|. On the other hand, it is sufficient for the proof of proposition 3.1, since it considers the diagonalizing transformation only at the formal level. Therefore, the proof of proposition 3.1 is complete. ▪

Remark 3.3. —

It is worth noting that proposition 3.1 automatically solves the above discussed problem of the non-uniqueness of regularizing transformation for the Burnett equation. The ‘optimal’ transformation is supposed not only to make the equations well-posed, but also to simplify them as much as possible. In our case, it makes them diagonal, i.e. transforms them to three independent equations: two Burgers–Korteweg–de Vries (KdV) equations for two sound modes and one heat equation for the thermal mode. Nothing simpler can be expected at the Burnett level.

In the next section, we shall clarify a connection of the above transformation with asymptotics for small ε of solutions of the Cauchy problem (3.2).

4. Asymptotic expansion for small Knudsen numbers

We consider problem (3.2) and pass to Fourier representation,

4.1

Note that we consider solutions that depend on v only through v_x and |v|. The null-space N(L) (3.5) of the self-adjoint operator L is three-dimensional. It is convenient to introduce a special orthonormal basis in N(L),

4.2

It is easy to verify that

4.3

The solution of the Cauchy problem (4.1) can be written as

4.4

The existence and uniqueness of the solution for any fixed ε>0 and real k follows from the general theory of semi-groups under very general restrictions on the scattering cross-section in the Boltzmann collision integral (see [24] for potentials with compact support and [27] for the more general class of cross-sections). Moreover, the corresponding semi-group is contracting (it follows from dissipativity of L: 〈g,Lg〉≤0 for any ‘nice’ real functions g(v)) and therefore

4.5

in the notation of (3.3). We shall consider below such initial conditions that is bounded on any compact set in k-space and

4.6

with some N>0 (to be fixed below). Then we can use the inverse Fourier transform and obtain

4.7

For any fixed real positive a, we can split this integral into two parts:

4.8

The second integral can be estimated as follows:

4.9

where N is the same as in inequality (4.6). For our goals, it is sufficient to have N=4. It remains to evaluate the first integral. For brevity, we assume that we consider the case of intermolecular potential with compact support (e.g. hard spheres). Then we can use the following result by Arsen’ev [24] (lemmas 13 and 15, simplified for our axially symmetric case, see also [28]).

Lemma 4.1. —

There exists a number r>0 such that

• (i) the eigenvalue problem

  4.10

  has exactly three linearly independent solutions, λ_j,ϕ_j, j=1,2,3, having the following form of convergent power series

  4.11

  in the notation of (4.2);
• (ii) the solution of problem (4.1) and (4.6) for real k, such that ε|k|<r, reads

  4.12

  where b(r)>0, B is a bounded operator such that . Operators P_j,j=1,2,3, are orthogonal projectors on corresponding subspaces of A (convergent power series in ε|k|).

Hence, we obtain the following general asymptotic formula valid for almost all :

4.13

where N is the same as in (4.6) and the constants in O symbols are bounded in norm (3.3) uniformly with respect to . It is also possible to prove (see lemma 12 in [24]) that the second term from (4.8) is in fact of order , but the simplified estimate (4.9) is sufficient for our goals. The constant r is not very important; the same formula (4.13) is valid for any independent of ε fixed constant r′<r. The final step is to consider various approximations of asymptotic formula (4.13). We consider equations (4.11) and truncate the power series there.

Let

4.14

where the connection of series (4.11) for eigenfunctions ϕ_j and corresponding projectors P_j will be explained below. Then by repeating almost without changes rather long, but straightforward calculations from [24] (pp. 878–882) we obtain the following result.

Proposition 4.2. —

If g₀(x,v) satisfies conditions

4.15

then for all integer n<N and l<N the following approximate formula is valid for the solution g(x,v,t) of the Cauchy problem (3.2)

4.16

where C>0 and b>0 are independent of x and t.

In order to clarify the series for projectors P(μ), we consider the eigenvalue problem (4.10) for real values of μ (note that the dependence of μ is analytic in some neighbourhood of zero for potentials with compact support [24]). Then the operator A=L−μv_x is real and self-adjoint in H. The eigenfunctions ϕ_j(v),j=1,2,3, are pairwise orthogonal. Therefore, we have a simple formula

4.17

The series for P_j(ikε) can be obtained by complex continuation in μ variable. We shall see below that the solutions of diagonal Burnett equations (3.10) approximate the hydrodynamic quantities with uniform-in-time error O(ε²), whereas the corresponding error for diagonal Navier–Stokes equations has the lower order O(ε).

5. Accuracy of equations of hydrodynamics and connection with the Chapman–Enskog expansion

We introduce a hydrodynamic vector

5.1

where e₁,e₂,e₃ are vectors (4.2) of the orthonormal basis in N(L) satisfying conditions (4.3). Note that all components of w are linear combinations of initial hydrodynamic variables (3.6). Since |〈h₁,h₂〉|≤∥h₁∥∥h₂∥ for any functions h_1,2(v), we can make the following conclusions from proposition 4.2. Under the conditions of proposition 4.2 we can construct corresponding approximations of the hydrodynamic vector w in the form

5.2

where n and l are the same as in proposition 4.2 and the functions g_n,l(x,v,t) are given in (4.14). Then of course we have also the same order of error (see (4.16)), i.e.

5.3

Here is a delicate point which shows a principal difference between Burnett’s level and higher levels for equations of hydrodynamics. It is clear from the above considerations that first we need to transform hydrodynamic equations of any level to diagonal form (as is done with Burnett equations in proposition 3.1). For brevity, we assume that we use for these equations correct initial conditions of right order l=Max(n−2,0). Then, we can expect that the solutions of the diagonal equations will be very close to w^(n,l)(x,t) and have the same order of approximation as in (4.16). Generally speaking, this is not true. The problem is that the integrals in (4.14) depend on a constant r>0 and do not coincide with usual formulas for inverse Fourier transform (see e.g. (4.7)). To estimate this additional error in modified formula (4.16) we need to estimate integrals like

5.4

where n≥2, m=1,2,3. It is well known that for any molecular model in the Boltzmann collision operator L. Therefore we obtain that . Hence, this additional error is negligible at the Navier–Stokes and Burnett level. However, it may happen for some molecular model that for some m=1,2 or 3. Then the corresponding (diagonal!) super-Burnett equations are ill-posed and cannot be used directly. On the other hand, their formal Fourier transform remains useful since the vector w^(4,2) given in equations (4.14) and (5.2) (with integrals over a compact domain in k-space) approximates the true solution with error of order O(ε³).

After some standard calculations, we obtain the following explicit formulas for components w_j,j=1,2,3, of hydrodynamic vector w(x,t) at Navier–Stokes (n=2,l=0) and Burnett level (n=3,l=1):

5.5

where

5.6

with the usual definition of the inverse operator L⁻¹ acting in the orthogonal complement H₁ of N(L). At the Burnett level, we obtain

5.7

where

5.8

It is clear that functions (5.5) can be expressed through solutions of diagonal equations of hydrodynamics with appropriate initial data. The next statement directly follows from equations (5.3) and (5.5).

Proposition 5.1. —

The above solutions of diagonal equations of hydrodynamics satisfy the estimates

5.9

where w(x,t) is given in equation (5.2), t≥ε^1+δ with any δ>0.

Proof. —

Indeed it follows from (4.14), (5.2) and (5.5) that

in the notation of (5.4). Exactly the same estimate is valid for |w^B(x,t)−w^3,1(x,t)|. Then, we combine these estimates with inequalities (5.3) and this completes the proof. ▪

Inequalities (5.9) clearly show that the Burnett equations do improve the results obtained at the Navier–Stokes level provided that we use them in the correct way. Finally, we briefly discuss the general structure of the Chapman–Enskog expansion (2.7) in the linearized case considered above. If we consider the problem (3.2), where and g depends on only through v_x and |v|, and choose the hydrodynamic variables in form (5.1), then we obtain the Fourier-transformed Chapman–Enskog series in the form

5.10

What can be said about the general structure of the matrix U?

Proposition 5.2. —

The following identities are valid:

5.11

The series

5.12

converge for such that |k|≤r/ε.

The proof follows directly from differentiation of the sum in equations (4.12) multiplied scalarly by unit vectors e_j,j=1,2,3. The convergence of series in equations (5.12) follows from convergence of similar series in equations (4.11). Hence, the Chapman–Enskog equations of hydrodynamics read as

5.13

where B(ε), Λ(ε), are analytical at ε=0. Diagonal equations can be obtained by obvious transformation

The connection with ‘changes of variables’ from proposition 2.1 which lead to diagonal hydrodynamic equations is obvious. A similar diagonalizing transformation can also be made for the general class of solutions of the linearized Boltzmann equation. In the general case, the null-space N(L) is five-dimensional and there are some minor technical difficulties which demand a longer presentation. For the sake of brevity and clarity, we decided to confine ourselves in this paper to a simpler class of axially symmetric solutions.

6. Conclusion

We briefly summarize below the results of the paper.

We have considered the problem of regularization of the Chapman–Enskog expansion (in particular, at the Burnett level). It was shown that the regularization can be done by transformation to new hydrodynamic variables. It was also shown that the way of truncation of the Chapman–Enskog series is not unique (even at the Navier–Stokes level). What is the meaning of these transformations? How to find the optimal transformation? Does it exist? All these questions were considered in detail for the linearized Boltzmann equation. For brevity only one-dimensional (in space) axially symmetric flows were considered.

It was shown that the Chapman–Enskog expansion has a special structure which allows one to pass to diagonal equations of hydrodynamics (three independent equations in the case under consideration or five independent equations in the general case). The diagonal Navier–Stokes equations are: two linearized Burgers (viscous) equations for two sound modes and the heat equation for one thermal mode. It is proved (proposition 5.1) that the uniform in time and space accuracy of the Navier–Stokes approximation is O(ε), where ε is the Knudsen number. The diagonal Burnett equations are: two independent linearized KdV–Burgers (viscous) equations for sound modes and the same heat equation for thermal mode. It is proved (proposition 5.1) that the uniform in time and space accuracy of the Burnett approximation is O(ε²). These estimates are valid for t>ε^1+δ,δ>0, i.e. outside of the initial layer.

Thus, for the linearized Boltzmann equation the modified Burnett equations really improve the Navier–Stokes approximation. On the other hand, the above stated questions remain open for the nonlinear case.

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Competing interests

I declare I have no competing interests.

Funding

The author gratefully acknowledges support from the Russian Foundation for Basic Research by grants 16-01-00256, 17-51- 52007 and the Program of the Presidium of RAS N 01 grant no. PRAS-18-01.