Group analysis of the generalized Burnett equations

Alexander V. Bobylev; Sergey V. Meleshko

doi:10.1080/14029251.2020.1757238

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Volume 27, Issue 3, May 2020, Pages 494 - 508

Group analysis of the generalized Burnett equations

Authors

Alexander V. Bobylev^a, Sergey V. Meleshko^b^{, *}

^aKeldysh Institute of Applied Mathematics, Russian Academy of Science, Miusskaya Pl. 4, Moscow, 125047, Russia,alexander.bobylev@kau.se

^bSchool of Mathematics, Institute of Science, Suranaree University of Technology, 30000, Thailand,sergey@math.sut.ac.th

^*Corresponding author.

Corresponding Author

Sergey V. Meleshko

Received 14 November 2019, Accepted 28 November 2019, Available Online 4 May 2020.

DOI: 10.1080/14029251.2020.1757238 How to use a DOI?
Keywords: Generalized Burnett equations; Lie group; group classification; conservation law; invariant solutions
Abstract: In this paper group properties of the so-called Generalized Burnett equations are studied. In contrast to the classical Burnett equations these equations are well-posed and therefore can be used in applications. We consider the one-dimensional version of the generalized Burnett equations for Maxwell molecules in both Eulerian and Lagrangian coordinates and perform the complete group analysis of these equations. In particular, this includes finding and analyzing admitted Lie groups. Our classifications of the Lie symmetries of the Navier-Stokes equations of compressible gas and generalized Burnett equations provide a basis for finding invariant solutions of these equations. We also consider representations of all invariant solutions. Some particular classes of invariant solutions are studied in more detail by both analytical and numerical methods.
Copyright: © 2020 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Symmetries have always attracted the attention of scientists. One of the tools for studying symmetries is the group analysis method [14, 17, 21, 26, 27], which is a general method for constructing exact solutions of partial differential equations. It is worth to mention here that applications of the group analysis method to a wide variety of models in science (up to year 1996) were collected in [16].

The group classification of the viscous gas dynamics equations under some restrictions on the viscosity coefficients was done in [11]. The group classification of two-dimensional steady viscous gas dynamics equations for an ideal gas was done in [22]. For some models of viscous gas dynamics equations, group analysis was applied in [10]. Unsteady two-dimensional steady viscous gas dynamics equations with arbitrary state equations were studied in [23]. Many of the invariant solutions of the viscous gas dynamics equations have also been obtained by other methods [1,2,9,12 ,13,30,33].

In this paper we study symmetry properties of equations of hydrodynamics (derived from the Boltzmann equation) at the Burnett level [15, 20]. This level of description of rarefied gases is important because it includes, for example, physical phenomena such as dispersion of sound waves that are absent at the Navier-Stokes level. However, the well-known instability [4] of the classical Burnett equations makes these equations ill-posed and therefore practically useless for applications. There are several methods to regularize these equations (see, for example, [5,19,32] and references therein). In this paper we use an approach developed by one of the authors, which is based on the idea of ’infinitesimal’ changes of variables. In other words, we consider the equations not for true hydrodynamic variables (density ρ^tr, bulk velocity v^tr, and temperature T^tr), but for slightly different quantities,

ρ=ρtr+O(ɛ2), v=vtr+O(ɛ2), T=Ttr+O(ɛ2),(1.1)

for which the standard notations (ρ, v, T) are used. The small parameter ε denotes the Knudsen number, i.e. mean free path divided by the typical macroscopic length. It was shown in [6] that this approach leads to what are called generalized Burnett equations (GBEs). Moreover, it was proven rigorously in [7] that solutions of ‘diagonal version’ of these equations are more accurate (the second order approximation in Knudsen number), than the solutions of the Navier-Stokes equations (the first order of approximation) in comparison with corresponding solutions of the Boltzmann equation in the vicinity of absolute Maxwellian. Note that nothing of this kind can be proven for classical (ill-posed) Burnett equations. Therefore it definitely makes sense to study the GBEs in more detail. In particular, the investigation of the shock wave profile for the GBEs was performed in [8]. The aim of this paper is to study the group symmetry properties of the GBEs for one-dimensional flows. We shall consider below the case of Maxwell molecules since this is the only model for which the classical Burnett equations are known in fully explicit form.

This paper is organized as follows. The generalized Burnett equations are introduced and briefly discussed in Section 2. Their group properties are described in Section 3. Section 4 is devoted to invariant solutions. Some non-trivial examples of invariant solutions of the GBEs and their comparisons with invariant solutions of the NSEs are given. The group analysis of the GBEs and NSEs in Lagrangian coordinates is performed in Section 5. Comparisons of invariant solutions in Lagrangian and Eulerian coordinates are presented. Conservation laws are discussed in Section 6.

2. Generalized Burnett Equations

We consider the following set of three equations for density ρ(x,t), bulk velocity v(x,t)= (v(x,t),0,0), and absolute temperature T (x,t) [8]

ρt+(ρv)x=0,ρ(vt+vvx)+(ρT)x+Πx=0,32ρ(Tt+vTx)+ρTvx+Πvx+Qx=0,(2.1)

where (x,t) are Eulerian coordinates, x ∈ ℝ, t ∈ ℝ₊, and fluxes Π and Q are given by

Π=−43ηTvx+2η23ρ(−3712T2ρρxx+256T2ρ2ρx2+43Tvx2+2Tx2−196TρρxTx),Q=−154ηTTx+η2T3ρ2(3178ρTx−53Tρx)vx.(2.2)

If the terms with η²are omitted in (2.2), then equations (2.1) correspond to the Navier-Stokes equations of compressible gas (NSEs).

As always, we set ε = 1 in the resulting equations. The constant positive factor η is given by

η−1=32π∫−11g(μ)(1−μ2)dμ, g(cosθ)=|u|σ(|u|,θ),(2.3)

where σ(|u|,θ) is the differential scattering cross-section for Maxwell molecules, |u| is the absolute value of the relative velocity of colliding particles, and θ ∈ [0,π] is the scattering angle.

equations (1.1) for the GBEs (2.1) read as

ρ=ρtr, u=utr, Ttr=T−η22(138T2ρxρ2+23TTxρ),(2.4)

We want to make use of this relation. Our aim is to investigate group properties of the GBEs (2.1). This will be done in next two sections.

3. Admitted Lie algebra and its analysis

In this section, group properties of equations (2.1) are studied. We use the standard terminology of group analysis [26, 27], assuming that the reader is familiar with it. Readers who are mainly interested in applications of group analysis such as self-similar solutions, etc., may proceed to the next Section.

3.1. Equivalence group

The first step of the group classification of the class (2.1) is to describe the equivalence among equations from this class, up to which the group classification is carried out.

The class of equations (2.1) is parameterized by the arbitrary element η. Equivalence transformations of the class preserve the structure of its equations, but are allowed to change the arbitrary elements.

Generators of one-parameter groups of equivalence transformations [24, 27] are assumed to be in the form

Xe=ξt∂t+ξx∂x+ζp∂ρ+ζv∂v+ζT∂T+ζn∂η,

where all the coefficients of the generator depend on (t,x,ρ,v,T,η).

The class of differential equations (2.1) is defined by auxiliary equations for the arbitrary elements η which are given by

ηt=0, ηx=0, ηρ=0, ηv=0, ηT=0.

For finding equivalence transformations we have used the infinitesimal criterion [27]. For this purpose the determining equations for the components of generators of one-parameter groups of equivalence transformations were derived. The solution of these determining equations gives the general form of elements of the equivalence algebra of the class (2.1). The basis elements of the equivalence algebra of the class (2.1) are

X1e=∂x, X2e=∂t, X3e=t∂x+∂v,X4e=x∂x+v∂v+2T∂T, X5e=t∂t+x∂x−ρ∂ρ, X6e=t∂t+x∂x+η∂η.

These generators define the equivalence group of equations (2.1). Because of the transformations corresponding to the generator X6e:

x′=xea, t′=tea, η′=ηea,

one can assume that η = 1 in (2.1). However, we keep η in the equations, as we also consider limits when η → 0.

For classifying subalgebras of the admitted Lie algebra, one can use the equivalence transformation corresponding to the involution which also an equivalence transformation^a

x→−x, v→−v.(3.1)

3.2. Admitted Lie algebra

A generator admitted by equations (2.1) is considered in the form

Xe=ξt∂t+ξx∂x+ζρ∂ρ+ζv∂v+ζT∂T,

where the coefficients depend on (t,x,ρ,v,T).

Usual calculations (see e.g. [26, 27]), which are omitted for brevity, show that the admitted Lie algebra L₅ is defined by the generators

X1=∂x, X2=∂t, X3=t∂x+∂v,X4=x∂x+v∂v+2T∂T, X5=t∂t+x∂x−ρ∂ρ.

One notices that Xie=Xi,, (i = 1, 2, …, 5). This is because equations (2.1) only contain the single arbitrary element η, which is constant.

It should be mentioned that the Lie algebra L₅ coincides with the Lie algebra admitted by the Navier-Stokes equations of a compressible gas.

Using the commutator table

	X₁	X₂	X₃	X₄	X₅
X₁	0	0	0	X₁	X₁
X ₂	0	0	X₁	0	X₂
X ₃	0	−X₁	0	X₃	0
X ₄	−X₁	0	−X₃	0	0
X ₅	−X₁	−X₂	0	0	0

one finds the automorphisms

A1:x¯1=x1+a(x4+x5),A2: x¯1=x1+ax3, x¯2+x2+ax5,A3: x¯1=x1−ax2, x¯3=x3+ax4,A4: x¯1=eax1, x¯3=eax3,A5: x¯1=eax1, x¯2=eax2,

where only the changeable coordinates are presented. Involution (3.1) gives the automorphism

x¯1=−x1, x¯3=−x3.

For high-dimensional Lie algebras one can use a two-step algorithm [28]. This algorithm reduces the problem of constructing an optimal system of subalgebras with high dimensions to a problem with low dimensions.

According to the theory of the group analysis method [27], all invariant solutions split into classes of equivalent solutions. The equivalence is considered with respect to the admitted Lie group corresponding to the Lie algebra L₅. For finding representatives of these classes one can use an optimal system of the subalgebras of the admitted Lie algebra L₅ [27].

The Lie algebra L₅ can be presented as the direct sum L₅ = L₂ ⊕ I₃ of the subalgebra L₂ = {X₄,X₅} and the ideal I₃ = {X₁,X₂,X₃}. First one construct the optimal system of subalgebras of the subalgebra L₂. As the Lie algebra L₂ is Abelian, its classification is trivial: an optimal system of one-dimensional subalgebras of the Lie algebra L₂ consists of the subalgebras

{X5+αX4},{X4},{0}.(3.2)

Here {0} is included in the list for subalgebras which do not include the generators from the Lie algebra L₂. The second step consists of joining the ideal I₃: each of the elements from the list (3.2) is extended by generators from the ideal I₃ using their stabilizer. Finally the optimal system of one-dimensional subalgebras consists of the subalgebras:

{X5+αX4},{X4+ɛX2},{X5−X4−X1}, {X5+X3},{X2+X3}, {X2}, {X4}, {X3}, {X1},

where ε = ±1, and α is an arbitrary constant.

4. Representations of invariant solutions

For finding invariant solutions one needs to choose a subalgebra from the optimal system of subalgebras of the admitted Lie algebra (see e.g. [27] for details). Then one finds all functionally independent invariants of the subalgebra. Setting the invariants for which the rank of the Jacobi matrix is equal to the number of the dependent variables by functions of the other invariants, one constructs a representation of a solution invariant with respect to the chosen subalgebra. Substituting the representation of the invariant solution into the original system, one derives a reduced system of equations. Notice that the reduced system of equations has fewer independent variables. Representations of all invariant solutions are presented in Table 1.

	Generator	ρ	v	T	z
1.	X₅ + αX₄,	t⁻¹R(z),	t^αV (z),	t^2α Q(z),	xt^{− (α+1)}
2.	X₄ + εX₂,	R(z),	e^εtV(z),	e^2εt Q(z),	xe^{− εt}
3.	X₅ − X₄ − X₁,	t⁻¹R(z),	t⁻¹V(z),	t⁻²Q(z),	x + lnt
4.	X₅ + X₃,	t⁻¹R(z),	lnt +V (z),	Q(z),	xt−lnt
5.	X₂ + X₃,	R(z),	t +V (z),	Q(z),	x−12t2
6.	X₂,	R(x),	V (x),	Q(x),	x
7.	X₄,	R(t),	xV (t),	x²Q(t),	t
8.	X₃,	R(t),	xt+V(t),	Q(t),	t
9.	X₁,	R(t),	V (t),	Q(t),	t

Table 1.

Representations of all invariant solutions.

Consider, for example, the subalgebra with the basis generator

X5+αX4=(α+1)x∂x+t∂t−ρ∂ρ+αv∂v+2αT∂T.

For finding invariants J(x,t,ρ,v,T) one should solve the equation

(X5+αX4)J=0.

A set of functionally independent solutions of the latter equation can be chosen as follows,

z=xt−(α+1), ρt, vt−α, Tt−2α.

A representation of the invariant solution is

ρ=t−1R(z), v=tαV(z), T=t2αQ(z).

Substituting this representation into equations (2.1), one obtains a system of ordinary differential equations. Because for the GBEs equations these ordinary differential equations are cumbersome, we only present them for the Navier-Stokes equations of compressible gas

R′(V−(α+1)z)+R(V′−1)=0,3R′Q+Q′(3R−4ηV′)−4ηV″Q+3V′R(V−(α+1)z)+3αRV=0,45η(Q″Q+Q′2)−18Q′R(V−(α+1)z)+4Q(4ηV′2−3V′R−9αR)=0.(4.1)

Solutions of equations (4.1) and the reduced equations of the GBEs were constructed numerically by a six-th order Runge-Kutta scheme. For testing the code, we used solutions of these systems with V = z and α ≠ 0.

The equation of conservation of mass gives that R′= 0, for example, R = q₁, where q₁ is constant. Hence, we have only one unknown function Q(z) and two free parameters q₁ and α. Note that equations (2.1) in the limiting case η = 0 become the usual Euler equations for monoatomic ideal gas. equations (4.1) for η = 0 and the above assumptions lead to

Q′q1=0, Q(1+3α)q1=0.

Thus we have two options:

(a)

non-trivial limit with
Q=q2>0, q1>0, α=−1/3,(4.2)
where q₂ is constant;
(b)

trivial limit with Q = 0 or/and q₁ = 0. Note that the non-trivial limit corresponds to adiabatic solution of the Euler gas dynamic equations
ρ=q1/t, v=x/t, T=q2t−2/3,(4.3)
which has clear physical meaning of one-dimensional spatially homogeneous expansion of gas in the whole space. This information is important because we will consider below two different classes of solutions to the NSEs and GBEs.

Case of the NSEs. The remaining equations of the reduced system become

Q′(3q1−4η)=0,(4.4)

45Q″ηQ+45Q′2η+18Q′αq1z+4Q(4η−3(3α+1)q1)=0(4.5)

If Q = 0, for example, Q = q₂, then equation (4.5) gives that

3(3α+1)q1=4η.

Then we have two options. If we want to have the non-trivial limit (4.2) at η = 0, we just define a new value of the exponent α

α=−13+4η9q1,

considering Q = q₂ and q₁ as free parameters. We denote this solution by Q_NSE = q₂. Alternatively we can choose to have a trivial limit at η = 0, then we obtain

q1=4η3(3α+1),

considering α as a free parameter. Similarly, the case of non-zero Q′(z) in (9), (10) also leads to trivial limit at η = 0.

If Q′ ≠0, then

q1=4η3,

and equation (4.5) becomes

15(QQ′)′+8α(Q′z−2Q)=0.

Note that solutions which have trivial limit at η = 0 can be also used as tests for computer codes and numerical methods.

Case of the GBEs. The remaining equations of the reduced system become

Q′(24Q″η2+8η2−12ηq1+9q12)=0(4.6)

3Q″ηQ(317η−90q1)+3Q′2η(349η−90q1)−108Q′αq12z+8Q(27αq12+8η2−12ηq1+9q12)=0(4.7)

If Q = q₂, then equation (4.7) gives that

α=−8η2−12ηq1+9q1227q12.(4.8)

We denote this solution by Q_GBE = q₂ This value of α with free parameters q₁ and Q = q₂ corresponds to solution of the GBEs, having the non-trivial limit (4.2) at η = 0. Other solutions of (4.6), (4.7) have trivial limit at η = 0. We present here an explicit example of such solution.

If Q′ ≠ 0, then finding Q″ from equation (4.6), integrating it, and substituting into (4.7), one finds that

q1=31790η, α=−69649301467,

and Q=q2−6964943200z2, where q₂ = Q(0) is a free parameter. Notice that α in this case also satisfies relation (4.8).

Numerical solutions of Q of system (4.1) for the NSEs and GBEs are presented in Figures 1–3. For testing the code we used exact solutions obtained above with V (z)= z. Figures 1–2 demonstrate a good agreement of numerical solutions with exact solutions. In Fig. 1 the exact solution is for the NSEs (Q_NSE), and in Fig. 2 the exact solution is for the GBEs (Q_GBE). Fig. 3 presents the numerical solution Q of equations (4.1) for the NSEs and GBEs with the following initial data: Q(.5) = 1, R(.5) = 1, Q′(.5) = 10, R′ (.5) = −5, v(.5) = .5. In all these cases the parameter α = −1.852.

Remark.

Solutions of the travelling wave type were applied in [8] for studying the structure of a shock wave. This type of solution is equivalent to the solution invariant with respect to the generator X₂, which corresponds to the set of stationary solutions. The conservative form of equations (2.1) provides three integrals of equations (2.1).

5. Analysis of equations (2.1) in Lagrangian coordinates

The Eulerian (x,t) and mass Lagrangian (ξ, t) coordinates are related by the relation x = φ(ξ, t), where the function φ satisfies the equations

ϕt(ξ,t)=v(ϕ(ξ,t),t), ϕξ(ξ,t)=ρ−1(ϕ(ξ,t),t).

In the mass Lagrangian coordinates (ξ, t) equations (2.1) take the form

ρ−2ρt+vξ=0,vt+(ρT+Π)ξ=0,32Tt+ρTvξ+Πvξ+Qξ=0,(5.1)

where

Π=−43vξρT+23η2S1, Q=−154ηρTTξ+13η2TS2,S1=2Tξ2ρ−196ρξTξT−2ρ−1ρξ2T2−3712ρξξT2+43vξ2ρT,S2=vξ(3178Tξρ−53ρξT).

5.1. Admitted Lie group

As the transition to the Lagrangian coordinates is not a point transformation, group analysis of these equations has to be performed independently of their representations in Eulerian coordinates. In particular, group classification of the admitted Lie algebra and invariant solutions of equations (2.1) in mass Lagrangian coordinates are obtained in this section.

Calculations show that the admitted Lie algebra L52 is defined by the generators

Y1=∂ξ, Y2=∂t, Y3=∂v,Y4=ξ∂ξ+v∂v+2T∂T, Y5=t∂t−ρ∂ρ.

The commutator table is

	Y₁	Y₂	Y₃	Y₄	Y₅
Y₁	0	0	0	Y₁	0
Y₂	0	0	0	0	Y₂
Y₃	0	0	0	Y₃	0
Y₄	−Y₁	0	−Y₃	0	0
Y₅	0	−Y₂	0	0	0

while the automorphisms are

A1:y¯1=y1+ay4,A2:y¯2=y2+ay5,A3:y¯3=y3+ay4,A4:y¯1=eay1, y¯3=eay3,A5:y¯2=eay2

There is also the involution^b

ξ→−ξ, v→−v,

which provides the automorphism y₁ →−y₁.

On the first step one classifies the subalgebra L₂ = {Y₄,Y₅} which is Abelian

{Y4+αY5},{Y5},{0}.(5.2)

Hence, the optimal system of one-dimensional subalgebras of Lie algebra L52 consists of the subalgebras

{Y4+αY5}α≠0, {Y4+ɛY2}, {Y4}, {Y5+Y1+αY3}, {Y1+Y2+αY3},{Y5+ɛY3}, {Y5}, {Y2+ɛY3}, {Y2}, {Y1+αY3}, {Y3}.

Representations of all invariant solutions are presented in Table 2. Notice that there are no solutions invariant with respect to Y₃.

	Generator	ρ	v	T	z	Tab.1
1.	Y₄ + αY₅,	t⁻¹R(z),	t^αV (z),	t^2α Q(z),	ξ t⁻^α	α = −1: X₅ − X₄ + X₁
2.	Y₄ + εY₂	R	ξV	ξ²Q	ξe^−εt	α ≠ −1: X₅ + αX₄ X₄ + εX₂
3.	Y₅ +Y₁ + αY₃,	t⁻¹R(z),	V (z) + α lnt,	Q(z),	ξ − lnt	X₅ + αX₃
4.	Y₁ + Y₂ + αY₃,	R(z),	V (z) + αt,	Q(z),	ξ−t	X₂ + αX₃
5.	Y₅ + εY₃,	t⁻¹R(z),	V (z) + ε lnt	Q(z),	ξ	X₅ + X₃
6.	Y₅,	t⁻¹R(z),	V (z),	Q(z),	ξ	X₅
7.	Y₂ + εY₃,	R(ξ),	V (ξ) + εt,	Q(ξ),	ξ	X₂ + εX₃
8.	Y₂,	R(ξ),	V(ξ),	Q(ξ),	ξ	X₂
9.	Y₁ + αY₃,	R(t),	V (t) + α ξ,	Q(t)	t	X₁\|_α=0; X₃\|_α≠0
10.	Y₄	R	ξV	ξ²Q	t	X₄

Table 2.

Representations of all invariant solutions.

5.2. Relations between invariant solutions in Lagrangian and Eulerian coordinates

Consider the generator

Y4+αY5=αξ∂ξ+t∂t+αv∂v+2αT∂T−ρ∂ρ.

It has the invariants

y=ξt−α, tρ, vt−α, Tt−2α.

A representation of the invariant solution has the form

ρ=t−1R(y), v=tαV(y), T=t2αQ(y).

Hence, the Lagrangian and Eulerian coordinates are related by the formula

ϕξ(ξ,t)=tR−1(ξt−α), ϕt(ξ,t)=tαV(ξt−α).(5.3)

Integrating the first equation of (5.3), one has that

ϕ=tα+1R^+h,

where R^′(y)=R−1(y), and h(t) is an arbitrary function of the integration. Substitution into the second equation of (5.3) gives that

tαV=(α+1)tαR^−αξR−1+h′,

which can be rewritten in the form

tα(V−(α+1)R^+αyR−1)=h′.

Thus, one obtains that

V−(α+1)R^+αyR−1=k,

and

h′=ktα,

where k is constant. Because of the equivalence transformation related with the shift of x, the function h can be found up to an arbitrary constant.

If α + 1 = 0, then h = k ln(t)+ k₀, where the constant k₀ can be assumed k₀ = 0. Hence, one obtains that

x−kln(t)=R^.

Using the inverse function theorem, one obtains that

ξt−1=F(z),

for some function F, where z = x − k ln(t). This gives that

ρ=t−1R˜(z), v=t−1V˜(z), T=t−2Q˜(z).

This is a representation of a solution invariant with respect to the subalgebra X₅ − X₄ + kX₁, which is similar to the subalgebra number 3 in Table 1.

If α + 1 ≠0, then h=k1α+1tα+1+k0. Hence,

x−1α+1ktα+1=tα+1R^

or

xt−(α+1)=R^+1α+1k.

Using the inverse function theorem, one obtains that

ξt−α=F(z)

for some function F, where z = xt⁻^(α+1). This gives that

ρ=t−1R˜(z), v=t−1V˜(z), T=t−2Q˜(z).

This is a representation of a solution invariant with respect to the subalgebra X₅ + αX₄, which is number 1 in Table 1.

Representations of all invariant solutions are given in Table 2. Their equivalence with invariant solutions in Eulerian coordinates is presented in the last column of Table 2.

6. Conservation laws

6.1. Eulerian coordinates

In conservative form equations (2.1) are rewritten as

ρt+(ρv)x=0,(ρv)t+(ρ(T+v2)+Π)x=0,(ρ(3T+v2))t+(ρv(5T+v2)+2(Πv+Q))x=0.

The general form of a conservation law is

DtBt+DxBx=0,(6.1)

where the functions B^tand B^xdepend on the independent and dependent variables, and the derivatives of the dependent variables with respect to the independent variables up to some order.

Conservation laws provide information on the basic properties of solutions of differential equations, and they are also needed in the analyses of stability and global behavior of solutions. Noether’s theorem [25] is the tool which relates symmetries and conservation laws. However, an application of Noether’s theorem depends on the following condition: that the differential equations under consideration can be rewritten as Euler-Lagrange equations using some Lagrangian. Among approaches trying to overcome this limitation one can mention here the approaches developed in [3,18,29,31 ]^c.

In the present paper we use the method applied in [31]. The method consists of substituting the functions B^tand B^xin general form into equation (6.1). Excluding the main derivatives of system (2.1), and splitting it with respect to the parametric derivatives, one derives an overdetermined system of linear partial differential equations for the functions B^tand B^x. The general solution of this system of equations provides the complete set of conservation laws.

The calculations show that the Navier-Stokes equations of a compressible gas only have one additional conservation law

Bit=ρ(x−tv), Bix=ρ(xv−t(T+v2))+43ηtTvx.

This conservation law can be written in the form

Bit=ρ(x−tv), Bix=xρv−t(ρ(T+v2)+Π).(6.2)

This conservation law in inviscid gas dynamics is called the conservation law of the center of mass. One can check directly that the (Bit, Bix) also provide densities of a conservation law for the GBEs.

6.2. Lagrangian coordinates

The conservation laws in Eulerian and mass Lagrangian coordinates are related by the formula

DtBLt+DξBLξ=ϕξ(Dx(ρvBt+Bx)+DtL(ρBt)),

where D_t and D_ξ are the operators of the total derivative in Lagrangian coordinates, DtE and D_x are the operators of the total derivative in Eulerian coordinates. Thus, the relations between coordinates of the conserved vectors in Eulerian and mass Lagrangian coordinates are

BLt=ρ−1Bt, BLξ=Bx−vBt.(6.3)

The conservation of mass becomes the identity in mass Lagrangian coordinates. The conservation laws of the momentum and energy become, respectively,

vt+(ρT+Π)ξ=0,12(3T+v2)t+(12ρv(7T+v2)+Πv+Q)ξ=0.

The conservation law of the center of mass is reduced to the conservation law of momentum.

7. Conclusions

In this paper we have studied group properties of the so-called Generalized Burnett equations. These equations are well-posed in contrast with the classical Burnett equations, and therefore, are used in various applied problems of rarefied gas dynamics. We considered the one-dimensional version of the GBEs in both Eulerian and Lagrangian coordinates and performed a complete group analysis of these equations. In particular, this includes finding admitted Lie groups and their analysis. The presented classifications of the Lie symmetries of the NSEs and the GBEs provides a basis for finding invariant solutions of these equations. Such solutions can be used for testing numerical schemes for these models. We also considered representations of all invariant solutions. As expected, the complete Lie group and the set of conservation laws are similar for the GBEs and for NSEs. However, there are important differences if we consider some concrete invariant solutions. On one hand, the GBEs are the more accurate equations with respect to the Knudsen number. On the other hand, they contain higher order derivatives and can therefore admit some non-physical solutions. Some classes of invariant solutions were considered in more detail by both analytical and numerical methods. It was shown, in particular, that the GBEs admit two different classes of solutions that can have (a) non-trivial or (b) trivial limits when the mean free path of gas molecule tends to zero. In case (a) the solution tends to the corresponding solution of Euler gas dynamics equations. In case (b) the density of gas tends to zero. This is a natural phenomenon, which shows that higher order PDEs can introduce some ‘spurious’ solutions, which can be removed by boundary conditions, etc. This and other questions related to the GBEs need further investigation and we plan to continue this work.

Acknowledgements

The research was supported by Russian Science Foundation Grant No 18-11-00238 ‘Hydrodynamics-type equations: symmetries, conservation laws, invariant difference schemes’. The authors thank V.A. Dorodnitsyn and E. Schulz for valuable discussions.

Footnotes

a

The transformation t →−t, v →−v, η →−η does not change the structure of equations (2.1) either.

b

There is also the involution t →−t, v →−v η →−η.

c

Therein one can find more details and references.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 3
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TY  - JOUR
AU  - Alexander V. Bobylev
AU  - Sergey V. Meleshko
PY  - 2020
DA  - 2020/05/04
TI  - Group analysis of the generalized Burnett equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 494
EP  - 508
VL  - 27
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