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Conservation Laws of the Two-Dimensional Gas Dynamics Equations

Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Using group classification and Noether's theorem, conservation laws are obtained. Their counterparts in Eulerian coordinates are given. Among these counterparts there are new conservation laws.


Lagrangian fluid dynamics
Physical phenomena in continuum mechanics are modeled in two distinct ways. The typical approach for fluid dynamics uses Eulerian coordinates, where the system describes fluid motion at fixed locations. Velocities, density and other properties of fluid particles in the Eulerian description are considered to be functions of time and of fixed space coordinates. In contrast, in the Lagrangian description, particles are identified by the positions which they occupy at some initial time. Typically, Lagrangian coordinates are not applied in the description of fluid motion. However, in some special contexts the Lagrangian description is indeed useful in solving certain problems. Analytically, both coordinate systems are capable of producing exact solutions of fluid flows, including discontinuous flows. They are regarded as equivalent to another, except that the Lagrangian system gives more information: 'in practice such description is often too detailed and complicated, but it is always implied in formulating physical laws' [1].
There is extensive literature on Lagrangian fluid dynamics (see, for example, [2,3,4] and references therein). These studies can be roughly separated out into two groups: the analysis of equations in the gas dynamics variables in mass Lagrangian coordinates [5,4] and analysis of Euler-Lagrange equations obtained by varying the Lagrangian in mass Lagrangian coordinates [3,2,6,7]. In the present paper the Euler-Lagrange equations in mass Lagrangian coordinates of the gas dynamics equations are studied.

Symmetries and conservation laws
Symmetries have always attracted the attention of scientists. One of the tools for studying symmetries is the group analysis method [8,9,10,11,12], which is a basic method for constructing exact solutions of partial differential equations. The group properties of the gas dynamics equations in Eulerian coordinates were studied in [8,13]. Extensive group analysis of the one-dimensional gas dynamics equations in mass Lagrangian coordinates was given in [14,15]. Here the results of [16] should also be mentioned, where nonlocal conservation laws of the one-dimensional gas dynamics equations in mass Lagrangian coordinates were found. The authors of [3,2] analyzed the Euler-Lagrange equations corresponding to the one-dimensional gas dynamics equations in mass Lagrangian coordinates and extensions of the known conservation laws were derived. These conservation laws correspond to special forms of the entropy. The group nature of these conservation laws is given in [17], where a complete group classification of these Euler-Lagrange equations is presented. Notice that the hyperbolic shallow water equations are equivalent to the isentropic gas dynamics equations of a polytropic gas with γ = 2. The complete group analysis of the Euler-Lagrange equations of the one-dimensional gas dynamics equations of isentropic flows of a polytropic gas with γ = 2 was given in [6]. Group properties of the two-dimensional shallow water equations describing flows over a bed which is rotating with position-dependent Coriolis parameter in Lagrangian coordinates are studied in [18]. Conservation laws were constructed by using a Lagrangian of the form presented in [19].
As mentioned above, besides assisting with the construction of exact solutions, the knowledge of an admitted Lie group allows one to derive conservation laws. 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 [20] is the tool which relates symmetries and conservation laws. However, the application of Noether's theorem depends on the following condition: that the differential equations under consideration can be rewritten as Euler-Lagrange equations with appropriate Lagrangian. Among approaches which try to overcome this limitation one can mention here the approaches developed in [21,22,23,24] 1 .

Noether's theorem
We begin with the background related to the application of symmetries for constructing conservation laws 2 . Let be a Lie-Bäcklund operator, and F = F (x, u, p). Here x = (x 1 , x 2 , . . . , x n ) are the independent variables, u = (u 1 , u 2 , . . . , u m ) are the dependent variables, and p denotes their derivatives u k i1...i l of finite order. Noether's theorem is based on two identities. The first identity is called the Noether identity [25], where are variational derivatives, W k = η k − ξ i u k i , (k = 1, 2, ..., m), and Here the variational derivatives δF δu k i1i2...is are obtained from (2) by replacing u k with the corresponding derivative u k i1i2...is . The second identity is [9,26] 3 : Theorem (Noether). Suppose, a generator Then the generator X is an admitted symmetry of the system of Euler-Lagrange equations δL δu k = 0, (k = 1, 2, ..., m), and the vector is a conserved vector. In the case B = (B 1 , B 2 , ..., B n ) = 0, the symmetry X is called a variational symmetry, otherwise for B = 0 the symmetry X is called a divergent one.

Objectives of the present paper
The present paper is focused on the two-dimensional Euler-Lagrange gas dynamics equations of a polytropic gas. Its objective is to make group classification of the Euler-Lagrange equations with respect to the function of entropy, and to construct conservation laws by applying Noether's theorem.

Gas dynamics equations of a polytropic gas
In this section the gas dynamics equations of a polytropic gas in mass Lagrangian coordinates are considered. The two-dimensional Euler-Lagrange equations are obtained by using a variational approach.

Gas dynamics equations in Eulerian coordinates
The gas dynamics equations of a polytropic gas are where ρ is the density, (u, v) is the velocity, p = Sρ γ , and S is a function depending on the entropyS of the polytropic gas. The function S is related with the entropỹ S of a polytropic gas as follows [27] where R is the gas constant, c v is the dimensionless specific heat capacity at constant volume, andS 0 is constant.
As there is no ambiguity, the sign˜is further omitted. Notice that in the one-dimensional case one assumes that the system of equations (10)-(13) is closed, and it becomes the classical one-dimensional gas dynamics equations in mass Lagrangian coordinates [5]:

Variational approach
Choosing the Lagrangian one derives the Euler-Lagrange equations δL δϕ 1 = 0 and δL δϕ 2 = 0: One can show that equations (16), (17) are reduced to the equations of linear momentum (7) and (8). Corresponding changes of derivatives are obtained from the identities In the present paper equations (16), (17) are called the gas dynamics equations in mass Lagrangian coordinates, whereas equations (10)-(13) are called the gas dynamics equations in the gas dynamics variables in mass Lagrangian coordinates.

Relations between conserved vectors
As there is the change Because of the following relations

Equivalence transformations
The group classification of equations depends on the representations of the arbitrary elements [8]. As the function S(ξ, η) is an arbitrary element of equations (16), (17), the group classification has to be made with respect to it.

Classifying equation
The admitted Lie group is derived by solving the determining equations where E i is the left-hand side of equations (16), (17), |E = 0 means that the expressions XE i are considered on the manifold defined by equations (16), (17), and the generatorX is the prolongation of the generator where all coefficients depend on (t, ξ, η, ϕ 1 , ϕ 2 ). Notice that the function S(ξ, η) is considered to be given. Calculations show that the admitted generator has the form and the classifying equation is where c 7 (γ − 2) = 0, and The analysis of the classifying equation is split into two cases: isentropic and nonisentropic flows.

Isentropic flows
In this case S ξ = 0, S η = 0, and one derives from the classifying equation that c 10 = 0.
Thus, for general choice of γ, the admitted Lie algebra is infinite dimensional: L 8 ⊕ {X h }, where a basis of L 8 consists of the generators X j (j = 1, 2, ..., 9; j = 7), and h(ξ, η) is an arbitrary function. For γ = 2 the basis of the finite part L 8 is extended by the generator X 7 .

Conservation laws in Lagrangian coordinates
For a symmetry to be divergent (variational) one needs to require that For a divergent symmetry one obtains This gives that each of the generators X j , (j = 1, 2, 3, ..., 7), X h and provides a conservation law. These conservation laws have the conserved vector coordinates: Here and further on the subscripts relate to the corresponding generator providing the conservation law.
In the case where γ = 2 there is one more conservation law in Lagrangian coordinates which corresponds to the generator X 7 :

Conservation laws in Eulerian coordinates
First of all, we should mention the conservation law of mass which is valid for isentropic as well as nonisentropic flows, ρ t + (ρu) x + (ρv) y = 0.
The latter conserved vectors (T t i ,T ξ i ,T η i ), (i = 1, 2, ..., 8) have the following forms in Eulerian coordinates The function 4 h(ξ, η) becomes a function h(t, x, y) which only satisfies the equation The corresponding conservation law is The conserved vector (T t 8 ,T ξ 8 ,T η 8 ) cannot be represented in Eulerian coordinates because of the presence of ξ. Also notice that for γ = 2 the term containing ξ vanishes.
For γ = 2 there are two more conserved vectors: the conserved vector (T t 7 , T ξ 7 , T η 7 ) becomes and, as noted above, for γ = 2 the term with ξ vanishes in the conserved vector (T t 8 ,T ξ 8 ,T η 8 ), and the conserved vector (T t 8 ,T ξ 8 ,T η 8 ) also has its representation in Eulerian coordinates:

Discussion
First of all we note that for deriving relation (20) it was important to consider generator X in its general form: not analyzing condition (4) for each specific generator.
The conserved vectors (T t i , T x i , T y i ), (i = 1, 2, ..., 8) are well-known [28]: the vectors (T t 1 , T x 1 , T y 1 ) and (T t 2 , T x 2 , T y 2 ) give conservation laws of linear momentum; the vectors (T t 3 , T x 3 , T y 3 ) and (T t 4 , T x 4 , T y 4 ) correspond to the conservation law of angular momentum motion of the center of mass; the vector (T t 6 , T x 6 , T y 6 ) corresponds to the conservation law of energy; the vectors (T t 7 , T x 7 , T y 7 ) and (T t 8 , T x 8 , T y 8 ) are extensions of the classical conservation laws for a polytropic gas with γ = 2 derived in [28].
The conserved vector (T t h , T x h , T y h ) with the Lagrangian invariant h(t, x, y) (satisfying equation (21)) provides a new conservation law. In contrast to [16] this conservation law is local. It should be also noted that this conservation law is naturally derived: its counterpart in Lagrangian coordinates was derived directly using Noether's theorem without any additional constructions.
Remark. The gas dynamics equations (6)-(9) for an isentropic flow and γ = 2, coincide with the hyperbolic shallow water equations. Group properties of the shallow water equations describing flows over a bed which is rotating with position-dependent Coriolis parameter in Lagrangian coordinates 5 are studied in [18]. Conservation laws were constructed by using a Lagrangian of the form presented in [19]. In the present paper the admitted Lie group is wider, and we also obtained more conservation laws. This can be explained by the presence of nonzero Coriolis parameter and different Lagrangian used in [18].
Notice that the general solution of (23) is ψ 0 = F (S), where the function F is an arbitrary function.

Conservation laws in mass Lagrangian coordinates
The condition for the generator X to be divergent (variational) gives that The conserved vectors for (T t i , T ξ i , T η i ), (i = 1, 2, ..., 6) are the same as in the isentropic case. The remaining conserved vectors are In the case γ = 2 there is one more conserved vector corresponding to the generator X 7 . This conserved vector is the vector (T t 7 , T ξ 7 , T η 7 ) as in the isentropic case for γ = 2:

Conservation laws in Eulerian coordinates
Representations of the conserved vectors (T t i , T ξ i , T η i ), (i = 1, 2, ..., 7) are of similar forms ( e T t i , T x i , T y i ) as in the isentropic case. The other conserved vectors in Eulerian coordinates have the form Here the function ψ 2 (t, x, y) satisfies the equations The conserved vector (T t 9 , T ξ 9 , T η 9 ) has no a representation in Eulerian coordinates.

Discussion
Continuing the discussions presented in the previous section, the conserved vectors ( e T t 8 , T x 8 , T y 8 ) and ( e T t F , T x F , T y F ) provide new conservation laws. The conserved vector ( e T t F , T x F , T y F ) is similar to the vector ( e T t h , T x h , T y h ) studied for isentropic flows.
The conserved vector ( e T t 8 , T x 8 , T y 8 ) contains the function ψ 2 (t, x, y) which satisfies the equations (24). The overdetermined system of the gas dynamics equations with (24) is involutive. This leads to a similar idea as in [16] to extend an original set of the dependent variables, and to seek for new conservation laws containing the added variables. However, in the present cases the conservation laws are naturally derived: their counterparts in Lagrangian coordinates were derived directly using Noether's theorem without any additional assumptions.

Conclusion
New conservation laws of two-dimensional gas dynamics equations of a polytropic gas are found in the present paper. The conservation laws are derived using corresponding Lagrangian and Noether's theorem. For constructing conservation laws we used the complete group classification of the Euler-Lagrange equations (16), (17). In contrast to the one-dimensional case of the Lagrangian gas dynamics equations [17], where ϕ 1η = 0 and ϕ 2 = η, the group classification only separates out the gas dynamics equations (16), (17) on analysis of isentropic and nonisentropic flows, and the admitted Lie group found has no constraints on the entropy 6 . The complete set of admitted generators allowed us to use Noether's theorem for deriving conservation laws in mass Lagrangian coordinates. Their corresponding counterparts in Eulerian coordinates were also constructed. Some of these counterparts contain Lagrangian invariants. Moreover, the conserved vector ( e T t 8 , T x 8 , T y 8 ) contains the function ψ 2 (t, x, y) which satisfies an overdetermined system of equations. One can show that the overdetermined system of equations consisting of these equations and the gas dynamics equations is involutive. Applying the group analysis method to this overdetermined system the admitted Lie group of the gas dynamics can be extended.
The involutiveness of this system also gives a similar idea as in [16]: to extend an original set of the dependent variables, and to seek for new conservation laws containing the added variables. However, in the present paper the conservation laws are naturally derived: their counterparts in Lagrangian coordinates were derived directly using Noether's theorem without any additional assumptions.
It should be also noted that there are conservation laws which have no their counterpart in Eulerian coordinates.