## Abstract

An explicit determination of all local conservation laws of kinematic type on moving domains and moving surfaces is presented for the Euler equations of inviscid compressible fluid flow in curved Riemannian manifolds in *n*>1 dimensions. All corresponding kinematic constants of motion are also determined, along with all Hamiltonian kinematic symmetries and kinematic Casimirs which arise from the Hamiltonian structure of the inviscid compressible fluid equations.

## 1. Introduction

The study of topological, geometrical and group-theoretic aspects of fluid equations in dimensions *n*>1 has attracted considerable interest [1–3] in the mathematical theory of fluid flow. Two central topics in studying these aspects are Hamiltonian structures [3,4] and conserved integrals [5,6].

For the Euler equations of inviscid compressible fluid flow in multi-dimensional flat manifolds *kinematic* conservation law, like mass, energy, momentum and angular momentum, refers to a continuity equation in which the conserved density and spatial flux involve only the fluid velocity, density and pressure, in addition to the time and space coordinates. By contrast, a *vorticity* conservation law, such as helicity in three dimensions as well as circulation and enstrophy in two dimensions, refers to a continuity equation where the conserved density and spatial flux have an essential dependence on the curl of the fluid velocity. These two classes of continuity equations comprise all of the local conservation laws found to date for inviscid compressible fluid flow in *n*>1).

An explicit classification of kinematic and vorticity conservation laws on moving domains is known [7] in the case of inviscid compressible fluid flow with a barotropic equation of state for the pressure. In particular, the vorticity conservation laws consist of helicity in all odd dimensions *n*≥3 and enstrophy in all even dimensions *n*≥2, while the only kinematic conservation laws apart from mass, energy, momentum and angular momentum consist of Galilean momentum which holds for all equations of state, plus a similarity energy and a dilational energy which arise for polytropic equations of state where the pressure is proportional to a special dimension-dependent power *γ*=1+2/*n* of the density. Here a moving domain refers to a closed volume in

A similar classification has been obtained recently [8] for inviscid non-isentropic compressible fluid flow in *n*>1), where the entropy is conserved only along streamlines and the pressure is given by an equation of state in terms of both the fluid density and the entropy. In this case, helicity and enstrophy are no longer conserved, but in all even dimensions *n*≥2 there is a vorticity conservation law given by an entropy circulation (which vanishes whenever the fluid is irrotational or isentropic), plus there is one extra kinematic conservation law consisting of volumetric entropy in any dimension. Both of these conservation laws hold for all equations of state.

Much less is known, however, about conserved integrals for inviscid compressible fluid flow in multi-dimensional curved manifolds. One general result is that all of the vorticity conservation laws on moving domains for fluid flow in

The present paper will settle the open question of explicitly determining all local conservation laws of kinematic type on moving domains and moving surfaces for inviscid compressible fluid flow in curved Riemannian manifolds. In particular, any such conservation laws will be found that hold only for (i) special dimensions of the manifold or the surface; (ii) special conditions on the geometry of the manifold or the surface; and (iii) special equations of state. Importantly, the general form of these kinematic conservation laws will be allowed to depend on the intrinsic Riemannian metric, volume form and curvature tensor of the manifold or the surface. All kinematic constants of motion that arise from the resulting kinematic conservation laws also will be determined.

A sequel paper will address the remaining open problem of determining whether the known local conservation laws of vorticity type on moving domains and moving surfaces are complete for inviscid compressible fluid flow in flat and curved manifolds.

Note, in all of this work, the fluid is assumed to fill the entire manifold. For results on conservation laws of fluid flow with a free boundary, see [13].

In §2, first a summary of the Euler equations of inviscid compressible fluid flow in *n*-dimensional manifolds is given. Next, the formulation of local conservation laws, conserved integrals and constants of motion is discussed for general hydrodynamic systems in *n*-dimensional manifolds, and this formulation is adapted to moving domains and moving surfaces. Finally, necessary and sufficient determining equations are presented for directly finding all conserved densities of kinematic type on moving domains and moving surfaces for the Eulerian fluid equations.

The main results giving an explicit classification of all kinematic conserved densities on moving domains and moving surfaces for inviscid compressible fluid flow in *n*-dimensional manifolds are presented in §3. A corresponding classification of kinematic constants of motion is also stated, along with Hamiltonian kinematic symmetries and kinematic Casimirs which arise from the Hamiltonian structure of the inviscid compressible fluid equations.

The proof of these results is carried out in §4, by solving the determining equations from §2. The steps are carried out using tensorial index notation, which is summarized in appendix B.

One interesting feature of the classifications is that special equations of state in which the pressure depends only on the entropy of the fluid are considered. For any such equation of state, new local conservation laws describing a generalized momentum and energy which depend on the entropy are found to arise for non-isentropic compressible fluid flow.

Some concluding remarks are made in §5.

## 2. Preliminaries

The Eulerian fluid equations in ** u**, the mass density

*ρ*, the entropy

*S*and the pressure

*P*by

*P*=

*P*(

*ρ*,

*S*).

To generalize the Eulerian fluid equations to an *n*-dimensional manifold *M* [3], the only structure needed on *M* is a Riemannian metric *g*. Let ∇ be the metric-compatible covariant derivative determined by ∇*g*=0, and write grad and div for the contravariant gradient operator and the covariant divergence operator defined by *ξ*⌋∇=*g*(*ξ*,grad) and *g*(grad,*ξ*)=div *ξ* holding for an arbitrary vector field *ξ* on *M*. These operators are the natural Riemannian counterparts of the gradient **∇** and divergence **∇**⋅ operators in ** ϵ** be the volume form normalized with respect to

*g*, and let

*ϵ*be the dual volume tensor, satisfying ∇

*ϵ*=0 and

*g*(

*ϵ*,

*ϵ*)=

*n*!. Let Riem =[∇,∇] be the curvature tensor determined from

*g*, and let

*R*be the scalar curvature. Also, let Grad and Div denote the total contravariant gradient and the total covariant divergence, and let

*D*

_{t}denote the total time derivative, which are respectively defined by grad, div, and ∂

_{t}acting via the chain rule.

In this geometric notation, the covariant generalization of the fluid velocity equation (2.1) from *M* is given by
*u* is the fluid velocity vector on *M*. Similarly the covariant equations for the fluid mass density *ρ* and entropy *S* on *M* are given by
*P* determines an associated internal (thermodynamic) energy which is defined by [14,15]

A transcription between geometric notation and tensorial index notation is provided in the beginning of appendix B.

### (a) Local conservation laws on moving domains

For any hydrodynamic system in a Riemannian manifold *M*, local conservation laws are described by a covariant continuity equation
*T* and *X* are some functions of the hydrodynamic variables and their spatial derivatives, as well as the time and space coordinates *t*,*x*. Physically, the scalar function *T* is a conserved density while the vector function *X* is a spatial flux. Note that, through their dependence on *x*, both *T* and *X* are allowed to depend on the metric tensor *g*, volume tensor *ϵ* and curvature tensor Riem.

Consider any domain (i.e. an orientable closed spatial volume) *M* through which the fluid is flowing, and let *V* =** ϵ** is the volume

*n*-form (dual of the volume tensor

*ϵ*), and

*n*−1-form in terms of the normal vector

A physically more useful form for expressing hydrodynamic conservation laws (2.9) and (2.10) is obtained by considering a domain *i*=1,…,*n*), where *u* is the fluid velocity vector and *x*^{i} are local coordinates on *M*. Introduce the material (advective) derivative
*u*. Then the spatial flux through the moving boundary *T* by the transport equation
*conserved integral on a moving domain* in the fluid. As shown by equation (2.15), the integral expression

Both the conserved integral (2.15) and the underlying transport equation (2.13) have an alternative formulation using differential forms, which generalizes in a simple way to moving surfaces. The following transport identity will be needed. Let *p*-dimensional submanifold transported along the fluid streamlines, with 1≤*p*≤*n*. Then for any *p*-form ** α**,

This identity can be applied to the volume integral in equation (2.15), while the hypersurface integral in equation (2.15) can be converted into a volume integral by Stokes’ theorem, yielding
*n*-form vanishes. Hence the density *T* and flux *Φ* satisfy
**d** is the exterior derivative acting as a total (spatial) derivative.

### (b) Local conservation laws on moving surfaces

Let 1≤*p*≤*n*−1 and consider any *p*-dimensional surface (i.e. an orientable submanifold) *M* that moves with the fluid, whereby each point *i*=1,…,*n*), in local coordinates on *M*.

A *conserved integral on a moving surface* *p*-form density ** α** and a

*p*−1-form flux

**that are some functions of the hydrodynamic variables and their spatial derivatives, and the time and space coordinates**

*β**t*,

*x*, holding for all formal solutions of the hydrodynamic system. The dependence of

**and**

*α***on**

*β**x*allows them to depend on any geometrical tensors defined on the surface

The integral expression ** β**=

**d**

**is an exact**

*γ**p*−1 form for all formal solutions of the hydrodynamic system, as thereby

The density *p*-form ** α** and the flux

*p*−1-form

**in the conserved integral (2.19) satisfy a transport equation that arises from converting the boundary integral into a surface integral through Stokes’ theorem and using the transport identity (2.16). This yields**

*β**p*-form vanishes,

Note that if the conserved integral (2.19) is extended to the case *p*=*n*, with ** α**=

*T*

**and**

*ϵ***=**

*β**Φ*⌋

**.**

*ϵ*### (c) Trivial conservation laws

A conserved integral (2.19) on a moving submanifold *p*≤*n*, reduces to a boundary integral iff the conserved density ** α**=

**d**

**is an exact**

*Θ**p*-form, holding for all formal solutions of the hydrodynamic system, where the

*p*−1-form

**is some function of the hydrodynamic variables and their spatial derivatives, and the time and space coordinates**

*Θ**t*,

*x*. The corresponding flux is given by the

*p*−1-form

**is non-zero then the resulting boundary integral has no physical significance, as the conservation equation for the integral is just an identity,**

*β**trivial*.

However, if the flux in a conserved boundary integral (2.22) is zero,
*p*−1, assuming *p*−1-form conserved density ** Θ** and with a vanishing

*p*−2-form flux.

When *p*=*n*, a trivial local conservation law on a moving domain is equivalent to a conserved density given by *T*=Div *Θ* in terms of the vector function *Θ*=*ϵ*⌋** Θ**. The corresponding flux is given by

*Φ*=−

*D*

_{t}

*Θ*−(Div

*Θ*)

*u*.

### (d) Determining equations

Necessary and sufficient equations will now be derived to determine all conserved integrals on moving domains and moving surfaces for the Euler equations (2.4)–(2.7) of inviscid compressible fluid flow.

For fluid flow in an *n*-dimensional manifold *M*, a scalar function *T* will be a density for a conserved integral (2.15) on a moving domain in the fluid iff *Φ* for some vector function *Φ*, where *T* and *Φ* depend on the time and space coordinates *t*,*x*, the fluid variables *u*,*ρ*,*S* and their spatial derivatives. Hence the defining equation for *T* and *Φ* to be, respectively, a conserved density and a moving flux is simply
*T*.

### Lemma 2.1

*Let v be a tensor field on a Riemannian manifold M, and let* ∇^{m}*v denote the mth order covariant derivatives of v. A scalar function f*(*x*,*v*,∇*v*,…,∇^{k}*v*) *is a total covariant divergence* Div *F*(*x*,*v*,∇*v*,…,∇^{k}*v*) *iff*
*where*
*is the covariant spatial Euler operator* (*variational derivative*) *with respect to v*.

Here Grad ^{m} denotes the *m*-fold product of the total gradient operator Grad; the superscript * denotes a formal adjoint defined by
*ξ*_{i} and an arbitrary scalar function *f* on *M*. A proof of lemma 2.1 employing index notation is given in appendix B.

Necessary and sufficient conditions for determining *T* are now obtained by applying this lemma to equation (2.24).

### Proposition 2.2

*All conserved densities T*(*t*,*x*,*u*,*ρ*,*S*,∇*u*,∇*ρ*,∇*S*,…,∇^{k}*u*,∇^{k}*ρ*,∇^{k}*S*) *on a moving domain for the Euler equations* (*2.4*)–(*2.6*) *of compressible fluid flow in ann-dimensional Riemannian manifold M are determined by the* (*necessary and sufficient*) *equations*
*Moreover, a density will be non-trivial iff it satisfies at least one of the conditions*

This characterization of conserved densities has a straightforward extension to moving surfaces.

A *p*-form function ** α** will be a density for a conserved integral (2.19) on a

*p*-dimensional moving surface in the fluid iff

**d**

**for some**

*β**p*−1-form function

**, where**

*β***and**

*α***depend on the time and space coordinates**

*β**t*,

*x*, the fluid variables

*u*,

*ρ*,

*S*and their spatial derivatives. The following result based on the variational bi-complex [16] gives necessary and sufficient conditions to determine

**.**

*α*

### Lemma 2.3

*Let v be a tensor field on a submanifold of a Riemannian manifold M with dimension n, and let* ∇^{m}*v denote the mth order covariant derivatives of v. A homogeneous p-form function* ** f**(

*x*,

*v*,∇

*v*,…,∇

^{k}

*v*)

*with*1≤

*p*≤

*n*−1

*is an exact p-form*

*d***(**

*F**x*,

*v*,∇

*v*,…,∇

^{k}

*v*)

*iff*

*where*

*is the total exterior derivative operator*.

Here ∧ denotes the antisymmetric tensor product. A proof of lemma 2.3 in local coordinates can be found in [16]. We remark that the homogeneity condition on ** f** is necessary, as otherwise the

*p*-form cohomology of

*M*must be taken into account.

From lemma 2.3, a necessary and sufficient condition for determining ** α** is given by applying

**d**to the transport equation

*p*-form densities.

### Proposition 2.4

*All conserved homogeneous p-form densities* ** α**(

*t*,

*x*,

*u*,

*ρ*,

*S*,∇

*u*,∇

*ρ*,∇

*S*, …,∇

^{k}

*u*,∇

^{k}

*ρ*,∇

^{k}

*S*),

*with*1≤

*p*≤

*n*−1,

*on a p-dimensional moving surface for the Euler equations*(

*2.4*)–(

*2.6*)

*of compressible fluid flow in an n-dimensional Riemannian manifold M are determined by the*(

*necessary and sufficient*)

*equation*

*Moreover, a homogeneous density will be non-trivial iff it satisfies the condition*

## 3. Main results

We begin by recalling the notion of symmetries for Riemannian manifolds.

A Riemannian manifold (*M*,*g*) possesses an isometry if there exists on *M* a vector field *ζ* satisfying the Killing equation
*ζ*=0. Here ⊙ denotes the symmetric tensor product. Similarly, a Riemannian manifold (*M*,*g*) possesses a homothety if there exists on *M* a vector field *ζ* satisfying the homothetic Killing equation
*ζ*=*λg*.

A vector field *χ* on *M* is curl-free (irrotational) if ∇∧*χ*=0. Locally on *M*, this condition is equivalent to *χ*=∇*ψ*, for some scalar field *ψ*. The identity 2∇*ζ*=∇⊙*ζ*+∇∧*ζ* shows that a curl-free vector field *χ* is a Killing vector *ζ* iff it is covariantly constant, ∇*χ*=0.

We now state the main classification results for kinematic conserved densities on moving domains and moving surfaces in compressible fluid flow in Riemannian manifolds. The results are obtained by directly solving the respective determining equations in propositions 2.2 and 2.4, as carried out in §4.

### (a) Conservation laws on moving domains

### Theorem 3.1

(*i*) *For compressible fluid flow* (*2.4*)–(*2.6*) *in a Riemannian manifold* (*M,g*) *of any dimension n*>1, *the non-trivial kinematic conserved densities T*(*t,x,u,ρ,S*) *admitted for a general equation of state P*(*ρ,S*) *comprise a linear combination of*
*where e is the thermodynamic energy* (*2.8*) *of the fluid, and f*(*S*) *is an arbitrary non-constant function*. (*ii*) *The only special equations of state P*(*ρ,S*) *for which extra kinematic conserved densities T*(*t,x,u,ρ,S*) *arise are the polytropic case*
*with dimension-dependent exponent γ=1+2/n, where σ(S) is an arbitrary function, and the isobaric-entropy case*
*where κ*(*S*) *is an arbitrary non-constant function. The extra admitted conserved densities consist of a linear combination of*
*in the polytropic case* (*3.8*), *and*
*in the isobaric-entropy case* (*3.9*), *where f*(*S*) *is an arbitrary non-constant function*.

The kinematic conserved integrals (2.15) corresponding to these conservation laws (3.3)–(3.13) on an arbitrary spatial domain

The classification presented in theorem 3.1 generalizes a recent classification [7,8] of kinematic conservation laws for the compressible fluid equations (2.1)–(2.3) in *P*(*ρ*,*S*) that have an essential dependence on the pressure, *P*_{ρ}≠0. In particular, the conserved integrals arising from the conserved densities (3.3)–(3.11) provide a covariant generalization (cf. equations (A 1)–(A 7)) of the well-known conserved integrals [5,6] for mass, volumetric entropy, energy, linear and angular momentum, Galilean momentum, similarity energy and Galilean energy in *M*,*g*), including the flat case *P*_{ρ}=0 were not considered).

As a corollary of theorem 3.1, note that there are no special dimensions *n*>1 in which extra kinematic conserved densities are admitted.

### (b) Conservation laws on moving surfaces

### Theorem 3.2

(*i*) *For compressible fluid flow* (*2.4*)–(*2.6*) *in a Riemannian manifold* (*M,g*) *of any dimension n*>1, *no non-trivial kinematic conserved p-form densities* ** α**(

*t,x,u,ρ,S*)

*are admitted for a general equation of state P*(

*ρ,S*). (

*ii*)

*The only special equations of state for which a non-trivial kinematic conserved p-form density*

**(**

*α**t,x,u,ρ,S*)

*arises is the barotropic case*

*The admitted conserved p-form density consists of*

*where*

*u**is the fluid velocity 1-form defined by the dual of u with respect to g*(

*namely, ζ*⌋

**=**

*u**g*(

*ζ,u*)

*for an arbitrary vector field ζ*).

The corresponding kinematic conserved integral (2.19) on an arbitrary curve (one-dimensional surface)

Unlike for conserved densities, no classification of kinematic *p*-form conservation laws for compressible fluid equations have previously appeared in the literature. As a corollary of theorem 3.2, there are no special dimensions *n*>1 in which extra kinematic conserved 1-form densities are admitted, and no conserved *p*-form densities for 2≤*p*≤*n*−1 are admitted.

### (c) Constants of motion

Finally, we state a classification of kinematic constants of motion on moving domains and moving surfaces in compressible fluid flow in Riemannian manifolds by examining when the net fluxes in the kinematic conserved integrals vanish for all solutions of the fluid equations.

A moving domain *Φ* itself is identically zero. By contrast, a *p*-dimensional moving surface *p*≤*n*−1) either has a boundary *p*−1-form ** β** is identically zero, whereas if

From the flux expressions in the kinematic conserved integrals given by theorem 3.1 (cf. equations (A 1)–(A 9) on moving domains) and theorem 3.2 (cf. equation (3.16) on moving curves), we immediately obtain the following result.

### Theorem 3.3

*For compressible fluid flow* (*2.4*)–(*2.6*) *in a Riemannian manifold* (*M,g*) *of dimension n*>1, *the only non-trivial kinematic constants of motion are a linear combination of mass* *and volumetric entropy* *on moving domains* *for any equation of state P*(*ρ,S*), *and circulation* *on closed moving curves* *for barotropic equations of state P*(*ρ*).

### (d) Hamiltonian symmetries and Casimirs

The well-known Hamiltonian formulation for the inviscid compressible Euler equations in *M*,*g*). In covariant form, the Hamiltonian fluid operator is given by
*F* and *G* are functions of *t*,*x*,*u*,*ρ*,*S* and covariant derivatives of *u*,*ρ*,*S*. Here *δ*/*δu*, *δ*/*δρ* and *δ*/*δS* denote variational derivatives, which respectively coincide with the spatial Euler operators *E*_{u}, *E*_{ρ} and *E*_{S} when acting on functions that do not contain time derivatives of *u*,*ρ*,*S*.

The covariant Eulerian fluid equations (2.4)–(2.6) in (*M*,*g*) are given by
*T*, where the components of the symmetry generator are given by

A conserved density *T* that lies in the kernel of the Hamiltonian operator (3.18) determines a conserved integral called a Casimir [16] of the Hamiltonian structure. Every Casimir corresponds to a trivial symmetry, **X**=0. From theorem 3.1, a simple calculation shows that the only Casimirs arising from kinematic conserved densities *T*(*t*,*x*,*u*,*ρ*,*S*) are linear combinations of the mass **X**=*τ*∂_{t}+*χ*⌋∂_{x}+*η*^{u}⌋∂_{u}+*η*^{ρ}∂_{ρ}+*η*^{S}∂_{S} on (*t*,*x*,*u*,*ρ*,*S*), where *η*^{u}, *η*^{ρ}, *η*^{S} are functions of *t*,*x*,*u*,*ρ*,*S*, while *τ*, *χ* are functions only of *t*,*x*.

In particular, the kinematic conserved densities for energy (3.5), (linear/angular) momentum (3.6) and Galilean momentum (3.7), which exist for a general equation of state (2.7), respectively yield the point symmetries

Note the symmetry (3.26) describes a space-translation symmetry if the Killing vector *ζ* is curl-free (irrotational) and non-vanishing at every point in *M*, or a rotation symmetry if the Killing vector *ζ* is not curl-free and vanishes at a single point (centre of rotation) in *M* around which its integral curves are closed. If the Killing vector *ζ* does not have either of these properties, then the physical interpretation of the symmetry (3.26) depends on the nature of the zeros and integral curves of *ζ* in *M*.

## 4. Solution of the determining equations

In index notation (B 9)–(B 11), the Euler equations (2.4)–(2.7) for inviscid compressible fluid flow in an *n*-dimensional Riemannian manifold *M* are written as
*D*^{i}*P*=*P*_{ρ}∇^{i}*ρ*+*P*_{S}∇^{i}*S*.

### (a) Moving domains

A general kinematic conserved density has the form
*E*_{ρ} and *E*_{S} are shown in index notation in equations (.28) and (.27). The proper setting for evaluating *E*_{ρ}(*D*_{t}*T*) and *E*_{S}(*D*_{t}*T*) is the first-order jet space *J*^{1}(*u*^{i},*ρ*,*S*) of the dynamical variables, which is coordinatized by _{i} and ∇^{i} will act on *T* only with respect to its explicit dependence on the coordinate *x*^{i}.

A straightforward calculation yields
^{k}*ρ*, ∇^{k}*S* and ∇^{j}*u*^{k}, we get the system of determining equations
*T*(*t*,*x*^{i},*u*^{i},*ρ*,*S*) and *P*(*ρ*,*S*). We are interested only in non-trivial solutions, such that *T* and *P* each have some homogeneous dependence on at least one of *u*^{i}, *ρ*, *S*. Note *P* can be determined only up to an arbitrary additive constant.

In these equations (4.11)–(4.17), note that *t*,*x*^{i},*u*^{i},*ρ*,*S* are regarded as independent variables, while *g*_{jk} is a function of *x*^{i} such that ∇_{i}*g*_{jk}=0. Hereafter we assume

To proceed, we contract equation (4.17) with *g*^{ij}, which yields
*ρ* first and *u*^{k} next, we obtain

Next, we find equations (4.13) and (4.15) simplify to give
*ρ* and *S*, we obtain
*u*^{j}, we now substitute equation (4.28) into equation (4.25) and integrate, giving
*e*(*ρ*,*S*) is the thermodynamic energy (2.8) defined in terms of *P*(*ρ*,*S*).

Then combining expressions (4.30), (4.26) and (4.22), we see that the solution of the determining equations (4.12), (4.13), (4.15) and (4.17), up to the case splitting (4.23), is given by

Substituting *T* from equations (4.31) and (4.32) into the remaining determining equations (4.11), (4.14), (4.16) and using equation (4.29), we get the system of equations
*C*_{j}(*t*,*x*^{i},*S*) and *C*_{0}(*t*,*x*^{i},*S*).

First, from equation (4.34), we have
*g*^{ij}, which gives
_{i} to equation (4.42), which gives ∇_{i}*C*_{0t}=0. Integrating this equation, we get
*t*, which yields
*t*, we get
_{ρ} of expression (4.47) yields ((*n*/2)*P*−*ρe*)_{ρρ}(*tI*_{1}+*t*^{2}*I*_{2})=0. After this equation is separated with respect to *t*, it is equivalent to the equation
*t*, yielding

The expressions (4.47), (4.46), (4.45), (4.44), (4.39), (4.31), together with equation (4.50), constitute the general solution of the determining equations (4.11)–(4.17), up to the case splittings (4.49), (4.48) and (4.23).

Finally, we consider the various case splittings. From equation (4.48) combined with expression (2.8), we directly have
*ρ* dependence of *P*. The remaining case splitting is given by equation (4.49), which has a more complicated form
*ρ* and *S* dependence of *P*.

*Case 1*: *P*_{ρ}=0.

From this condition, we have that *P* is given by the equation of state *P*=*P*(*S*) and hence *e*=−*ρ*^{−1}*P*(*S*) is the thermodynamic energy, where *P*_{S}≠0 due to condition (4.20). Then equations (4.51) and (4.52) yield no conditions on *C*_{k}, while equation (4.53) reduces to the condition

As there are no further conditions, the expression (4.31) for the conserved density *T* becomes (after an integration by parts in the integral term)
*I*_{0}(*S*), *J*(*S*), *μ*_{(a)}(*S*), *ν*_{(a)}(*S*), and with arbitrary Killing vectors *ψ*_{(a)}(*x*^{i}) for arbitrary curl-free Killing vectors. From the transport equation (2.24), a straightforward calculation now yields the moving flux associated to *T*,

*Case 2*: *P*_{ρ}≠0 and (*n*/2)*P*_{ρρ}−*ρ*^{−1}*P*_{ρ}=0.

By solving these conditions, we find that *P* is given by the equation of state *P*=*σ*(*S*)*ρ*^{γ}+*σ*_{0}(*S*), with *γ*=1+2/*n*, and hence *e*=(1/(*γ*−1))*σ*(*S*)*ρ*^{γ−1}−*ρ*^{−1}*σ*_{0}(*S*) is the thermodynamic energy. Then equations (4.51)–(4.53) yield
*C*_{k}, respectively, into equation (4.61), we get

If *σ*_{0}=const., then from equation (4.50) combined with expressions (4.63), we have

As there are no further conditions, the expression (4.31) for the conserved density *T* in this case is given by
*σ*_{0}, *I*_{0}, arbitrary functions *σ*(*S*), *J*(*S*), and with an arbitrary homothetic Killing vector *ξ*^{j}(*x*^{i}), an arbitrary Killing vector *ζ*^{j}(*x*^{i}), in addition to a potential *θ*(*x*^{i}) for an arbitrary curl-free homothetic Killing vector, and a potential *ψ*(*x*^{i}) for an arbitrary curl-free Killing vector, where *T* is a trivial conserved density (namely, it does not satisfy condition (4.19)), and hence we will put *σ*_{0}=0. Then, from the transport equation (2.24), the moving flux associated to *T* is given by

Instead, if *σ*_{0}≠const., then we have *ζ*_{j}, *ψ* satisfy equation (4.68). As there are no further conditions, in this case the expression (4.31) for the conserved density *T* becomes
*I*_{0}, arbitrary functions *σ*(*S*), *σ*_{0}(*S*), *J*(*S*) and with an arbitrary Killing vector *ζ*^{j}(*x*^{i}), in addition to a potential *ψ*(*x*^{i}) for an arbitrary curl-free Killing vector. From the transport equation (2.24), the moving flux associated to *T* is given by

*Case 3*: *P*_{ρ}≠0 and (*n*/2)*P*_{ρρ}−*ρ*^{−1}*P*_{ρ}≠0.

In this case, *P* is given by a general equation of state *P*=*P*(*ρ*,*S*) other than the form arising in case 2, and hence the thermodynamic energy *e* has the general form (2.8). We then find equations (4.51)–(4.53) yield the conditions
*C*_{k} into these conditions, we get
*T* is therefore given by
*I*_{0}, an arbitrary function *J*(*S*), and with an arbitrary Killing vector *ζ*^{j}(*x*^{i}), in addition to a potential *ψ*(*x*^{i}) for an arbitrary curl-free Killing vector. Similarly to the previous case, the moving flux associated to *T* is given by

### (b) Moving surfaces

A general kinematic *p*-form conserved density, with 1≤*p*<*n*, has the form
*α*_{i1⋯ip}=*α*_{[i1⋯ip]} is totally antisymmetric. From proposition 2.4 combined with the identity (B 15), all kinematic conserved densities (4.79) are determined by the necessary and sufficient equation
*J*^{2}(*u*^{i},*ρ*,*S*) of the dynamical variables, which is coordinatized by _{i} and ∇^{i} will act on *α*_{i1⋯ip} only with respect to its explicit dependence on the coordinate *x*^{i}.

It will be useful to work with the dual of equation (4.80) as follows. Let
*α*_{i1⋯ip}, with
*ϵ*^{jj1⋯jq−1i1⋯ip} to equation (4.80), yielding
*q*.

The total divergence of *T*^{j1⋯jq} is given by

We now substitute expressions (4.86)–(4.92) into the determining equation (4.84), combine terms after use of the derivative identities (4.81) and split the resulting expression with respect to _{j}∇_{k}*ρ*, ∇_{j}∇_{k}*S*, _{j}*u*^{l}∇_{k}*ρ*, ∇_{j}*u*^{l}∇_{k}*S*, ∇_{j}*ρ*∇_{k}*ρ*, ∇_{j}*S*∇_{k}*S*, ∇_{j}*ρ*∇_{k}*S*, ∇_{j}*u*^{l}, ∇_{k}*ρ*, ∇_{k}*S*. This yields the system of determining equations
*T*^{j1⋯jq}(*t*,*x*^{i},*u*^{i},*ρ*,*S*) and *P*(*ρ*,*S*). We are interested only in non-trivial solutions, such that *T*^{j1⋯jq} and *P* each have some homogeneous dependence on at least one of *u*^{i}, *ρ*, *S*, where *P* can be determined only up to an arbitrary additive constant. In these equations (4.93)–(4.104), note that *t*,*x*^{i},*u*^{i},*ρ*,*S* are regarded as independent variables, while *g*_{jk} is a function of *x*^{i} such that ∇_{i}*g*_{jk}=0. Hereafter we assume the conditions (4.18)–(4.20), as before.

To proceed, we contract equation (4.93) with *δ*_{j}^{l} and integrate with respect to *ρ*, which yields
_{S} of equation (4.106) , we obtain

Next, we see equation (4.99) splits with respect to *u*^{i}, yielding
*δ*_{k}^{l}, and using condition (4.83), we get

We now note equation (4.94) becomes *δ*_{k}^{m}, we obtain

Then combining expressions (4.112), (4.109), (4.105), we find that the solution of the determining equations (4.93)–(4.100), up to the case splittings (4.115) and (4.113), is given by
*T* is a trivial conserved density (namely, it does not satisfy condition (4.19)), and hence we will put

Substituting expressions (4.116) and (4.117) for *T* into the remaining determining equations (4.101)–(4.104), and splitting with respect to *u*^{i}, we get the system of equations

From condition (4.20), we have that equations (4.118) and (4.119) yield

Finally, we consider the case splittings. Clearly, we want *T* will be trivial. Hence equations (4.113) and (4.115) directly give *P*_{S}=0 and *q*=*n*−1. In this case, we have that *P* is given by the barotropic equation of state *P*=*P*(*ρ*) and hence the thermodynamic energy *e* has the general barotropic form *q*=*n*−1, the covariantly constant skew tensor *ϵ*^{kj1⋯jn−1},

From equation (4.82), the dual *p*-form density corresponding to the skew tensor density *T*^{j1⋯jn−1} is given by
*p*−1-form flux in the transport equation (2.32) is given by
*p*−1=0).

## 5. Concluding remarks

The classification results in §3 apply generally to inviscid compressible fluids when the fluid flow is either isentropic (in which case *S* is constant throughout the fluid domain *M*) or non-isentropic (in which case *S* is constant only along fluid streamlines in *M*). This is a consequence of the case-splitting method that is used in §4 to solve the determining equations. More specifically, although the isentropic fluid equations (2.1)–(2.2) could possibly admit additional conservation laws that do not hold for non-isentropic fluid flow, the determining equations show that this possibility does not occur. As a consequence, it turns out that all kinematic conservation laws of isentropic fluid flow arise from the kinematic conservation laws of non-isentropic fluid by simply restricting the entropy *S* to be constant in *M*.

It is worth emphasizing that these classification results are complete for conservation laws of kinematic form on moving domains and moving surfaces. Kinematic conservation laws are a physically important but limited class. In a subsequent paper, it is planned to classify all conservation laws of vorticity form on moving domains and moving surfaces. This class of conservation laws turns out to be much larger than the class of kinematic conservation laws, as shown by the examples of new conserved vorticity integrals found in recent work [12].

A fully complete classification of fluid conservation laws will require going beyond those two forms for conserved densities and spatial fluxes, in particular by allowing dependence on arbitrary high order derivatives of all the fluid variables. This open problem will remain a hard challenge for future work.

## Competing interests

We declare we have no competing interests.

## Funding statement

S.C.A. is supported by an NSERC research grant. N.T. thanks HEC, Pakistan, for providing a fellowship grant to support a research visit to Brock University. A.D. thanks the Department of Mathematics and Statistics at Brock University for partial support during the period when this research was completed.

## Appendix A

Let

For a general non-isentropic equation of state (2.7), the kinematic conserved densities (3.3)–(3.7) yield the conserved integrals
*λ*=0. The conserved integrals yielded by the extra kinematic conserved densities (3.12) and (3.13) for the isobaric-entropy equation of state (3.9) are given by

Note the conserved integral (A 4) describes linear momentum if the Killing vector *ζ* is curl-free (irrotational) and non-vanishing at every point in *M*, or angular momentum if the Killing vector *ζ* is not curl-free and vanishes at a single point (centre of rotation) in *M* around which its integral curves are closed. If the Killing vector *ζ* does not have either of these properties, then the conserved integral (A 4) can be viewed as describing a generalized momentum whose physical interpretation depends on the nature of the zeros and integral curves of *ζ* in *M*.

## Appendix B

To begin, a complete transcription between geometric notation and tensorial index notation will be listed.

**(a) Notation**

Vector product operations:
*a*,*c* are arbitrary vector fields and *b* is an arbitrary covector field, on *M*.

Tensor product operations:
*A*,*B* and *C* are arbitrary tensor fields on *M*.

Geometrical structures and operators:

Fluid velocity, pressure gradient and their covariant derivatives:

Covariant derivative identities:
*f* is an arbitrary scalar function, *a* is an arbitrary vector field and *b* is an arbitrary covector field, on *M*.

Lie derivative identities:
*A* is an arbitrary skew tensor field, on *M*.

Symmetries of Riemann tensor:

**(b) Euler operator and its properties**

The covariant spatial Euler operator (2.26) is given by
*v*, and
*v*^{i1⋯ip}_{j1⋯jq}. In all cases, the Euler operator is uniquely determined by the following variational identity:
*w*^{i1⋯ip}_{j1⋯jq} and an arbitrary scalar function *v*^{i1⋯ip}_{j1⋯jq}. There is an explicit expression for *Θ* in terms of *w*^{i1⋯ip}_{j1⋯jq} and partial derivatives of *f*, which we will not need here. The identity (B 19) leads to a simple proof of lemma 2.1.

If *f*=Div *F* then *w*^{i1⋯ip}_{j1⋯jq} is arbitrary.

Conversely, if *f*. Let *v*_{(λ)}^{i1⋯ip}_{j1⋯jq} be a one-parameter homotopy such that *f*=Div *F* with *Θ*_{0} is any vector field satisfying Div *Θ*_{0}=*f*|_{v=v0}.

- Received April 4, 2015.
- Accepted July 21, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.