## Abstract

For the Euler equations governing compressible isentropic fluid flow with a barotropic equation of state (where pressure is a function only of the density), local conservation laws in *n*>1 spatial dimensions are fully classified in two primary cases of physical and analytical interest: (i) kinematic conserved densities that depend only on the fluid density and velocity, in addition to the time and space coordinates, and (ii) vorticity conserved densities that have an essential dependence on the curl of the fluid velocity. A main result of the classification in the kinematic case is that the only equation of state found to be distinguished by admitting extra *n*-dimensional conserved integrals, apart from mass, momentum, energy, angular momentum and Galilean momentum (which are admitted for all equations of state), is the well-known polytropic equation of state with a dimension-dependent exponent, γ=1+2/*n*. In the vorticity case, no distinguished equations of state are found to arise, and here the main result of the classification is that, in all even dimensions *n*≥2, a generalized version of Kelvin’s two-dimensional circulation theorem is obtained for a general equation of state.

## 1. Introduction

Conservation laws and Hamiltonian structures are central to the mathematical study of fluid flow and have long been known for both the incompressible (ideal fluid) and compressible (inviscid fluid) Euler equations governing fluid flow in two and three dimensions. Over the past few decades there has been considerable mathematical interest in studying the Eulerian fluid equations in *n* dimensions (Arnold & Khesin 1998).

One strong motivation came from the work of Arnold (1966, 1969) showing that the Euler equations for incompressible fluids in *n*-dimensional spatial domains have an elegant geometric formulation as the geodesic equation on the Lie group of volume-preserving diffeomorphisms of the given domain of the fluid flow. This formulation gives an interesting geometrical significance to fluid conservation laws by interpreting them as geodesic first integrals related to invariance properties of the geodesic Lagrangian. Subsequently, the main group-theoretic aspects of Arnold’s work were extended to the compressible Euler equations, first (Guillemin & Sternberg 1980; Marsden *et al*. 1984) for isentropic fluids (whose entropy is constant throughout the fluid domain) in which the pressure is specified to be a function only of density as given by an equation of state, then later (Novikov 1982; Arnold & Khesin 1998) for adiabatic non-isentropic fluids (in which the entropy is conserved only along streamlines) where the pressure is given by a dynamical equation.

The aims of the present paper and a sequel will be to give a complete picture of the conservation laws of kinematic type and vorticity type for general compressible fluids in *n*>1 dimensions for both isentropic and non-isentropic cases. By a *kinematic* conservation law we will mean a local continuity equation where the conserved density and flux depend only on the fluid velocity, pressure and density (but not their spatial derivatives), in addition to the time and space coordinates. Such conservation laws encompass the familiar physical continuity equations in two and three dimensions for mass, momentum and energy (Landau & Lifshitz 1968; Chorin & Marsden 1997; Batchelor 2000). In contrast, a *vorticity* conservation law will refer to a local continuity equation for a conserved density and flux that have an essential dependence on the curl of the fluid velocity in a form exhibiting odd parity under spatial reflections. Examples of conservation laws with this form are helicity in three dimensions as well as circulation and enstrophy in two dimensions, which are well-known for incompressible fluid flow (Shepherd 1990; Majda & Bertozzi 2002).

To date, all of the known *n*-dimensional fluid flow conservation laws (Ibragimov 1973, 1994–1996; Khesin & Chekanov 1989) belong to these two classes but have been derived through special methods that fall short of providing a complete classification. An interesting open question we will settle in this paper for isentropic compressible fluid flow is to find all particular equations of state for which the *n*-dimensional Eulerian fluid equations admit vorticity conservation laws or extra kinematic conservation laws.

In §2, as preliminaries, the general formulation of local continuity equations and integral conservation laws for the Euler equations for compressible isentropic fluids in *n*>1 dimensions is reviewed. In particular, we introduce necessary and sufficient determining equations for directly finding conserved densities of any specified form. By solving the determining equations for kinematic conserved densities, we obtain a complete classification showing that, apart from mass, momentum and energy, the only additional conservation laws of kinematic form consist of Galilean momentum (connected with centre-of-mass motion) and angular momentum holding for any equation of state, plus dilational-type energies arising for polytropic equations of state where the pressure is proportional to a particular dimension-dependent power of the density.

Next in §3, we solve the determining equations to find all vorticity conservation laws, starting from the transport equation for the curl of the fluid velocity. This classification yields an odd-dimensional generalization of helicity and an even-dimensional generalization of circulation and enstrophy, which are found to hold for any equation of state. We show that the generalized circulation has an equivalent formulation as a constant of the fluid motion defined on the boundary of any spatial domain that is transported in the fluid. This new result gives a generalization of Kelvin’s two-dimensional circulation theorem to all even dimensions *n*≥2.

Finally, in §4, we use the well-known Hamiltonian structure (Verosky 1985) of the compressible Euler equations to classify all Hamiltonian symmetries corresponding to the kinematic and vorticity conservation laws. In §5, we make some concluding remarks, including a summary of our classification results stated in index notation.

A corresponding treatment of kinematic and vorticity conservation laws for adiabatic non-isentropic compressible fluids in *n*>1 spatial dimensions will be given in a separate paper (Anco & Dar in preparation).

## 2. Compressible isentropic flows

The Euler equations for compressible isentropic fluids in (in the absence of external forces) with velocity **u**(*t*,**x**) and density ρ(*t*,**x**) consist of
2.1
2.2
with a barotropic equation of state for pressure
2.3
Throughout, we will use bold notation to denote vector or tensor variables and operators. A dot will denote the Euclidean inner product as well as stand for contraction between vectors and tensors, while a wedge will denote the antisymmetric outer product of vectors and/or antisymmetric tensors.

Fluid conservation laws are described by a local continuity equation (Olver 1993; Anco & Bluman 2002*b*; Bluman *et al*. 2009)
2.4
holding for all formal solutions of equations (2.1)–(2.3), where *T* and **X** are some functions of *t*,**x**,ρ,**u** and their **x**-derivatives. Here *D*_{t} and *D*_{x} denote total time and space derivatives, respectively. Physically, *T* is a conserved density with **X** being a corresponding spatial flux. In the integral form, the continuity equation (2.4) is equivalently described by
2.5
where *V* is any spatial domain in through which the fluid is flowing and is the outward unit normal on the domain boundary ∂*V* . A physically more useful form for expressing fluid conservation law equations (2.4) and (2.5) is obtained by considering a spatial domain *V* (*t*) that moves with the fluid (Ibragimov 1994–1996). Then the flux through the moving boundary ∂*V* (*t*) is **ξ**=**X**−*T***u**, which is related to the conserved density *T* by the transport equation
2.6
where *D*_{t}+**u**⋅*D*_{x} represents the total convective (material) derivative and ∇⋅**u** represents the expansion or contraction of an infinitesimal volume moving with the fluid. The corresponding integral form of a fluid conservation law in a moving domain is accordingly expressed as
2.7
whereby will be a constant of the fluid motion in if the net flux across ∂*V* (*t*) vanishes.

The determining equations for finding conserved densities *T* are given by
2.8
where *E*_{ρ} and *E*_{u} are spatial Euler operators (Anco & Bluman 2002*b*) with respect to ρ and **u**, and is the total time derivative evaluated on solutions of the Euler equations (2.1)–(2.3). (The explicit form of these operators is shown using index notation in §2*b*.) The equations (2.8) arise from the fact that spatial divergences have a characterization (Olver 1993; Bluman *et al*. 2009) as functions of *t*,**x**,ρ,**u** and **x**-derivatives of ρ, **u** that are annihilated by both of the spatial Euler operators.

A conservation law is locally trivial if the conserved density and spatial flux have the form
2.9
whereby the continuity equations (2.4) and (2.5) hold as identities for some vector function **Θ** and antisymmetric tensor function **Ψ** of *t*,**x**,ρ,**u**, and **x**-derivatives of ρ, **u**. The corresponding identity holding in a moving domain *V* (*t*) takes the form
2.10
where **ξ**=−*D*_{t}**Θ**−(*D*_{x}⋅**Θ)****u** is the spatial flux through the moving boundary ∂*V* (*t*). As it stands, equation (2.10) has no physical content. However, an interesting observation is that if the moving-flux **ξ** is divergence-free for all formal solutions of the Euler equations (2.1)–(2.3) then the quantity will be a non-trivial constant of motion. Namely, when **u**(*t*,**x**) and ρ(*t*,**x**) satisfy equations (2.1)–(2.3), vector functions **Θ** that satisfy the condition
2.11
lead to non-trivial conservation laws of the form
2.12
for any boundary hypersurface ∂*V* (*t*) that is transported in the fluid. In particular, it is sufficient for **Θ** to satisfy the condition of vanishing flux, −**ξ**=*D*_{t}**Θ**+(*D*_{x}⋅**Θ)****u**=0, for all formal solutions of equations (2.1)–(2.3). We will call equation (2.12) a *moving-boundary* conservation law.

### (a) Classification of kinematic conservation laws

We first consider kinematic conservation laws as defined by the form
2.13
for the conserved density. In the case of polytropic equations of state,
2.14
where pressure is proportional to a power of the density, all of the known local conservation laws (2.13) of the polytropic Euler equations (2.1), (2.2), (2.14) in *n*>1 dimensions are summarized in table 1 (Ibragimov 1973).

We begin by stating a general classification of kinematic conservation laws with respect to all equations of state given by the form (2.3). Note that any conservation law of this form is locally non-trivial since it does not contain **x**-derivatives of ρ or **u**.

### Theorem 2.1

(*i*) *For a general equation of state of the form* (*2.3*), *the fluid conservation laws* (*2.13*) *in any dimension n*>1 *comprise a linear combination of mass, momentum, angular momentum, Galilean momentum and energy. In particular, for any spatial domain* *transported in the fluid*,
2.15
2.16
2.17
2.18
2.19
where
2.20

(*ii*) *Modulo a constant shift in p, the only equation of state for which extra conservation laws* (*2.13*) *arise is the polytropic case with dimension-dependent exponent* ,
2.21

*The admitted conservation laws consist of a linear combination of a similarity energy and a dilational energy. In particular*,
2.22
2.23
*where*
2.24
*is the polytropic energy density.*

The proof of this classification theorem is given in §2*b* using index notation. A summary of the conservation laws (equations (2.15)–(2.24)) written in explicit component form is presented in §5.

We remark that these conservation laws were first derived (Ibragimov 1973) for the case of irrotational fluid flow with a polytropic equation of state (2.14). The Euler equations for such fluids in *n*>1 dimensions turn out to have a Lagrangian formulation when a velocity potential is introduced (i.e. ∇∧**u**=0 implies **u**=∇Φ), which allows local continuity equations to be classified in terms of point symmetries by means of Noether’s theorem (Olver 1993; Bluman *et al*. 2009). In particular, mass conservation arises from invariance of the Euler–Lagrange fluid equations under shifts in the velocity potential. Invariance under space translations, rotations, Galilean boosts, and time translation yields, respectively, conservation of momentum, angular momentum, Galilean momentum, and energy. For the special polytropic equation of state (2.21), the similarity energy arises from a particular combination of scaling and dilation invariance that produces a variational symmetry, while the dilational energy corresponds to an extra symmetry (Ovsyannikov 1962) that is admitted only for this equation of state.

### (b) Classification proof

The proof of theorem 2.1 is based on explicitly solving the determining equations (2.8) by tensorial index methods. We introduce the following index notation: , , and (using a subscript comma to denote partial derivatives), and , where *i*=1,2,…,*n*; indices will be freely raised and lowered via the Kronecker symbols δ_{ij} and δ^{ij} (which are components of the Euclidean metric tensor and its inverse on in Cartesian coordinates). The summation convention will apply to repeated indices.

In index notation,
2.25
are the Euler equations (2.1)–(2.3). The spatial Euler operators with respect to ρ and *u*^{i} are given by
2.26

For a conserved density of the kinematic form *T*(*t*,*x*^{i},ρ,*u*^{i}), we have
2.27
First applying the Euler operator *E*_{ρ} to equation (2.27) we get
which is a linear inhomogeneous scalar expression in *u*^{i}_{,}^{j}. Its coefficient must vanish, yielding the two equations
2.28
2.29
Next, we apply the other Euler operator *E*_{ui} to equation (2.27), obtaining
which is a linear inhomogeneous expression in ρ_{,}^{j}, *u*^{j}_{, j} and *u*^{j}_{,i}. First, we see the coefficient of ρ_{,}^{j} yields the same terms as in equation (2.28). Next, since *u*^{j}_{, j} and *u*^{j}_{,i} are linearly independent in *n*>1 dimensions, their coefficients must separately vanish, which yields
2.30
This leaves the inhomogeneous terms
2.31

Hence, the determining equations consist of equations (2.28)–(2.31) to be solved for *T*. We start from equation (2.30), which is first-order linear in ρ. By integrating with respect to ρ, and then doing a trivial integration with respect to *u*^{i}, we obtain
2.32
Substituting equation (2.32) into equation (2.28) gives
2.33
which separates with respect to ρ,*u*^{i} into two equations
2.34
where *c*(*t*,*x*^{i}) is a constant of separation. Integration of equation (2.34) yields
2.35
2.36
2.37
where *c*_{0}, *c*_{1}, *c*_{2} and are constants of integration with respect to ρ and *u*^{i}. Thus, we have
2.38
where . Since the term *c*_{0} is trivially conserved, we can put *c*_{0}=0.

By substituting equation (2.38) into equations (2.29) and (2.31), in each case we get a cubic polynomial in terms of *u*^{i} whose separate coefficients must vanish. This leads to the following system of equations:
2.39
2.40
2.41
2.42
To proceed, we note equation (2.39) immediately implies *c*=*c*(*t*). Then equation (2.40) has the form of a time-dependent dilational Killing vector equation on . To derive the solution, we first take the antisymmetrized derivative of equation (2.40), i.e. differentiating with respect to *x*^{k} followed by antisymmetrizing in *j* and *k*, which yields
2.43
Similarly, by taking the curl of equation (2.41), i.e. differentiating with respect to *x*^{j} and antisymmetrizing in *i* and *j*, we obtain
2.44
Hence, equations (2.43) and (2.44) give
2.45
Adding equation (2.45) to (2.40), we get
2.46
and thus, by integration with respect to *x*^{j},
2.47
Then equation (2.41) becomes
2.48
which yields
2.49
Finally, from equation (2.42), using equation (2.49) and the trace of equation (2.40), we get
2.50
Splitting equation (2.50) with respect to *x*^{i} yields
2.51
2.52
From equation (2.51) we have
2.53
with constants *a*_{0i}, *a*_{1i}, *b*_{0}, *b*_{1} and *b*_{2}. Differentiating equation (2.52) with respect to ρ gives
2.54
which leads to the following two cases.

*Case*. *c*′=0: Hence *c*=*b*_{0} and *b*_{1}=*b*_{2}=0, which implies *C*_{2}=const. from equation (2.52). Then equations (2.47) and (2.49) yield
2.55
whence from equation (2.38) we obtain
where *C*_{2}, *C*_{1ij}=−*C*_{1ji}, *a*_{0i}, *a*_{1i}, *b*_{0} are arbitrary constants and *e* is a function of ρ given by equation (2.37).

*Case.*: Simplifying (ρ*e*)′′=ρ*e*′′+2*e*′=ρ^{−1}*P*′ through equation (2.37), we get a linear ODE
2.56
which has the general solution *P*(ρ)=*d*_{0}+*d*_{1}ρ^{1+2/n}, where *d*_{0}, *d*_{1} are constants. Note that we can put *d*_{0}=0, since the pressure *p*=*P*(ρ) can be shifted (without loss of generality) by an arbitrary constant. This implies
2.57
from equation (2.37), with a constant of integration *d*_{2}. Thus, equation (2.52) becomes *C*_{2}′+*d*_{2}*c*′=0, whence
2.58
where *b*_{3} is an integration constant. Then equations (2.47) and (2.49) yield
2.59
Substituting equations (2.58) and (2.59) into equation (2.38), we find that the terms involving *d*_{2}*c*(*t*) cancel out, giving
with , where *d*_{1}, *b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}, *a*_{0i}, *a*_{1i}, *C*_{1ij}=−*C*_{1ji} are arbitrary constants.

## 3. Vorticity conservation laws

In *n* >1 dimensions, the curl of the fluid velocity is the antisymmetric tensor
3.1
which satisfies the identities ∇∧**ω**=0 and ∇⋅**ω**=Δ**u**−∇(∇⋅**u**). There is a natural odd-parity expression that can be constructed purely out of products of **ω** and the spatial orientation tensor **ε** as follows. (Recall, **ε** is a rank *n* totally skew-symmetric tensor whose components in Cartesian coordinates for are given by the Levi-Civita symbol. In particular, up to a choice of sign, **ε** is determined by its properties ∇**ε**=0 and |**ε**|^{2}=**ε**⋅**ε**=*n*!.)

When the spatial dimension is even, say *n*=2*m*,
3.2
defines a vorticity scalar that has odd parity since **ε** changes sign under spatial reflections. Similarly, when the spatial dimension is odd, say *n*=2*m*+1, the analogous expression
3.3
defines a vorticity vector with odd parity under spatial reflections. These expressions (3.2) and (3.3) can be written in the more compact notation *(**ω**^{m}), where * is the Hodge dual operator (acting by contraction with respect to **ε**) and *m*=[*n*/2] is a positive integer.

We now define vorticity conservation laws to have the form
3.4
for the conserved density, such that the expression (3.4) possesses odd parity under spatial reflections as follows. Let be a reflection operator defined with respect to a spatial unit vector **ℓ** in . Specifically, reverses all vectors parallel to **ℓ** while leaving invariant all vectors in the hyperplane orthogonal to **ℓ** through the origin. Note extends to act on tensors by multi-linearity and acts as the identity on scalars. Then for any choice of **ℓ**, satisfies the properties and , so consequently the parity of a conserved density (3.4) will be odd under spatial reflections iff
3.5
Note that, in contrast, the parity of all the kinematic conserved densities (2.15)–(2.23) is even,
3.6

Our main result will now be a complete classification of vorticity conservation laws for all equations of state given by the form (2.3).

## Theorem 3.1

*For any equation of state* (*2.3*), *the only non-trivial fluid conservation laws* (*3.4*) *and* (*3.5*) *in dimensions n* > 1 *are given by helicity*
3.7
*for odd dimensions n* = 2*m*+1, *where*
3.8
*and generalized enstrophy*
3.9
*for even dimensions n* = 2*m, where f is any nonlinear odd function of ϖ/ρ. In particular, there are no special equations of state that admit extra vorticity conservation laws.*

The helicity and enstrophy conservation laws were first derived by means of a Hamiltonian Casimir analysis (Khesin & Chekanov 1989; Arnold & Khesin 1998), as we will discuss further in §4, which is a more restrictive analysis than directly solving the determining equations (2.8) for conserved densities. By comparison, our classification has more generality and actually holds without imposing the odd-parity condition (equation (3.5)) if we consider conserved densities (3.4) that just have an essential dependence on **ω**. The proof is given in §3*a* and a summary of the conservation laws (3.7)–(3.9) in explicit component form is shown in §5.

The local continuity equations underlying these conservation laws (3.7)–(3.9) are readily obtained from the transport equations satisfied by **ϖ** and ϖ. Through equation (3.1), we first note the identity **u**⋅**ω**=**u**⋅∇**u**−1/2∇(**u**⋅**u**), whereby the Euler equation (2.1) for the fluid velocity can be written in the form
3.10
where *f*(ρ) is given by equation (3.8) through the equation of state (2.3). Taking the curl of equation (3.10) we get
3.11
Substitution of equation (3.11) into the time derivatives of equations (3.2) and (3.3) then yields the respective transport equations
3.12
or, equivalently,
3.13
3.14

The vorticity transport equations (3.13) and (3.14) each have the form of a local continuity equation holding in even (*n*=2*m*) and odd (*n*=2*m*+1) dimensions, with respective conserved densities, *T*=ϖ and *T*=**ϖ**. As we will now show, both of these vorticity conservation laws are locally trivial and therefore fall outside of our classification theorem. From the explicit expressions given by equations (3.2) and (3.3) for the vorticity densities in terms of the curl of the fluid velocity **ω**=∇∧**u**, we find that in both even and odd dimensions
3.15
is a spatial divergence. The fluid equations (3.10) and (3.11) yield the time derivative of **Θ**=*(**u**∧**ω**^{m−1}) to be
3.16
where is an antisymmetric tensor. Since in both even and odd dimensions the expression
3.17
is the spatial flux, **X**=ϖ**u** and **X**=**u**∧**ϖ**, arising from the vorticity transport equations (3.13) and (3.14), we then see equations (3.17) and (3.15) have the form (2.9) of a locally trivial spatial flux and a locally trivial conserved density.

It is interesting to investigate the corresponding vorticity flux **ξ**=**X**−*T***u** through a moving domain boundary in the fluid.

In odd dimensions *n*=2*m*+1, we can write
3.18
where **W** is a totally skew-symmetric tensor of rank 3. Hence, for any domain *V* (*t*) transported in the fluid, we obtain
3.19
for all formal solutions of the dynamical equations (3.10) and (3.14). Since the moving-flux through ∂*V* (*t*) fails to vanish, equation (3.19) does not yield a constant of motion.

For even dimensions *n*=2*m*, we have
3.20
where **w** is an antisymmetric tensor. Then for all formal solutions of the dynamical equations (3.10) and (3.13), we obtain
3.21
Thus, equation (3.21) yields a constant of motion for the moving boundary ∂*V* (*t*) in . It describes an even-dimensional generalization of Kelvin’s circulation for isentropic fluid flow in two dimensions (Batchelor 2000). Specifically, when *n*=2 (*m*=1), the vorticity scalar is given by
3.22
and hence **w**=**ε** is the spatial orientation tensor. This yields , and then the conservation law equation (3.21) becomes Helmholtz’s circulation theorem (Chorin & Marsden 1997),
3.23
where ∂*V* (*t*) is a closed curve in , is a unit tangent vector along ∂*V* (*t*) and dσ is the arclength element. By writing , we see equation (3.23) states that the line integral defining the circulation of the fluid velocity around a curve transported in the fluid is a constant of the fluid motion. For higher dimensions *n*=2*m* (*m*>1), we can write the moving-boundary conservation law equation (3.21) in an analogous form by noting
3.24
where is the volume tensor for the boundary hypersurface ∂*V* (*t*) in . Then, equation (3.21) becomes
3.25
which states that the (hyper)surface integral
3.26
is a constant of the fluid motion, where denotes the volume element for the moving-boundary (hyper)surface ∂*V* (*t*) in even dimensions *n*=2*m*, analogous to for moving-boundary curves in two dimensions.

## Proposition 3.2

*The only moving-boundary conservation law* (*2.12*) *of vorticity type is the generalized circulation given by equation* (*3.25*) *holding for any equation of state in all even dimensions.*

The proof of this classification is given in the next section.

### (a) Classification proof

We will use the same index notation introduced in §2*b* for the proof of theorem 2.1. To begin, we write out the component form of the fluid curl, the vorticity scalar and vector, along with their transport equations:
In addition, we will need the component form for
Here , are the components of the spatial orientation tensor and the Euclidean metric tensor; round brackets denote symmetrization of the enclosed indices, and square brackets denote antisymmetrization.

The proof of theorem 3.1 and proposition 3.2 proceeds by explicitly solving the determining equations (2.8) for conserved densities of vorticity type in even and odd dimensions *n*>1. Recall, the Euler equations (2.1)–(2.3) are given by
3.27
where
3.28

*Case*. *n*=2*m*: For a conserved density of the form *T*(ρ,*u*^{i},ϖ), its time derivative is given by
3.29
In equation (3.29), we see the coefficient of *u*^{i} equals *D*_{i}*T* by the chain rule, which allows us to write the corresponding terms as *u*^{i}*D*_{i}*T*=*D*_{i}(*u*^{i}*T*)−*u*^{i}_{,i}*T*. Hence
3.30
where
3.31

To begin, we substitute equation (3.30) into the first determining equation,
3.32
where we have introduced the notation
3.33
Since *T* does not contain any derivatives of ϖ, the coefficient of ϖ_{,}^{i} in equation (3.32) must vanish,
3.34
Integrating equation (3.34) and dropping a kinematic term *c*(ρ,*u*^{i}) that does not involve ϖ, we get
3.35
Then equation (3.32) reduces to 0=tr *u**A*_{ρ} and hence we obtain
3.36
This is a first-order linear PDE for *a*_{ρ}, which yields
3.37

Thus, from equations (3.35) and (3.37) we have
3.38
and
3.39
Now we substitute equation (3.39) into the second determining equation,
3.40
where
3.41
Note that *b*(ϖ) has no dependence on tr *u* since ϖ does not contain *u*^{ij} (which is linearly independent of ϖ_{ij}). Consequently, the coefficients of all terms in equation (3.40) must vanish, yielding *B*=*B*′=0 so thus *b*′′=0. Hence, we obtain
3.42
with constants *b*_{0}, *b*_{1}. Therefore, equations (3.42) and (3.38) give the result
3.43
with
3.44
from equation (3.39), where *b*_{1}, *b*_{0} are arbitrary constants and *c* is an arbitrary function of ϖ/ρ. Since the term *b*_{0} is trivially conserved, we can put *b*_{0}=0.

We now note that the first term in equation (3.43) is a trivial conserved density
3.45
Its corresponding moving-flux vanishes, ξ^{i}=0, owing to the form of equation (3.44). Similarly, if is a linear function of ϖ/ρ, where is a constant, then the second term in equation (3.43) becomes a trivial conserved density, . This proves theorem 3.1 and proposition 3.2 in the even-dimensional case.

*Case*. *n*=2*m*+1: Here we will need the identities
3.46
3.47
3.48
which are consequences of
3.49
3.50
3.51
where equation (3.51) follows from the fact that there are no totally skew-symmetric tensors of rank 2*m*+2>*n*=2*m*+1.

Now for a conserved density *T*(ρ,*u*^{i},ϖ^{i}), by the same steps followed in the previous case to evaluate the time derivative, we have
3.52

To proceed we substitute equation (3.52) into the first determining equation, . Using the notation (3.33), this yields the terms
3.53
To start, we observe that the first term in equation (3.53) vanishes by the identity (3.48). Next, since *T* does not contain any derivatives of ϖ^{i}, the last term in equation (3.53) must vanish modulo the identity (3.47). This implies
3.54
Applying the derivative operator ∂_{uk} to equation (3.54) and antisymmetrizing in [ *j**k*], we get
3.55
The trace of this equation with respect to (*i**j*) yields (*n*−1)*a*_{uk}=0, whence in *n*>1 dimensions we obtain
3.56
By also applying the derivative operator ∂_{ϖk} to equation (3.54), we similarly get
3.57
Integration of equations (3.54), (3.56), (3.57) then yields
3.58
Here we have dropped an integration constant *c*(ρ,*u*^{i}) since it does not involve ϖ^{i} (i.e. it is of kinematic form). Hence the determining equation (3.53) reduces to
3.59
with coefficients
3.60
3.61
in terms of the notation *b*_{i}=*b*_{ϖi}. Since ϖ^{i} does not contain *u*^{ij} we see that *b*(ρ,ϖ^{i}) has no dependence on *u*^{ij} (or *t**r* *u*). Consequently, equation (3.59) implies that the coefficient of *u*^{ij} must vanish,
3.62
By considering the product of equation (3.61) with ϖ_{k}ϖ_{l} and antisymmetrizing in [*i**l*] and [*j**k*], we find ϖ_{[l}*B*_{i][j}ϖ_{k]}=0. The same antisymmetric product applied to equation (3.62) then implies *A*ϖ_{[l}δ_{i][j}ϖ_{k]}=0, which gives
3.63
and hence
3.64
Now by taking the product of equation (3.64) with ϖ_{k} antisymmetrized in [*j**k*], we get
3.65
This can hold only if *a*′=0 and *b*_{jρ}=ϖ_{j}*c*(ρ,ϖ^{i}). Then, equation (3.61) becomes
3.66
whence equation (3.64) implies *c*=0 and so *b*_{jρ}=0. Thus, we have
3.67
whose solution is . Since contributes only a kinematic term in equation (3.58), it will be dropped hereafter. Then
3.68
satisfies both equations (3.63) and (3.64).

Thus we have, from equations (3.58) and (3.68), 3.69 and from equation (3.52), 3.70 Through equation (3.47), we note and , while by equation (3.48). Thus we have 3.71

The second determining equation thereby yields the terms
3.72
where
3.73
with the notation , . Using the identities
and collecting like terms in equation (3.72), we get a linear homogeneous expression in *u*^{jkl}, ω^{k(l}_{,}^{j)}, ω^{k(l}_{,}^{j)}*u*^{gh}:
3.74
Since ω^{k(l}_{,}^{j)} is linearly independent of *u*^{jkl} in *n*>1 dimensions, their coefficients in equation (3.74) must vanish. From the coefficient of ω^{k(l}_{,}^{j)}*u*^{gh}, we have
3.75
which yields
3.76
These two equations imply that *B*_{kji} must be a constant tensor and *A*_{i} must be a constant vector. Next, the coefficients of *u*^{jkl} and ω^{k(l}_{,}^{j)} yield the equations
3.77
3.78
We note that antisymmetrizing equation (3.77) in [*i**j*] leads to equation (3.78) after indices are renamed. Hence, only equation (3.77) needs to be considered. Taking the trace of equation (3.77) in (*i**j*) and using the identity (3.46), we obtain
3.79
Such an algebraic equation can hold only as a consequence of the skew-symmetry property (3.49), so equation (3.79) is satisfied only if *B*_{ijh}=*B*_{(ij)h} is of the form
3.80
where *c*_{i}, are constant vectors and is a trace-free totally symmetric constant tensor. Hence, equation (3.77) reduces to
3.81
The trace of equation (3.81) in (*k**l*) leads to (since and ), which implies
3.82
By the same argument that led to equation (3.80), the only totally symmetric tensor that can satisfy equation (3.82) is , which is trace-free only if . Thus, we have
3.83
giving the solution of equation (3.74). Substitution of expressions (3.73) into equation (3.83) then gives us
3.84
Antisymmetrizing equation (3.84) in [ *j**k*] yields
3.85
Taking the product of equation (3.85) with ϖ_{l} antisymmetrized in [*j**k**l*], followed by taking the trace in (*i**k*), we get . This implies
3.86
whence equation (3.85) yields
3.87
and thus
3.88
Substituting equations (3.87) and (3.88) into equation (3.84), we get
3.89
which directly implies
3.90
From equation (3.87), we thus have and so, by direct integration,
3.91
where we have dropped a constant term since it does not involve ϖ^{k}.

As a result, from equation (3.69) we have
3.92
with arbitrary constants *a*, , while from equation (3.71)
3.93
where the last term comes from
3.94
owing to equations (3.47) and (3.48).

To conclude the proof of theorem 3.1 and proposition 3.2, we note that in equation (3.92) the term 3.95 is a trivial conserved density, while its corresponding moving-flux from equation (3.93) is given by , which fails to be divergence-free,

## 4. Correspondence between conserved densities and Hamiltonian symmetries

The *n*-dimensional compressible Euler equations (2.1)–(2.3) have the well-known Hamiltonian formulation (Verosky 1985)
4.1
where is the internal energy density and is called a Hamiltonian operator. This means determines a Poisson bracket (Olver 1993)
4.2
having the properties that (modulo divergence terms) it is antisymmetric and obeys the Jacobi identity, for arbitrary functionals and where *F* and *G* are functions of *t*,**x**,ρ,**u** and their **x**-derivatives. Here δ/δ**u** and δ/δρ denote variational derivatives, which coincide with the spatial Euler operators *E*_{u} and *E*_{ρ} when acting on functions that do not contain time derivatives of ρ and **u**.

To check that the Hamiltonian structure (4.1) produces equations (2.1)–(2.3), we note δ*E*/δ**u**=ρ**u** and , which yields
whereby we obtain
4.3

Through this formulation, the Hamiltonian operator gives rise to an explicit mapping that produces symmetries of the compressible Euler equations from conservation laws as follows. Recall, fluid symmetries (Ibragimov 1994–1996) are described by an infinitesimal transformation
4.4
on all formal solutions of equations (2.1)–(2.3), where and are some functions of *t*,**x**,ρ,**u**, and **x**-derivatives of ρ, **u** determined by infinitesimal invariance (Olver 1993; Bluman & Anco 2002) of the Euler equations (2.1)–(2.3),
Now if *T* is a conserved density of the Euler equations (2.1)–(2.3), then the mapping
4.5
can be shown (cf. general results in (Olver 1993)) to yield a symmetry , given by
In particular, as seen from equation (4.3), the conserved energy density equation (2.19) yields a time translation symmetry
4.6

For the other kinematic conserved densities listed in theorem 2.1, we find the following correspondences: mass density (equation (2.15)) yields a trivial symmetry
4.7
momentum densities (equation (2.16)) yield space translation symmetries
4.8
angular momentum densities (equation (2.17)) yield rotation symmetries
4.9
Galilean momentum densities (equation (2.18)) give rise to Galilean boost symmetries
4.10
Here **g** is Euclidean metric tensor on (recall, in Cartesian coordinates, the components of **g** are given by the Kronecker symbol).

In the case of a polytropic equation of state (2.14) with special exponent γ=1+2/*n*, the similarity energy (equation (2.22)) yields
4.11
which is a scaling (i.e. similarity) symmetry; and the dilational energy (equation (2.23)) yields
4.12
which we call a Galilean dilation symmetry because it preserves **x**−*t***u** and **x**/*t*. Here, *E* is polytropic energy density (2.24).

In contrast, for the helicity and generalized enstrophy densities (equations (3.7)–(3.9)), we get, 4.13 4.14 Thus, we have the following classification result.

## Proposition 4.1

*In all dimensions n* > 1, *the nontrivial infinitesimal symmetries produced from the kinematic conserved densities* (*2.15*)–(*2.23*) *under the Hamiltonian mapping* (*4.5*) *consist of space translations* (*4.8*), *rotations* (*4.9*), *Galilean boosts* (*4.10*), *and a time translation* (*4.6*) *for general equations of state, plus a similarity scaling* (*4.11*) *and a Galilean dilation* (*4.12*) *for the special polytropic equation of state* (*2.21*).

All of these Hamiltonian symmetries (4.6)–(4.14) can be seen to have the form of infinitesimal point transformations
4.15
so thus
4.16
where **η**,η are functions of *t*,**x**,**u**,ρ, while τ,**ξ** are functions only of *t*,**x**. These symmetries have the following geometrical description (proven in §5).

## Proposition 4.2

*Let ζ*(

*)*

**x***be any solution of the dilational Killing vector equation*, Ω=

*const., on*

*and let*(

**χ***)*

**x***be any irrotational solution of the same equation*, , , ∇∧

**χ**=0,

*on*.

*Then the Hamiltonian symmetries*(

*4.8*)–(

*4.12*)

*corresponding to the kinematic conserved densities*(

*2.16*)–(

*2.18*), (

*2.22*)

*and*(

*2.23*)

*have the form*

A comparison of these Hamiltonian symmetries with all of the well-known point symmetries (Ovsyannikov 1962; Ibragimov 1994–1996) admitted by the compressible Euler equations (2.1)–(2.3) in dimensions *n*>1 gives the classification results listed in tables 2 and 3.

Finally, we remark that conserved densities *T* with the property
are known as a *Hamiltonian Casimir*. They are distinguished by having no correspondence to any symmetry of the Euler equations (2.1)–(2.3). Our classification shows that the only vorticity Casimirs and kinematic Casimirs admitted by the fluid Hamiltonian (4.1) consist of the conserved densities for helicity (3.7), enstrophy (3.9), and mass (2.15).

## 5. Summary and concluding remarks

For the Euler equations (2.1)–(2.3) governing compressible isentropic fluid flow in *n*>1 dimensions, we have directly classified all non-trivial kinematic and vorticity conservation laws (2.4) by solving the determining equations (2.8) for conserved densities of the respective forms (2.13) and (3.4). Alternatively, our classification of conservation laws can be carried out by means of multipliers (Anco & Bluman 1997, 2002*a*,*b*; Bluman *et al.* 2009). This approach is most easily presented as follows using index notation (cf. §2*b*).

Let *T*(*t*,*x*^{j},ρ,*u*^{k},ρ_{,j},*u*^{k}_{,j},…) be a non-trivial conserved density and let *X*^{i}(*t*,*x*^{j},ρ,*u*^{k}, ρ_{,j},*u*^{k}_{,j},…) be a spatial flux vector, given by some functions of the time and space coordinates *t* and *x*^{i}, the fluid density ρ and fluid velocity *u*^{i}, and their spatial derivatives ρ_{,i}, *u*^{j}_{,i}, etc. with respect to *x*^{i}, which satisfy a local continuity equation
5.1
where
5.2
is the time derivative defined through the Euler equations (2.25). If we express equation (5.1) in terms of the total derivative , then we obtain an equivalent equation
5.3
holding for ρ and *u*^{i} given by *arbitrary* functions of *t* and *x*^{i}, where
5.4
are functions of *t*, *x*^{j}, ρ, *u*^{k}, ρ_{,j}, *u*^{k}_{,j}, etc., and where differs from *X*^{i} by terms that are linear homogeneous in the Euler equations (2.25) and their total spatial derivatives. This equation (5.3) is called the *characteristic form* of the conservation law equation (5.1). It establishes, first, that every non-trivial local conservation law of the Euler equations (2.25) arises from multipliers given by equation (5.4). Second, since the spatial Euler operators *E*_{ρ} and *E*_{ui} annihilate divergences *D*_{i}Θ^{i} for any vector function Θ^{i}(*t*, *x*^{j}, ρ, *u*^{k}, ρ_{,j}, *u*^{k}_{,}_{j},…), the relation (5.4) shows that any two local conservation laws differing by a trivial conserved density of the form *D*_{i}Θ^{i} have the same multipliers. Thus, there is a one-to-one correspondence between non-trivial conserved densities (modulo spatial divergences) and non-zero multipliers.

Necessary and sufficient equations for determining multipliers (Anco & Bluman 2002*b*; Bluman *et al.* 2009) are given by applying variational derivative operators δ/δρ and δ/δ*u*^{i} to the characteristic equation (5.3), yielding a linear homogeneous polynomial system in ρ_{t}, , ρ_{t,j}, , etc., whose coefficients must separately vanish. The resulting determining equations for *Q*(*t*, *x*^{j}, ρ, *u*^{k}, ρ_{,j}, *u*^{k}_{,}_{j}, …) and *Q*_{i}(*t*, *x*^{j}, ρ, *u*^{k}, ρ_{,j}, *u*^{k}_{,}_{j}, …) consist of
5.5
5.6
5.7
where and denote linearization operators (Frechet derivatives) with respect to *u*^{i} and ρ; and denote the adjoint linearization operators (Anco & Bluman 2002*b*; Bluman *et al.* 2009). We now note, first, equation (5.7) provides the necessary and sufficient conditions (Olver 1993; Bluman *et al.* 2009) for *Q* and *Q*_{i} to have the form of variational derivatives of some function with respect to ρ and *u*^{i}. (Moreover, this function can be constructed explicitly from *Q* and *Q*^{i} by means of a homotopy integral formula (Olver 1993; Anco & Bluman 2002*b*; Bluman *et al.* 2009) or by an algebraic scaling formula (Anco 2003) based on invariance of the Euler equations under dilations *t*→λ*t*, *x*^{i}→λ*x*^{i}.) Second, we note that equations (5.5) and (5.6) constitute the adjoint of the determining equations for symmetries
where
5.8
is the symmetry generator in characteristic form. Thus, multipliers can be characterized as adjoint-symmetries that have a variational form (Anco & Bluman 2002*b*; Bluman *et al.* 2009). In particular, this formulation reduces the determination of multipliers and hence of conservation laws to an adjoint version of the determination of symmetries.

We now list the multipliers for the kinematic conservation laws (2.15)–(2.23) and vorticity conservation laws (3.7)–(3.9) in tables 4 and 5.

The adjoint relation between multipliers and symmetries can be expressed in an explicit form through the Hamiltonian formulation of the Euler equations (4.1),
5.9
where is the Hamiltonian operator. As shown in §4, the mapping defined by equation (5.9) annihilates the multipliers for Hamiltonian Casimirs consisting of the conserved densities for mass, helicity and enstrophy. From the specific form of the multipliers for the remaining conserved densities—momentum, angular momentum, Galilean momentum, energy, similarity energy and dilational energy—given in the preceding two tables, we find
5.10
and
5.11
where ξ_{i}, τ, σ are shown in table 6.

It is straightforward to show that ξ_{i}, τ and σ satisfy the following system of equations:
5.12
In particular, we note
5.13
with *c*=const., where
5.14
is the equation defining dilational Killing vectors ζ_{i}(*x*^{j}) on , and
5.15
are the equations defining irrotational dilational Killing vectors χ_{i}(*x*^{j}) on . As a result, the symmetries corresponding to the multipliers for the non-Casimir conserved densities under the Hamiltonian mapping (5.10)–(5.11) have the form of geometrical point symmetries
given in terms of ξ^{i}(*t*,*x*^{j}) through equations (5.13)–(5.15).

## Acknowledgements

S.C.A. is supported by an NSERC research grant. A.D. thanks HEC, Pakistan, for providing a six-month fellowship grant and the Department of Mathematics at Brock University for additional support during the extended period of a research visit when this paper was completed.

## Footnotes

- Received February 10, 2009.
- Accepted April 9, 2009.

- © 2009 The Royal Society