CONSERVATION LAWS FOR INCOMPRESSIBLE FLUIDS

By means of a direct approach, a complete set of conservation laws for incompressible fluids is determined. The problem is solved in the material (Lagrangian) description and the results are eventually rewritten in the spatial (Eulerian} formulation. A new infinite family of conservation laws is determined, besides those for linear momentum, angular momentum, energy and helicity.

In this paper we consider the incompressible fluid model and look for the full set of independent conservation laws, without any restrictive assumption on the form of such laws.In this case no variational formulation, and thus none of the various forms of Noether's theorem, is available.That is why we adopt a direct approach and seek solutions to the vanishing of a 4-dimensional divergence.We find it convenient to solve the problem in the material description and, eventually, to rewrite the results in the spatial (Eulerian) one.
As a result of our approach, the complete set of conserved vectors is given and, as a by- product, a new infinite class of conservation laws is determined explicitly.This class extends to incompressible fluids a remarkable result discovered very recently for compressible fluids [9].
Incidentally, the procedure elaborated in this paper is likely to work for more involved situations.
To set up a general framework for conservation laws in fluid dynamics we consider a system described by n functions #, 1, 2, n, of the (space-time) variables Xv, 1,2, m.The functions b# satisfy the second order system of differential equations F(X,,,v,,:) 0, 1,2,...,n, ( where a comma followed by a letter, E say, denotes the partial derivative with respect to X:.The functions F are supposed to be of class C with respect to their argument.Let DE denote the (total) derivative with respect to Xm.A conservation law for the system (2.1) is a second order differential equation Dr Ir "-0 (2.2) for suitable functions Is of the form The symbol -" is a reminder that equality is required to hold in connection with the solutions b to (2.1) only.Operatively, determining a conservation law amounts to finding a m-tuple of functions I,. such that (2.2) is satisfied identically by the solutions to (2.1).
Trivial conservation laws arise when the Iv's vanish for all solutions to (2.1) or when (2.2) holds for all functions @# regardless of whether they solve the system (2.1) [10].Such trivial conservation laws do not provide any information about the properties of the solutions and then two conservation laws are regarded as equivalent whenever they differ by a trivial conservation law.Non-equivalent conservation laws are said to be independent.Accordingly we are only interested in the determination of independent and non-trivial conservation laws.
Let v be the velocity field of the fluid and p the pressure.Moreover let latin indices run over 1,2, 3 and denote Cartesian components.So x is the i-th component of the position vector x in the three-dimensional Euclidean space ,3.The motion of an inviscid, incompressible fluid is governed by the system of differential equations [11] c cgv 1 -" "{-v' "z y + p x, 0, (3.1) ; o, (3.) in the unknown functions v, p. Equations (3.1)-(3.2) reflect the Eulerian description of motion where v and p are viewed as C* functions of x 6 3 and time 6 .
As it often happens, here it is convenient to account also for the motion of a fluid particle, usually denoted by a function x(t).To avoid any confusion about the meaning of the symbol x (independent variable or function of time) one might adopt the Lagrangian description where the particles are labelled by the position X they occupy in a reference configuration Y" .a.In that case the independent variables are X Y" and and the unknown functions are the motion x(X, t) and the pressure p(X, t).Accordingly, adopting the Lagrangian description in the standard way [11] makes (3.1)-(3.2)ina second order differentiM system.
For the present purposes the Lagrangian description such h another drawback, besides that of increing the order of the system, because it makes less immediate to contrt our results with tho obtained by Olver [5] within the Eulerian description.That is why eventually the results determined via the material formulation will be given the corresponding Eulerian form.
For convenience in calculations we denote by upper ce indices the Cartesian components of X.Moreover, in connection with the function x(X, t) and its inverse X(x, t) we let a a ax x, X, 0, (3.4) where the condition of ms conservation p po/J, h been used with J det(z,).Really J 1 but, to simplify the comparison with the compressible ce, we prefer to write just J.
For later convenience we write now some identities that will be freely used without further reference: @J 1 4. GENERAL FORM OF THE CONSERVATION LAWS.
Back to the general framework of section 2, we identify the unknown fields # with xi, 1,2,3, and p.The independent variables are X, H 1,2,3, and t.Then we search for conservation laws (2.2) in the form Dt I(Xz t, z, p, x,t X,H + DZ< IK (X t, z, p, z,t x,H 0 .
where I is the conserved density and IK is the associated flux.As a consequence, the helicity integral [1,4,5] is ruled out, since it involves the curl of the velocity field.In view of the previous discussion on equivalence, it is also to be observed that the density is only defined up to the diver- gence of a spatial vector; this remark will lead to a simplification in the subsequent calculations.
The explicit form of" (4.1) reads where (3.3) hs been taken into account.Here p,, x,,, and x, Hr are regarded arbitrary quantities while x, tH satisfies the constraint equation (3.4).Hence  where 6, denotes as usual the Kronecker symbol.Eqs.(4.3), (4.10), and (4.11) allow the determi- nation of the explicit dependence of I on p, xt, and xig.For example, since I is independent of p in view of (4.3), it follows in particular from (4.10) that may be represented as #, v[, + &, with/ and as functions of Xz, t, x, x,t.Then, after substitution into (4.11),we can use the arbitrariness in A to set A6, 8/ax,, so as to eliminate dependence on p in the contribution proportional to Xc.Back to more general considerations, when we impose the standard integra- bility conditions on (4.3), (4.10), and (4.11) we find further restrictions on the arbitrary functions at hand so that we arrive at , p(Bzu + B,) + bzu + b, (4.12) with B, B, b, b, C, CKn, an, and an depending on XH, t, and z,; CKn --CjHK.An obvious integration provides 1 I--2po(-Bx,,x,, + B,x,, +d) + poagx, nx,, -C, KJXK,, with d-d(Xn ,t,x.), whereas the expression for In becomes Further information on I and In follows from the analysis of (4.6).
The vanishing of the overall contribution of the terms line in p and ter of second and higher- order in the components xi leads to B constant, B, w, (t)x + m, (t),.
Then the vanishing of the terms in the products x,XR provides ca H b't 0,.b b(.z,,.t.,.C,.tc Co:. Further, the vanishing of the coefficient of xt yields 0 (2p0d + 2gb) -2po 0----" Taking the c3/c3x derivative of (4.20) and requiring the symmetry in and j of the resulting expkession we find, in view of (4.18), const.
Equation (4.20) can now be solved to give 2p0d -2Jb-2pot:nx + We now observe that b and a may always be represented as b a,.,, (x, t) g (X, t) Oz then (4.21) can be given the form 2p0d D (-2,JXg 2p,zxJXg, + ) thus showing that the contribution of d to the conserved density (4.15) is represented in the form of a divergence and consequently may be disregarded, independently of the explicit form of , p, and , that is of b and a.Similarly, we have where the condition Ci Ci (t) h been taken into count, whence it follows that the contri- bution of CiK to the conserved density can be disregarded well.
a consequence, the explicit determination of the unknowns b, a, Cig, Cgu, az, and b is not required, since we are allowed to consider an equivalent conservation law where the expression of the density does not involve d and Ci; at most the preceeding unknowns can influence the expression of the flux, and no essential information is lost if we set them equal to zero.conclusion, we can write the set of conservation laws through the 4-tuple (I, Iz I 2po (Bztxt + xt.z.+ m(t)x,) + .oa(X)z,tz., 1 a being a divergencfree vector.

EXPLICIT FORM OF THE CONSERVATION LAWS.
The direct approach of the previous section has provided an exhaustive set of independent conservation laws for incompressible fluid motions.To discuss the physical interpretation of the results we observe that the arbitrariness of mh, wh, and B leads to the conservation laws for the linear momentum (in a generalized form), the angular momentum, and the energy, respectively.
Indeed, the law for linear momentum derived here, where each conserved component-say PoXht is multiplied by an arbitrary function of the time m(t), constitutes an extension of the usual formulation which corresponds to rnu being constant.As a direct calculation can show, the greater generality is due to the incompressibility constraint; this interpretation is further enforced by the observation that a similar result also holds for incompressible viscous fluids [8].
As to a, we have I poaHv, x,H, Ix a(p--pv)J. (5.1) The law (5.1)mirrors the constraint of mass conservation for the coordinate transformation in the reference configuration r, as follows from a comparison with the results obtained in [9].
Each of the above conservation laws has a counterpart holding for compressible fluids, with the only exception of the conservation of linear momentum which requires a constant rn [9].In this connection it is rather unexpected that no center-of-mass theorem has been found, although it has been shown to hold for compressible fluids [8,9] as a continuum counterpart of a relation derived by Hill [12] for a system of N particles.In the present case this result is a consequence of the conservation law of linear momentum.In fact, it suffices to set mh(t) riaht/2, with constant, and to let the remaining arbitrary functions and constants vanish.Then the expression of the conserved density is I pofnhxhtt and may be trivially modified by addition of a divergence term of the form Dn(--VnXhXhpJXni/2) --poVnhxh.The resulting expression of the conserved density turns out to be given by I hPo(X,t-Xh) thus yielding the motion of the center-of-mass [9].Now we are ready for the final step toward the formulation of conservation laws in the Eulerian description.To this aim we observe that the densities in the Eulerian and the Lagrangian descrip- .tionsare related through multiplication by the factor J, as follows from the change of variable theorem within multiple integrals [13].We also make use of the correspondence between material and spatial descriptions of vector fields, Y, y, namely [13] YIg J lth X n h and we observe that D//l/"H JDy, [13], where D denotes the total derivative with respect to xh.Then, on examining separately the various contributions to the general form of the conservation laws in the Lagrangian formulation, we are led to the following independent contributions in the Eulerian description: D,(pw,,v,x) + D(pw1,vlxv + w,px,) O, D,(pv ) + Da( pvav + pun) O, (5.4) where constant factors have not been considered and the contributions to the conserved flux without analogue in in the expression for Iu are due to the fact that we consider fixed regions in the physical space.It has already been pointed out that eq.(5.2) generalizes the usual conservation law of linear momentum, m being an arbitrary function of time.Equations (5.3) and (5.4) are the usual conservation laws of angular momentum and energy.In this connection the .)ter-,-nass theorem takes the form D,[p(x vt)] Dh[p(x vt)vh tpSh] O, and may be regarded as equivalent to the conservation law (5.2).
To find the analogue of the conservation law determined by (5.1) it is appropriate to introduce the Eulerian counterpart of Jan, say a, which is given by a, a H X,H. Accordingly, we find and hence the Eulerian form of the conservaUo law rc,ds I-pa, vi and Ig (p-v2)JatXgt (5.6) The conserved density is just the projection of the linear momentum on the field ai.The vector ai is divergence free, since au is, and satisfies the further condition aa aa a-r + -ff, o which is the Eulerian correspondent of the condition that au is independent of t.In practical terms however, it is easily seen that within the usual Eulerian framework eq.(5.7) admits only the vanishing solution.This shows that the conservation law (5.6)can only be arrived at within the Lagrangian formulation and explains why Olver [5] could not find it.
Substitution for I and In