Conservation laws for the Navier-Stokes equations

1111.J Engag Sci Vol. 24, No. 8, pp. 1295-1302, Printed in Great Britain

0020-7225/86 Pergamon

1986

CONSERVATION NAVIER-STOKES

$3.00 + .oO Journals Ltd

LAWS FOR THE EQUATIONS

GIACOMO CAVIGLIA Istituto Matematico deIl’Universit& Via L. B. Alberti 4, 16 132 Genova. Italy (Communicated by E. S. Suhubi) Abstract-A variety of conservation laws for the Navier-Stokes equations are derived. The conserved currents are described in terms of Lie symmetries and of adjoint variables, which give rise to a complementary variational principle for the Navier-Stokes equations. I. INTRODUCTION

is devoted to an investigation of conservation laws for the Navier-Stokes equations. It is based on a systematic extension to the framework of viscous fluid dynamics of a method which has already been successfully applied to find first integrals of motion for mechanical systems in a finite number of degrees of freedom [ 1, 21. Specifically, the aim of this work is threefold. The primary aim is to construct a class of generators of conserved quantities. The second aim is to describe new general procedures aiming at the generation of conserved currents. A third, minor, aim is to investigate the existence of a so-called composite variational principle [3] leading to the Navier-Stokes equations: in particular we discuss the meaning of the additional variables and the related partial differential equations, which are introduced to ensure the existence of a potential for the Navier-Stokes equations and turn out to be strictly connected with conservation laws. The mathematical foundations of the present approach are briefly reviewed in Section 2, where the equations of variation for a given system of partial differential equations are introduced [4, 51 and it is shown that every infinitesimal symmetry transformation of the given equations identifies a solution to the equations of variation. Our basic result is that any solution to the adjoint system of the equations of variation gives rise to conserved currents; in addition, higher order conservation laws are established through a suitable process of derivation depending on the knowledge of symmetry transformations (Section 3). For these reasons, after a brief review of some basic features of the Navier-Stokes equations and their invariance transformations (Section 4), we determine the explicit form of the general solution to the adjoint of the equations of variation (Section 5). The resulting conservation laws are then analyzed in some detail (Section 6), whereas some concluding remarks can be found in the last section. THIS PAPER

2. MATHEMATICAL

PRELIMINARIES

Consider the following system of partial differential equations, namely

(2.1) where Greek (Latin) indices run from 1 to n(m); I$: = &p’/aY and &a = d2@/dx‘9xa. An infinitesimal transformation of the form X-OL=

xa + dyx”,

g = 4’ +

@b)

@(x0, f#lb)

(2.2) (2.3)

is called a symmetry transformation (Lie symmetry) iff it leaves (2.1) invariant up to terms of order e*, that is iff any solution to (2.1) is mapped into a solution of the same equation, 1295

I296

G. CAVIGLIA

written in terms of the variables X, 3 [5-S]; notice that, e.g., X refers to the set X’, . . . , 2”. It is found that the transformation (2.2) (2.3) is a Lie transformation iff the condition

holds on every solution to (2.1) [5]: here D, denotes the total derivative with respect to xn and DolB= Dm*Ds. Thus the generators T and [ of a Lie transformation are determined as solutions to (2.4) [5, 61. Conversely, any given Lie transformation may be used to give rise to particular solutions to (2.1) [6-lo]. We shall now rewrite (2.4) in an equivalent form which is particularly well suited for the subsequent investigation. Namely, partial differentiation of (2.1) with respect to A? yields

(2.5) Substituting into (2.4) the expression for dF,/axn which is obtained from (2.5) we find that (2.4) may be written as [S]

where we have set (2.7)

qc= 5”-

(2.8)

F#Z.

Equations (2.6) are the so-called equations of variation for (2. l), and t is referred to as a variation of (2.1) [4, 51. We have thus shown that to each Lie symmetry there corresponds a variation of (2.1). These considerations will provide an adequate basis for the study of conservation laws which is developed in the subsequent section. 3. ADJOINT

SYSTEM

OF

THE

EQUATIONS

OF

VARIATION

Let us denote by %.&(;i)the adjoint system of (2.6), which is given by Ili,(ij) = Ta,

- D,&jcb:o) + D,a(;i’c;f) = 0.

(3.1)

If A&(v) = M,(q) for all admissible variations then the system (2.1) is said to be self-adjoint. In that case equations (2.1) admit a variational formulation; i.e. they describe the stationary points of a suitable action functional [ 3,4, 111. Notice that the Navier-Stokes equations are not self-adjoint [3]. We shall now describe a method of deriving conservation laws in terms of the functions q, ij, and their derivatives [5]. To this aim we need the Lagrange identity, which reads G“Mz(o) - #&z(ii) = DJ”,

(3.2)

with

and is obtained as an immediate consequence of the definition of M,(q) and A?&). Then we consider a set of functions ;i’ = iiyx@, @) such that ;t’(xO, I*) yields a solution to (3.1) in correspondence with every solution (pb(xa)to the field equations (2.1). Likewise we

Conservation laws for the Navier-Stokes equations

1291

assume that a set $(xB, 4b) satisfying the equations of variation (2.6) on every solution to (2.1) is given; for example, 7 is generated by a Lie transformation through (2.8). Then it follows from (3.2) that the corresponding field J identifies a conservation law. Additional conserved currents determined by ;i may be constructed as follows. Multiply (3.1) by 4:: and rewrite it as ;i’u&:

+ D,[Ds(;icc:f)

- Tb:&$:: = 0.

(3.4)

An equivalent formulation of (3.4) is given by D,K:+VDcF&~=O

(3.5)

with

as it is easily seen by substitution of (3.6) into (3.5) and explicit evaluation of the total derivatives. Now observe that D,F, = 0 on every solution to (2.1); as a consequence it follows that whenever F, does not depend explicitly on xb, the corresponding vector K, yields a conserved current in connection with every solution to (3.1). A conservation law can be constructed-at least in principle-even though dF,ldx” does not vanish, provided one is able to find a solution X:(x6, c#J~, c#$)to Gj’dF,ldx” + 0,X: = 0. In this case it follows from (3.5) that the conserved current is given by K,, + A,. This algorithm to find conservation laws is the natural extension to field theories of the procedure already examined in [2] for mechanical systems having a finite number of degrees of freedom. In that context, it was shown that n identifies a differential form of the evolution space, which is invariant along the trajectories of the given dynamical system. Therefore, the existence of a first integral related to the invariance properties of a suitable geometric object was, in a sense, very natural. As a final comment on the meaning of (3.1) it is to be pointed out that if one considers the action functional J(W’),

ii’(

= s, F,ijc dK

(3.7)

where V is any suitable domain, the corresponding Euler-Lagrange equations are given by (2.1) and (3.1), where of course the definitions (2.7) are also taken into account. Action functionals of the type (3.7) have been referred to as composite variational principles [3], because they yield a variational formulation for any system of the form (2.1), but at the same time they require the introduction of a suitable set of auxiliary variables-namely ;i in the present case-in addition to the original variables. Thus we conclude that the T’s may be regarded as auxiliary variables for the construction of a composite variational principle. On the other hand, the present analysis brings also into evidence the fact that the auxiliary variables are “adjoint variables,” in the sense that they are determined by (3.1) which is the adjoint equation to (2.6) which in turn is a restatement of the definition of Lie symmetry. This observation clarifies a somewhat misleading interpretation that was discussed in [3]. Notice also that the variational principle (3.7) does provide an efficient algorithm leading immediately to the construction of the system (3.1). Of course, it is understood that in our search for conservation laws we need solutions to (3.1) that are constructed by restriction of suitable functions T(xa, c#J~) to solutions of (2.1). In the subsequent sections we shall describe a general procedure which leads to the determination of these adjoint variables. A third algorithm that leads to the construction of conservation laws and does not rely on (3.1) will now be described. The essential idea has already been examined in [5], in a completely different framework, where it was shown that the Lie derivative of any given conservation law along the direction of an isovector identifies a new conservation law. The

1298

G. CAVIGLIA

procedure which is described here emphasizes the fact that one may obtain, at least in principle, an infinity of higher order conservation laws. Consider the following derivation operators:

(3.8)

where DA yields the total derivative with respect to x” of any function depending on (xa, @, $:, d:,), whereas Y identifies an higher order extension of the Lie symmetry (2.2), (2.3) in the sense that, to first order in t, the a/&$: and a/&#$$ components of Y yield respectively the variations of $‘, and I#& induced by (2.2), (2.3). Iff(x”, $“, 4:) is any regular function, then it may be verified by a straightfonvard calculation that

VW, - QDx)f

= 0,

(3.10)

( YDA - DxY)f = -DxFD,f: Using (3.10) it is also easily seen that in correspondence @, 4:) the following identity holds: D,(X”D,#

(3.11)

with any set of functions XX(xa,

- ,%?D,T’) = DAXxD,r8 - DAr”Dz”.

(3.12)

Then, on comparing (3.12) with (3.1 l), where f is to be replaced by X’, it is found that + D,XxD8~ lo= DJ Y(X’) + X’D,T” - XsD,gx].

Y(D,X’)

(3.13)

Observe now that the left hand side of (3.13) vanishes on every solution to (2.1) under the assumption that Xx is a conserved current, because DxXx = 0 and Y is the prolongation of a Lie symmetry. Accordingly, it follows that L” = Y(Xa) + X”D,r@ - XBD,r”

(3.14)

yields a conserved current. Of course, the previous procedure can be extended to the case when X is allowed to depend on higher derivatives of 4, provided we consider corresponding higher order derivatives in (3.8) and (3.9). 4.

THE

NAVIER-STOKES

EQUATIONS

On the basis of the previous considerations we plan to deal with the problem of generating conserved currents for the Navier-Stokes equations. Thus we consider an incompressible viscous fluid whose velocity field v = uel + ve2 + we3 and pressure a satisfy vuy + wuz + px - v(uxr + uyy + u,;) = 0

(4.1)

v, + uv, + vvy + WV, + py - Y(V, + v, + v,,) = 0

(4.2)

w, + uw, + “WY+ ww, + p* - Y(Wxx+ wJy+ w,,) = 0

(4.3)

u, + v, + WZ= 0

(4.4)

ut + uu, +

where v is the (constant) kinematic viscosity and el , e2, e3 are the unit vectors of the given Cartesian axes, The system (4.1)-(4.4) can be written in the form (2.1) by setting (x, y, z, t) Z (x’, x2, x3, x4) and (u, v, w, p) = (I$‘, 42, 43, $4), whereas F,, F2, F3, and F4 are identified

I299

Conservation laws for the Navier-Stokes equations

with left hand side of (4.1)-(4.4), respectively. In particular, it turns out that c$ = 0 for any allowable (Y,,8, and c, because (4.4) does not depend on second order derivatives. The full group of Lie transformations for the Navier-Stokes equations has already been determined [7, 121. Here, we reproduce the final form of the generators T and [, which is given by (q

=

(a2x

-

(M = (-ff2zi -

a3y - a4.2 +f;

a321 -

a4w

+A,

a2y +

a3.x -

-U2Y

-a2w

+

+

a52 + g, a22 +

a324 -

a4ahu+

a5w

f.z5v+

a&x -I-a5y +

h, a1 +

2a2t)

(4.4)

.A)

(4.7)

f g,,

hl, -2azp

+j

- xf;r - ygfc -

where a,, u2, a3, u4, a5 are five arbitrary parameters andJ; g, h, j are arbitrary functions of t. A detailed analysis of the transformations corresponding to each arbitrary parameter or function in (4.6) and (4.7) can be found in [7, 121. Therefore, we only notice for future reference that substitution of (4.6) and (4.7) into (3.8) yields the explicit expression of the variations n.

5. ADJOINT

VARIABLES

FOR

THE

NAVIER-STOKES

EQUATIONS

We now proceed to the determination of the form of the adjoint variables, which are needed in order to construct conserved currents. As a preliminary step we write down the adjoint system (3.1) corresponding to the Navier-Stokes equations. Using the techniques described in section 3 it may be shown that the required adjoint system reads:

v.jj*-

w,;i3+ DxG4+ Y(D$jk + Dyyij’ + D,,ij’) = 0,

(5.1)

D,ij2 - uy;i’ - u$j2 - wy;i’ + Dyij4+ P(Dsij2+ Dwij2+ Dzz;i2)= 0,

(5.2)

D*ij3- u*Fj’- v,jj2 - wzij3+ D,;i4 + v(D,Tj3+ Dyyij3+ Dzzij3)= 0,

(5.3)

22,;i’ - u$j’ -

D,;i' + D,,;i2 + D,ij3 = 0,

(5.4)

where the operator .Q is defined as a>, = Dl + uD, + vD,, + wD,. According to the previous discussion we must find a set of functions ii’, ii*, ii3, ;i4depending on (x, 4: z, t, u, v, w, p) such that (5.1)-(5.4) are identically fulfilled on each solution (u, v, w, p) to the Navier-Stokes equations. We shall now give an outline of the procedure leading to the determination of the most general set of functions Gjsatisfying the required conditions, although many of the details of this calculation are omitted, owing to their length. Consider first the ‘divergence’ condition (5.4). By writing down the explicit expression of the derivation operators involved, we obtain a linear form of the derivatives of u, v, w, and p that may be represented as

(ii + ;i: + ?I>+

ijfu,+ i&4, + ;i$& + ijtv,+ ijfv,+ ij;lv, + ;ilw, + ;i’,w, + ;iEw, + jjfrpx+ +;pY+

ij;p= = 0.

(5.5)

If (5.5) is to hold on any solution to the Navier-Stokes equations, then we must have

In particular, notice that $fi, $2, and ;ii do not necessarily vanish since u,, v,, and w, are related by the incompressibility condition (4.4). However, the requirement that (5.7) holds for any allowable choice of u,, u, and w, leads to

1300

G. CAVIGLIA ijt, = ijf = $’

= c(x,

y, z, t),

(5.9)

where (5.8) have also been taken into account. In view of (5.8), it follows in particular by integration of (5.9) that

ij’=cu+b

(5.10)

with b = b(x, y, z, t). After insertion of (5.10) when written out in full condition (5.1) takes the following form: cu, +

UC,

+

6, + u( UC,

+

cu, + b,) + v( ucy + cuy + by) + w( UC, + cu, + bz)

- (cu + b)ux - jj’v, - fi3wx+ ;i: +

fi:u,+ ;i:u,+ ij”ww,+ i&x

+ v[b, + byy+ b,,

+ 2(C,U, + C,U, + c,u,) + C(U,, + U,, + U,,) + u(cxx + c,, + c,,)] = 0.

(5.11)

Therefore, substitution into (5.11) of the expression for V(U, + u,, + u,,) which is obtained from (4.1) shows that the coefficient of uI reduces to 2c; since it must vanish, we conclude that c is to vanish as well. Thus (5.8), (5.9) and (5.11) imply that ii’, ij2, and ;i3 depend only on the variables x, y, z, and t. Accordingly, equation (5.11) simplifies to ij: +

uij;+ I$;

+

w;i;+ ;i:+ (ij”, -

;i’)u, + (6%- ;i2)vx+ (;i”w- ;i3)wx+ i$px + VA+ = 0; (5.12)

the similar expressions originating from (5.2) and (5.3) are not reproduced. By the usual procedure it follows that we have to find the general integral of the system consisting of (5.6) and ;I: =

;i”,= ;I’;

62;

;l”w

=

53;

ij;

=

0;

(5.13)

;i; + u$ + t$ + wGj;+ $ + uA;i’ = 0;

(5.14)

;i: + z&f + vij; + w;i,’ + ;i; + vAG2= 0;

(5.15)

ij: + u$ + u;i; + w;1; + ;i; + VALE= 0.

(5.16)

Equation (5.13) can be integrated to the form

ij4 = $24+ ;j2v+ ;i’w + a(x, y,

z, t)

(5.17)

where (Yis an arbitrary function. Substitution of (5.17) into (5.14)-(5.16), in view of the fact that ii’, G2,ij3,and cxcannot depend on U, V,and w, leads to the compatibility conditions

It is straightforward consequence of conditions (5.18) that ii’, G2,and ;i3 are given by ;i2 = -ax

ij*=ay+bz+k;

+ cz + m;

i-j3= -bx - cy + n;

(5.19)

where a, b, c, k, m, n, are arbitrary functions oft. The form of CY has not yet been determined; in order to find it, let us introduce (5.19) and (5.17) into (5.14)-(5.16). We get ax = -(a,~ + b,z + k,); ffz = -(-b,x

-

cly

+

n,).

ay = -(-a[~

+ c,z + m,)

(5.20)

The requirement of compatibility between (5.20) yields a, = bt = c, = 0; hence we arrive at the final conclusion

Conservation

laws for the Navier-Stokes $

=

1301

equations

(5.2 1)

b,y + bzz + k,

;i2 = -b,x

+ b3z + m,

(5.22)

;i3 = -b2x - bfy + n,

(5.23)

ij4 = (b,y + b2z + k)u + (-blx + b3z + m)u + (-b2x where b, , b2, b3 are arbitrary time t. 6.

CONSERVED

parameters;

CUKRENTS

FOR

- b3y + n)w - (xk, + ymt + zn,) + o,

(5.24)

k, m, n, and o are arbitrary

of the

THE

NAVIER-STOKES

functions

EQUATIONS

We are now in a position to write down conserved currents for the Navier-Stokes equations. To begin with, let us remark that the equations of motion for a viscous incompressible fluid do not depend explicitly on the space and time coordinates. Accordingly, if we insert (5.21)-(5.24) into (3.6) we get four conserved vector fields depending on second order derivatives of v, whose expression is given by

K, = -(ij - v,)v - ij4v, - p,;i + v(Vv, - 9) -

4(&b + ;i~wo)el + C;i:us + G:wb)e2 + (fib, + ?~v,hl, K4,= -jj*v,,

(6.1) (6.2)

where u goes from 1 to 4; it has been set K, = Kke, + K?e2 + K?ej and i = ij’ei + ;i2e2 + ij3e3; finally, Vv, - ij is given by (Vv, * ij) - ei = (&,/dx’) - ij. It is to be pointed out that the conserved currents identified by (6.1) and (6.2) depend on three arbitrary parameters and four arbitrary functions of time. The expression of the conserved current J can be determined by comparison of (2.8), (3.3) (4.6), (4.7) (5.21)-(5.24). A particular example has been exhibited in [5]. It is clear by inspection that J depends on eight arbitrary parameters and eight arbitrary functions of time. Furthermore, it is worth noting that J, as well as K, involves second order derivatives of v. A natural question then arises as to the possibility of finding conserved currents K depending on higher order derivatives of v. A possible answer consists in looking for adjoint variables of the form ;i” = ?(x”, @, I$:). However, it may be shown through rather long calculations-that are not reported here-that dependence on the derivatives is not admitted. In a sense, this result is somewhat unexpected, since 4 does depend on the derivatives of the field functions, as it is implied by (2.8) (4.6) and (4.7). Actually, it seems that this strange result is the analogue of the already known property that the set of generators of infinitesimal invariance transformations, that is r and E, is not enlarged if dependence on the derivatives is allowed [ 121. To complete the analysis of this point it is to be observed that the construction of a huge variety of conservation laws can be achieved by means of (3.14) where X is any conserved current, for example it has been generated according to the previously described procedures. 7.

COMMENTS

AND

CONCLUSIONS

In this work we have explicitly determined a wide set of conservation laws for the NavierStokes equations, depending on derivatives of the velocity field of the second and higher orders. Our approach has been based on the extension to field theories of specific procedures that have been recently developed within the framework of mechanical systems in a finite number of degrees of freedom, in order to find first integrals of motion essentially derived from the knowledge of the so-called Jacobi fields [ 1, 2, 5, 131. The paper has been organized so as to be relatively self-contained. Accordingly, Section 2 and the first part of Section 3 have been devoted to a brief resume of some relevant properties of second order systems of partial differential equations, that have been extensively

1302

G. CAVIGLIA

discussed in [5]; for example, no mention has been made of the connections between generators of symmetry transformations and isovectors of the ideal corresponding to the given system [5]. On the contrary, we have commented on the new insight into the relationships between adjoint variables and composite variational principles that can be achieved in terms of the present approach. Similarly, we have emphasized the existence of new conservation laws canonically associated with any set of adjoint variables ;i and the algorithm which leads to the generation of higher order conservation laws. It is to be pointed out explicitly that the present construction may be extended systematically to any system of nonlinear field equations. In practice, one has to determine 9 and ;i as solutions to suitable systems of linear partial differential equations generated by the given field equations; then the conservation laws are easily found. From this point of view, our analysis of the Navier-Stokes equations is simply a specific application of the method. However, the techniques that have been used to find the adjoint variables are quite general and can serve as a guide to deal with more involved situations. It is also to be observed that the generators of Lie transformations, and consequently 4, have already been calculated for a number of significant equations, on account of their well known role of generators of exact solutions [7-lo]. Future applications of this procedure to the analysis of the three-dimensional Euler equations are already planned. In particular it is worthwhile to compare the present approach to conservation laws with the results recently obtained by Suhubi [ 141 for isoentropic onedimensional gas flows, to the ultimate aim of clarifying the connections between completely integrable systems and conservation laws. The possible role of boundary conditions and of external volume forces in the present framework is also to be investigated. Acknowledgments-This work has been performed under the auspices of GNFM-CNR through the MPI 40% research project “Problemi di evoluzione nei fluidi e nei solidi”.

and partially

supported

REFERENCES Inverse Problems 1, L13 (1985). G. CAVIGLIA, Internat. J. Theoret. Phys. 25, 147 (1986). R. W. ATHERTON and G. H. HOMSY, Stud. Appl. Math. 54, 31 (1975). R. M. SANTILLI, Ann. Phys. (N. Y.) 103, 357 (1977). G. CAVIGLIA, J. Math. Phys., In press. W. F. AMES, Nonlinear partial d@erential equations in engineering Vol. II, Academic Press, New York (1972). R. E. BOISVERT, W. F. AMES and U. N. SRIVASTAVA, J. Engng. Math. 17,203 (1983). M. LAKSHMANAN and P. KALIAPPAN, J. Math. Phys. 24, 795 (1983). D. FUSCO, A. DONATO and W. F. AMES, Wave motion 6, 517 (1984). 0. P. BHUTAN1 and P. MITAL, Int. J. Engng. Sci. 21, 555 (1983). F. BAMPI and A. MORRO, J. Math. Phys. 23, 2312 (1982). S. P. LLOYD, Acta Mech. 38, 85 (198 I). G. CAVIGLIA, J. Math. Phys. 24,2065 (1983). E. S. SUHUBI, Int. J. Engng. Sci. 22, 119 (1984).

[I] G. CAVIGLIA, [2] [3] [4] [5] [6] [7] [8] [9] lo] 1 I] 121 131 141

(Received 29 July 1985)