Group analysis of three-dimensional equations of an ideal fluid in terms of trajectories and Weber potential Текст научной статьи по специальности «Математика»

УДК 532.516

Group Analysis of Three-dimensional Equations of an Ideal Fluid in Terms of Trajectories and Weber Potential

Daria A. Krasnova*

Institute of Computational Modelling, Siberian Branch of the Russian Academy of Sciences, Akademgorodok, 50/44, Krasnoyarsk, 660036

Russia

Received 06.08.2013, received in revised form 16.09.2013, accepted 20.11.2013 Lie group analysis of equations of an ideal fluid written in variables of trajectories and Weber's potential was conducted. It was shown that the use of volume conserving arbitrary Lagrangian coordinates is in fact an equivalent transformation for the equations. The defining Lie algebra equations for the initial velocity distribution were obtained. The basic Lie group and its extensions were found.

Keywords: equations of an ideal fluid, equivalent transformation, Lagrangian coordinates, defining equations.

Introduction

In describing the motion of an ideal incompressible fluid with a free boundary one needs to find the solution of the Euler equations subject to kinematic and dynamic conditions at the free boundary. The kinematic condition allows us to transform the initial problem to the problem with the fixed domain. This is achieved with the use of the Lagrangian coordinates £ = (£, n, Z) which are the coordinates of fluid particles at the initial point in time t = 0: x = The particle coordinates x = x(£, t) are defined by the equation dx/dt = u(x,t). A system of equations of the following type is considered [1]

xt = (ynzz - znyz- u0) + zz + ziyzX^n - v0) + (y^- ziyn)(^C - wo), (0-l) yt = (-Xnzz + Znxz)(y - uo) + zz - z5xz)(y>n - vo) + (-x^zv + z?xv)(yz - wo), (0.2) Zt = (xnyz - ynxz)(<£>£ - uo) + (-x5yz + y5xz)(y>n - vo) + (x5yn - y^xn)(yz - wo), (0.3)

xe(ynzz - yzzn) + xn(-y«zz + yzz«) + xz(y«zn - ynz«) = 1 (°.4)

where (x, y, z) = x(£, t) are the fluid particles coordinates, y((£, t)) is the required function that arises from the transformation of the equations of motion and uo(£, n, Z), vo(£, n, Z), wo(£, n, Z) are the components of the particle velocity vector at t = 0. The transformation of equations of motion with respect to variables x and y was first discovered by G. Weber [2]. Equation (0.4) describes the volume conservation, detM = 1, where M = d(x)/d(£) is the Jacobi matrix.

To study the group properties of equations (0.1)-(0.4) the following index designations are introduced

x1 = x2 = n, x3 = Z, x4 = t,

12 3 4

u = x, u = y, u = z, u = y,

uk = uk(£, n, Z, t), k = 17^.

* krasnova-d@mail.ru © Siberian Federal University. All rights reserved

Let us rewrite the system of equations using the index designations = («2^3 — M2Mi) (Mi + Mo) + (—u2«3 + M1M3) («2 + vo) + (m2m2 — (M3 + wo) , (0.5)

«3 = (—u1«3 + w3«1) (u3 + «o) + («i«3 — u3«1) («3 + vo) + (—uj«3 + «3«2) («3 + wo) , (0.6)

«3 = (u2«3 — u2«3) («i + «o) + (—u2«3 + u2«3) («3 + vo) + (u];«2 — u2«2) («3 + wo) , (0.7) «2 («2«3 — u3«3) + «2 (—u2«3 + u3«3) + «3 (u2«3 — u3«2) —1 = 0. (0.8)

Here uo, vo, wo are the functions of the initial velocity distribution at t = 0. They depend on (x2, x2, x3) and are found to be the defining functions for the given system.

It can be shown that transition to arbitrary Lagrangian coordinates (a, P, 7) = a(£) which conserves the volume (detJ = 1, where J = d(a, P, y)/9(£, n, Z) is the Jacobi matrix) is the equivalent transformation for equations (0.1)-(0.4). The structure of equations (0.1)-(0.4) is not changed after such transformation. The components of the initial velocity vector are changed and they are described by the following formulas

Uo = (Pn 7C — Pz Yn )uo — (P« Yc — 7« Pz )vo + (P« Yn — Pn 7« )wo> Vo = (Yn ac — Yz an )u° — (Y« aC — a« YC )vo + (Y« an — Yna«

Wo = (an Pz — az Pn )«o — (a« Pz — P« az )v° + (a« Pn — an P« )w°,

where («o, v°, w°) = uo (£(a, P, y)) and divu0 = 0.

1. Formulation of the problem and group analysis of equations

It is necessary to find the kernel of basic Lie algebra of the transformation of system (0.1)-(0.4) and all specifications of the elements uo, vo, wo that give us an extension of the Lie algebra [3]. We define the infinitesimal operator for system (0.5)-(0.8) in the following way

x = e— + nfc—,

dx® duk

i =1, 4, k = 1, 4. From this point on we assume summation for all repeating indices. We assume that elements , £2, £3) depend on (x2, x2, x3) and depends only on x4. We also assume that (n2, n2, n3, n4) depend on (£2, £2, £3, £4, u2, u2, u3, u4).

System of equations (0.5)-(0.8) has first order derivatives. To construct the determining equations it is necessary to extend the operator X to the first order derivatives

X = X + Z k Z k = +un dn! — (dej+un aj A

X = X + duk , = dx® + du" «H dx® + ® du^ ,

n =1, 4, j = 1, 4.

Let us use the criterion of invariance [3] when the action of the operator X on equations

(0.5)-(0.8) gives us zero. It means transition to a manifold of equations (0.5)-(0.8). By expressing elements «4, «4 and «3 from equations (0.5)-(0.7) in terms of the remaining variables we determine the manifold M. We express element ^ from equation (0.8) in the following way

«2 = (u2«3— «I«!) 2+«2 (u2u3 — u3«3) («I«3 — u3u3) 2 — «3 (u2u3 — «3«!) («2«3 — «I«!) 2.

Let us consider each of the equations.

The result of action of the operator on equation (0.8) gives

dn1 d«4 dn2 = d«4 dn3 = d«4 = 0, dn1 dx1 dn1 dx2 dn1 dx3

dn2 dx1 dn2 dx2 dn2 dx3 = 0, dn3 dx1 dn3 dx2 dn3 dx3

X («1 («2«3 - «3«2) + «1 (-«2«3 + «3«3) + «1 («2«2 - «3«2) -1) = 0.

The extended version of the previous expression is

/=1 ( 2 3 3 2) , 1 /V2 3 i 2A3 a3 2 3/=2) A1 ( 2 3 3 2) Cl l«2«3 - «2«3j + «1 )Z2 «3 + «2C3 - C2 «3 - «2C3J - C2 l«1«3 - «1«3 J -

-«2 (Ci2«3+«2d - d«3 - «3c3) + z3 («2«i - «1«2) + «3 (Ci2«3+«1d - C22«3 - «2c3) = 0.

Let us switch to manifold M which is defined by equations (0.1)-(0.5). In what follows we split the resulting equation with respect to the independent variables «j, i, j = 1, 4 (we exclude variables «4, «4, «3, «1 ). The results of the splitting are presented below.

We notice that the derivatives of coordinates of the operator X are equal to zero for several variables, namely,

1 2 3 1 1 1

0, (1.1) 0. (1.2)

dx ux~

The following equation for the coordinates is valid

V + + dn3 - di! - dH - = 0 (13)

d«1 d«2 d«3 dx1 dx2 dx3

Let us analyze equation (0.5). The result of action of the extended operator on this equation with respect to the coordinates used in equations (1.1)-(1.2) is

x ((«2«3 - «3«i) («1 + «0) + (-«2«3+«3«3) («2 + vo) + («2«3 - «1«2) («4 + wo) - «4) = 0.

The extended version of the previous expression is

-c!+(C22«3+«2d -d«3 - «3d) («1+«0) + («2«3 - «3«2) (c4+i1 dxo+i2S+i3S)+ + (-d«3-«1d + z3«3+«3d) («4+vo) + (-«1«3+«1«3) (d+i1 dX1+i2 dX2+i3 S)+ + (c?«2 + «1zf - z3«2 - «1d)(«3 + wo) + («2«3 - «3«2) (d+i1 +i2+i3S) =0.

We obtain equation in which the transition to manifold M is realized. The resulting system of equations is split with respect to independent variables shown above. The following conclusions are made as a result of the splitting of the equation. First of all, we found out that

M = ^ =0 (14)

d«1 dx4 (1.4)

Second, we obtained the following type of equations

+ + dn2 - - - - + = 0 (15)

d«4 d«3 d«2 d«1 dx1 dx2 dx3 dx4 '

/V + =0 w^^n! + ^ =,

V0Ur + a^J =0' Md^ + a^J =0' (1.6)

' Brf Bn1\ (d-rf Bn1\

vo[ ^ + dU3) = 0 + a^) =0-

(1.7)

From equations (1.6)-(1.7) four cases follow: 1) vo = 0, wo = 0; 2) vo = 0, wo = 0; 3) vo = 0, wo = 0; 4) vo =0; wo = 0. The first three cases result in the following equations

dn2 dn1 dn3 dn1

—-—|--— = 0, —-—|--— = 0.

du1 du2 ' du1 du3

(1.8)

The last case is not taken into account.

Third, we obtained equation that contains components of the velocity vector (uo, vo, wo) in explicit form:

, du

duo

dn4 + £1 duo + + +

dx1 dx1 dx2 dx3

+uo (- + - dnL + ^ ^ + vo dt+wo = 0

\ du2 dx2 du3 dx3 du1 dx4 J dx1 dx1 '

dn4 + £1 d^o + £2 d^o+£3 d^o+

dx2 dx1 dx2 dx3

+uo +vo (- + - - dnL + ^ ^ + wo = 0

dx2 \ du2 dx1 du3 dx3 du1 dx4 J dx2 '

(1.9)

(1.10)

3w,

dwo

, dwo

dn4 + + £2'^fo + £3'^fo +

dx3 dx1 dx2 dx3

+uo +vo +wo( - + - - dnt + =0

dx3 dx3 I du2 dx1 du3 dx2 du1 dx4 J

(1.11)

Similarly to the previous case, the result of action of the extended operator on equation (0.6) is

-—4+(-$4 - u2d+^2^3+4ci)(ui+uo)+(-ulu3 + u32u1) (tf + £1+ £2+£3+ + (CM + u1$ - C3ui - u3d) (u4 + vo) + (u1u3 - ufui) + £1 + £2+£30) + + (-Hu3-u1c3 + Z3u\ + ufdM^+wo) + (-u1u3 +u3u1) (d +£1 dxo + £2dB +£3S) =0.

As a result of the splitting with respect to independent variables we obtained equation

dn4 dn2

du2 dx4

0

and the following equations

dnf dnf dn^ dn^ d^2 d^

du4 du3 du2 du1 dx1 dx2 dx3 dx4 '

dn3 dn2 dn4 dn2

—-—|--— = 0, —— +--— = 0.

du2 du3 du2 dx4

(1.12)

(1.13)

(1.14)

The last equations were obtained after considering the mentioned above cases: 1) vo = 0, wo = 0;

2) vo = 0, wo = 0; 3) vo = 0, wo = 0; 4) vo = 0; wo = 0.

We have also obtained the following equations

du0

. dw0

• dun

JUL. I + £2^. + £3^i +

Q„1 1 > Q„1 1 > Q„2 1 > Q„3 1

dx1

dx1

dx2

dx3

dn2

, d£2 dn3 d£3 <V <9£4\ d£2 d£3

du2 dx2 du3 dx3 du1 dx4/ 0 dx1 0 dx1

dn4 +£1 d^Q +£2 +£3 dvo + dx2 dx1 dx2 dx3

d£1 ( dn2 d£1 dn3 <9£3 dn1 ö£4 \ d£3

--1- Vn I--—--1--—--1---1--I + Wn-

(1.15)

+uo^T + vo dx2

+

+

+ wq

dx2

du2 dx1 du3 dx3 du1 dx4/

dn4 + £1 + £2 + £3 + dx3 dx1 dx2 dx3

+uo d£1+V0 + W0 (_ — d£1 + — + + ^ _0 dx3 dx3 V du2 dx1 du3 dx2 du1 dx4 /

(1.16)

(1.17)

Let us note that equations (1.15)-(1.17) differ from equations (1.9)-(1.11) by the terms in parentheses.

Finally, the result of action of the extended operator X on equation (0.7) is

-cf+(c1«3+«2d -c22«3 - u2c3) («4+«0) + («2«3 - «2«3) (ci+e1dXn+e2S+e3S)+ + (-c1 «2 - «id + c2«1 + «?&) («2+vn) + (-«!«2+«2«3) (c24 + e1 ^ + e2J£0r + e3+ + (ci«2+«1c2 -c2«2-«2c1) («3+wn)+(«1«2-«?«*) (c4+e1 dw+e2+e3= °-

As in previous cases, the splitting of the equation with respect to independent variables leads to the following result

dn4 dn3 d«3 dx4

0,

(1.18)

dn3 ön2 ön1 d£1 d£2 d£3 d£4 _ 0 du4 du3 du2 du1 dx1 dx2 dx3 dx4 '

dn4

dn3 dn2 du2 du3

dn4 dn3

0, —!—|--—

du3 dx4

0,

(1.19)

(1.20)

given that 1) v0 = 0, w0 = 0; 2) v0 = 0, w0 = 0; 3) v0 = 0, w0 = 0; 4) v0 = 0; w0 = 0.

Besides, there are equations which contain the coordinates of the operator and the components of initial velocity:

dn4 + ei +e2 +ei + dx1 dx1 dx2 dx3

+uo f _ _ dn! _ + + + V0 + W0 _ 0

\ du2 dx2 du3 dx3 du1 dx4/ dx1 dx1 '

dn4 + £1 dvo+£2 dvo+£3 dvo+

dx2 dx1 dx2 dx3

+uo d£1 + vo (_ d£1 _ _ + + ^ + Wo _ 0 dx2 l du2 dx1 du3 dx3 du1 dx^ dx2 '

(1.21)

(1.22)

0

^ + ^ dx3

dwo dx1

d^1 fdn2

+uo -5-3 + vo^-^ + wo -w-j dx3 dx3 V du2

+ e' del

dx1

2 dwo o < „

2 o + e3^3-+

dx2

i dwo dx3

dn3 du3

de2 <V se4

dx2 du1 dx4

0.

(1.23)

As in the previous cases, let us note that equations (1.21)-(1.23) differ from equations (1.15)— (1.17) by the terms in parentheses.

Therefore, the determining equations for the coordinates of the operator X that are admitted by system (0.5)—(0.8) are

dn1 dn2 dn3 du1 du2 du3'

+ dn4 - - - + = 0

du1 du4 dx1 dx2 dx3 dx4 '

5n4 t1 duo ,2 duo .3 dun i Sn1 de2 de3 de4 \ d£2 de3

--+ e--+ e--+ e--1- u^ 1 ------1--1 + nn--1- wn-

dx1

dx1

dx2

+e3

dx3

+ uo

du1 dx2

dx3 + dx4 ] + o dx1 + o dx1

+e1 dvo+e2 dvo+e3 dvo+uo +vof - - + + wo = 0

dx2 dx1 dx2 dx3 dx2 \ du1 dx1 dx3 dx4/ dx2 '

dn4 dwo a2 dwo dwo de1 de2 (<V de1 de2 de4 \ —— + e1 —o + e2 —o + e3 —° + uo ——— + vo ——— + wo 1 —'-------— +—— 1

■+e2

■+e3

■ + uo

+ vo-

- + wo

dx3 dx1 dx2 dx3 dx3 dx3 y du1 dx1 When vo =0 and wo = 0 the following equations are valid

dx2 + dx4 )

dn2 dn1 du1 du2

dn3 dn1

du1 du3

dn3 dn2

du2 du3

0.

0.

1.24)

1.25)

1.26)

1.27)

1.28)

1.29)

1.30)

1.31)

According to equations (1.1), (1.2), (1.4), (1.12.) and (1.18) the following relation are true:

n1(u1, u2, u3), n2(u1, u2, u3), n3(u1, u2, u3), n4(x1, x2, x3, u4), ej(x1, x2, x3), j = 1,2,3 and e4(x4).

The analysis of the equations which do not contain initial velocity gives the preliminary structure of the sought-for functions:

ei = ei(x1, x2, x3), i = 1,2,3, e4 = e4(x4),

n1 = C1u1 - C2u2 - C3u3 + C4, n2 = C2u1 + C1u2 + C5u3 + C6, n3 = C3u1 - C5u2 + C1u3 + C7, n4 = (2C1 - &) u4 + $(x1,x2,x3,x4), here C1... C7 are constants. Then we obtain the equation that links e1, e2 and e3:

C1 + C2 + e°3 = 3C1.

(1.32)

0

0

The remaining three determining equations are in fact classifying equations for the functions «o(£), "o(£), wo (£):

, duo o duo o duo ( „ d^1 d^4 N d^2 d^3 d—

e1 d"T + e2 d"0 + e3 + uo -2Cri + A + A + "o ^ + wo ^ + ^ =0, (1.33)

dx1 dx2 dx3 \ dx1 dx4/ dx1 dx1 dx1

^ dvo - dvo ~ dvo d^1 ( ^ d£2 d^4 N de3 d-e1 «"J + e2 «4 + e3 «"3 + uo A + "o -2C1 + ^ + ^ + wo^ + =0, (1.34)

dx1 dx2 dx3 dx2 \ dx2 dx4 J dx2 dx2

^ dwo - dwo ~ dwo de1 de2 ( ^ de3 de4 N d-

e1^ + + e3^T + uoA + "oA + wJ -2C1 + ^ + ^ =0. (1.35).

dx1 dx2 dx3 dx3 dx3 \ dx3 dx4/ dx3

Let uo(£), vo(£) and wo(£) be arbitrary functions then e1 =0, e2 =0 and e3 = 0. Therefore, we have C1 = 0, d^4/dx4 = 0, d-/dx1 = 0, d-/dx2 = 0 and d-/dx3 = 0. It means that e4 = C9 = const and — = h(x4). Then the basis of the main operators of the Lie algebra consists of the following operators

d ^ 2 d 1 d v 3 d 1 d v 2 d 3 d T—4, X2 = -u —— + u —^, X3 = -u —— + u —r, X4 = -u —r + u —^, dx4 du1 du2 du1 du3 du3 du2

d d d d

X = dur, X6 = du", X = du, X8 = h(x4) du.

It should be noted that equations (1.33)-(1.35) do not contain terms which depend on x4, with the exception of the term d£4/(dx4). Then we can conclude that d£4/(dx4) = C8 = const so e4 = Qx4 + C9.

It turns out that the analysis of system (1.33)-(1.35) is reduced to the analysis of three equations for the components of the initial vortex w = rotuo, where uo = (uo, vo, wo), w = (w 1, w2 , w3), w1 = wox2 - "ox3, w2 = uox3 - woxi and w3 = "oxi - uox2. Indeed, considering the condition of compatibility of equations (1.19)-(1.21) we get the system of equations that contains only components of vector w and the operator coordinates e1, e2 and e3

c - 2C1 + § + S) U1 - gw2 - gu3 + e^ 1i + e2^ + e3w 13 =0, (o6) C8 - 2C1 + g + §) w2 - gw1 - g^3 + e1* Xi + e2 ^ + e3w 23 = 0, (1.37) C8 - 2C1 + g + dS) w3 - gw1 - gw2 + e1* Xi + e2 ^ + e3w 33 = 0. (1.38)

Equations (1.36)-(1.38) supplemented by equation (1.32) are classifying equations for system

_1 _2 _3

(0.1)-(0.4). With the change of variables e1 = C^1 + e , e2 = Cix2 + e , e3 = Cix3 + e equation (1.32) becomes homogeneous one

—1 —2 —3

e*i + ex^ + ex3 =0. (1.39).

As a result of this change of variables, equations (1.36)-(1.38) take the form

-(Cix1 +C1)wXi +(C1x2 +e2)w12 + (C1x3+e3)wX3 =0, (1.40)

-eV Xi+(C1x2+e2) +(C1x3+e3)w 33

)w1 - d!-w2 1 dx2 - dU

dx1 , dx3

di!^ \ 2d?2 1 r - dxi" - del

dx2 . dx3

3

1

—3 \ —3 —3

V.3 . .1 . .2 , (C _1 , .3 , (C _2 , + (C _3 , ^

Cs .3-^v-^v.2+(Cix1+e1).3i+(Cix2+e2).32+(Cix3+e3).33 = 0. (1.42)

dx3 I dx1 dx2 We need to add equation

+ <2 + .33 =0. (1.43)

to equations (1.39)—(1.42).

Therefore, we have five equations (1.39)—(1.43) for six functions e , wj, j = 1, 2, 3.

2. The solution of classifying equations

From equations (1.40)—(1.42) the first classifying relation w% = const can be derived. Then system of equations (1.40)—(1.42) can be rewritten as

d?1 i dj1 2 d^1 3 „ i 7TTw1 + w2 +1T3 w3 = Cs^1,

dx1 dx2 dx3

—2 —2 —2 de i de 2 de 3 „ 2

TTTw1 + TT2 w2 + TTs w3 = Cs^2,

dx1 dx2 dx3 —3 —3 —3

de i de 2 de 3 „ 3

w1 + w2 + w3 = Csw3. dx1 dx2 dx3

Thus, from equations (1.40)—(1.42) we obtain the following equations

vf • w = Cswj, (2.1)

j = 1, 2, 3, w = (w1, w2, w3) is constant vorticity of the fluid at the initial point in time. Let us assume that w = 0 and without loss of generality we can also assume that w1 = const = 0. Then equation (2.1) can be divided by w1 and vorticity takes the form w = (1, w2, w3) with new constants w2 and w3.

Let us rewrite system of equations (2.1) in extended form, assuming that w1 = 1,

— + dO + w3 = C (22)

dx1 dx2 dx3 '

—2 —2 —2

de + de w2 + de w3 = C w2 (2 3)

TTT + w + 3w = Csw , (2.3)

dx1 dx2 dx3

—3 —3 —3

de + dLw2 + dLw3 = Csw3. (24)

dx1 dx2 dx3

—1 —2 —3

We obtain three first order partial differential equations in variables e , e and e . One can suggest the following solution of equations (2.2)—(2.4)

e1 = Csx1 + f ^a, ft), (2.5)

f = Csw2x1 + f2(a, ft), (2.6)

f = Csw3x1 + f3(a, ft), (2.7)

where a = x2 - w2x1, ft = x3 - w3x1.

If we assume f2 (a, ft) = /2 (a, ft) + aCs then instead of Csw2x1 we can write Csx2 in equation (2.6). Similarly, instead of Csw3x1 we can write Csx3 in equation (2.6).

Remark. When w2 = 0, w3 = 0; w2 =0, w3 = 0; w2 = w3 =0 we have special cases of relations (2.5)-(2.7):

a) w2 =0, w3 = 0: J1 = Qx1 + f 1(a, x3), J2 = C8x2 + f2(a, x3), J3 = f3(a, x3).

_i _2 _3

b)w2 =0, w3 =0: J1 = C8X1 + f 1(x2, P), J2 = f2(x2, P), J3 = C8x3 + f3(x2, P)

c)w2 = w3 = 0 J1 = C8x1 + f 1(x2, x3), J2 = f2(x2, x3), J3 = f3(x2, x3).

After substituting expressions (2.5)-(2.7) into equation (1.39), we obtain the following equation

3C8 + (f2 - w2f1 )a + (f3 - w3f% = 0. (2.8)

Let us introduce the following designations in equation (2.8): dg = f2 — w2f1 and da = f3 — w3f1 + 3C8p. Then we find that f2 = w2f1 + dg, f3 = w3f1 — 3C8P + da.

Conclusion

For the vorticity vector w = (1, w2, w3) = const we have the following coordinates of the operator

J1 = C1x1 + C8x1 + f 1(a, P),

J2 = C1x2 + C8x2 + w2f1 + dg,

J3 = C1x3 + C8 x3 + w3f1 — 3C8p + da,

J4 = C8x4 + Cg,

n1 = C1u1 — C2u2 — C3 u3 + C5,

n2 = C2u1 + C1u2 + C4 u3 + C6,

n3 = C3U1 — C4U2 + C1 u3 + C7,

n4 = (2C1 — C8)u4 + h(x\ x2, x3) + ^(x4),

where a = x2 — w2x1, P = x3 — w3x1 and functions ^(x1, x2, x3), <£>(x4), f 1(a, P), d(a, P) are arbitrary functions.

The basic Lie algebra Lq is extended by operators

d 2 d 3 d ! d 3 d 3 d 4 d

T + x 9 + x ^ Q + u ^ T + u ^ Q + u Q + 2u ^ T 7

dx1 dx2 dx3 du1 du3 du3 du4

X = 2 d 2 d 3 d 4 d 4 d X2 x Tj T + x Tj o" + x Tj q" + x Tj T « Tj T,

dx2 dx2 dx3 dx4 du4

d 2 d 3 d d d 2 2 3 d X3 = + w "oT + w ^r, X4 = dg—■ — «a^r, X5 = h(x , x , x ) —-j. dx2 dx2 dx3 dx2 dx3 du4

Another possibility to obtain classifying equations is given by the function w. Let us present system of equations (1.40)-(1.42) in the form

dx • V^k + £ • Vwk + C8^k — v£k • W = 0, (2.9)

k = 1, 2, 3, C2 and C8 are some constants,

div£ = 0, divw = 0. (2.10)

Equations (2.9) and (2.10) allow one to determine the coordinates of the operator £ in terms of w.

The basis of the Lie algebra for any function w that extends Lo can be found in the following cases:

1) if Ci = 1, C8 = 0;

2) if C1 =0, C8 = 1;

3) if C1 =0, C8 = 0.

In the first case the coordinates of the operator are derived from system (2.10) and equations

x • +1 • - V£fc • ш = 0,

k =1, 2, 3. In the second case the coordinates of the operator are derived from system (2.10) and equations

| • V^fc + -v£fc • ш = 0,

k =1, 2, 3. In the third case the coordinates of the operator are derived from system (2.10) and equations

i • v^fc - vefc • ш = 0,

k = 1, 2, 3. A special solution of the latter system of equations is | = ш. For this solution we have the following coordinates of the operator

f = ^(x1, x2, x3), г =1, 2, 3, £4 = e4(x4),

П1 = -C2U2 - C3M3 + C4, n2 = C2U1 + +C5M3 + Ce,

n3 = C3u1 - C5u2 + +C7, n4 = u4 + ф(x1,x2,x3,x4).

Author wants to thank Professor Viktor Andreev for problem formulation and helpful discussions.

This research was supported by the Siberian Branch of the Russian Academy of Sciences under project No. 44.

References

[1] V.K.Andreev, Stability of unsteady fluid flows with a freeborder, Novosibirsk, Nauka, 1992 (in Russian).

[2] D.Serrin, Mathematical foundations of classical fluid mechanics, Moscow, Izd. Fiz.-Mat. Lit., 1963 (in Russian).

[3] L.V.Ovsyannikov, Group analysis of differential equations, Moscow, Nauka, 1978 (in Russian).

Групповая классификация уравнений трёхмерной идеальной жидкости в терминах траекторий и потенциала Вебера

Дарья А. Краснова

Проводится групповой анализ уравнений движения идеальной жидкости в переменных траекторий — потенциал Вебера. Показано, что переход к произвольным лагранжевым координатам, сохраняющий объем, является преобразованием эквивалентности для этой системы. Получены классифицирующие уравнения на функции начального распределения скорости. Вычислена основная группа Ли и указаны её 'расширения.

Ключевые слова: уравнения идеальной жидкости, преобразование эквивалентности, лагранжевы координаты, классифицирующие уравнения.