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?1/2 2

3 1 3

??c exp ? ??c ?

f =g=

2 6 2

(2.4)

2

5 11 3

?w ? , , ??c ,

12 4 3 2

3

h = ? + c, ? = c? ? ? 2 + c1 ,

2

where w(·, ·, ·) is solution of the Whittaker equation (2.10). The above solution 17?

(2.4) is a particular solution of equations 17? (2.3) with ? = 1. When ? is an arbitrary

constant, the general solution of 17? (2.3) has the form

?1/2 2

3 1 3

??

??c exp ? ??c ?

17 . g =

2 6 2

2

5 11 3 (2.4)

?w ? , , ??c ,

12 4 3 2

3

h = ? + c, ? = c? ? ? 2 + c1

2

and f satisfies the ODE

3 1

? ? ? c f? + f ? ?g = 0.

f+

2 2

The general solution of 18? (2.3) is

?1/2 2

5 1 5

?

??c exp ? ??c ?

18 . f =g=

2 10 2

2

27 1 1 5 (2.4)

?w ? , , ??c ,

20 4 5 2

5

h = ?2? + c, ? = c? ? 3? 2 + c1 .

2

The general solution of 19? (2.3) is

?1/2 2

3 1 3

?

??c exp ? ??c ?

19 . f =g=

2 6 2

2

1 11 3 (2.4)

?w ? , , ??c ,

12 4 3 2

3

h = ?? + c, c? ? ? 2 + c1 .

?=

2

In 17? –19? (2.4) w(·, ·, ·) is an arbitrary solution of the Whittaker equation (2.10).

316 W.I. Fushchych, W.M. Shtelen, S.L. Slavutsky

Remark 1. The solutions of reduced ns equations 1? –19? (2.3) given in 1? –19? (2.4)

should be considered together with the corresponding ansatze of table 1; then one

gets solutions of the NS equations (1.1).

The solutions of the ns equations (1.1) obtained above can be used in a basic way

to construct multiparameter families of solutions. A procedure for generating new

solutions from a known one is based on the well known fact of Lie theory according

to which symmetry transformations transform any solution of a given differential

equation into another solution. For example, if transformations

xµ > xµ = fµ (x, ?), (µ = 0, n ? 1 ),

u(x) > u (x ) = R(x, ?)u(x) + B(x, ?),

where the ? are parameters, u = column (u1 , u2 , . . . , uk ), R(x, ?) is a non-singular

matrix k ? k, R(x, 0) = I, fµ , B (column) are some smooth functions, fµ (x, 0) = xµ ,

B(x, 0) = 0 leave considered PDEs invariant, then the function

uII (x) = R?1 (x, ?)[uI (x ) ? B(x, ?)] (2.27)

will be a new solution of the equation provided uI (x) is any given solution. Formulae

like (2.27) we call formulae of group multiplication of solutions (GMS) (Fushchych et

al [9]). So, to construct the formulae of GMS for the NS equations one has to find,

first of all, the final transformations generated by symmetry operators (1.2), (1.3) and

then, according to (2.27), construct the formulae. The results of this is given in the

table 2.

Note that in 1–11 p (x ) = p(x) and therefore pII = pI (x ). In this table ?0 , ?a ,

?a , ?a , ?, ?, ? are arbitrary constants, ? = (?1 + ?2 + ?3 )1/2 ; f and g are arbitrary

2 2 2

differentiable functions of t. The formulae of GMS stated above allow to construct

new solutions uII (x) of the NS equations (1.1) starting from a known one uI (x).

Table 2. Final symmetry transformations and the corresponding formulae

of GMS for the NS equations (1.1)

Final transformations

x>x u(x) > u (x )

N Operator Formulas of GMS

t = t + ?0 x = x u (x ) = u(x) uII (x) = uI (x )

1 ?t

t =t xa = xa + ?a u (x ) = u(x) uII (x) = uI (x )

2–4 ?a

t =t x = x cos ? + u a (x ) = ?ab cos ? + ua (x) = ?ab cos ? +

5–7 Jab II

+ (x ? ?) sin ? + +?abc ?c sin ? + +?abc ?c sin ? +

? ? ?

+ ?(? · x) 1?cos ? + ?a ?b 1?cos ? ub (x) + ?a ?b 1?cos ? ub (x )

?2 ?2 ?2 I

t =t x = x + ?t u (x ) = u(x) + ? uII (x) = uI (x ) ? ?

8–10 Ga

t = e2? t u (x ) = e?? u(x)

x = e? x uII (x) = e? uI (x )

11 D

p (x ) = e?2? p(x) pII (x) = e2? pI (x)

? ?

t =t x = x + ?f(t) u (x ) = u(x) + ?f(t) uII (x) = uI (x ) ? ?f(t)

12 Q

? ?

p (x ) = p(x) ? ?x · f(t) pII (x) = pI (x ) + ?x · f(t)

t =t x =x u (x ) = u(x) uII (x) = uI (x )

13 R

p (x ) = p(x) + ?g(t) pII (x) = pI (x ) ? ?g(t)

Reduction and exact solutions of the Navier–Stokes equations 317

Remark 2. It will be noted that operator Q given in (1.3) generates transformations

(N 12 in table 2) which can be considered as an invariant transition to a frame of

reference which is moved arbitrarily: xref = ?f (t).

Let us give some examples of the application of formulae of GMS. Having applied

formulae 5–7 of table 2 to solution 16? (2.4) we get a new multiparameter solution

for the NS equations (1.1)

? ?

1 ?? 2 2 2

u(x) = v e es ds + b ?3 + ?4 es ds

a ?1 + ?2 +c ,

t

(2.28)

c·x 1 c·x

? = v ? 1, p(x) = v + ?5 ,

t 2t

2t

where ?1 , . . . , ?5 are arbitrary constants, a, b, c are arbitrary orthonormal constant

vectors

a · b = a · c = b · c = 0.

a2 = b2 = c2 = 1, (2.29)

Further application of the formulae of GMS N 8–10 to (2.28) gives rise to the

following solution of the NS equations

y y

1 2

s2 2

u(x) = v e?y a ?1 + ?2 + c ? ?,

es ds

e ds + b ?3 + ?4

t

(2.30)

c · (x + ?t) c · (x + ?t)

1

v v

? 1, p(x) =

y= + ?5 ,

t

2t 2t

where the ? are arbitrary constants, the rest are the same as in (2.28).

The procedure of generating solutions by means of symmetry transformations can

be continued until one gets an ungenerative family of solutions, that is the family

which is invariant (up to transformation of constant parameters) with respect to the

total GMS procedure. Without doubt, the reader can carry out this procedure by

analogy with the above examples, for any solution 1? –19? (2.4) of the NS equations.

3. Examples of non-Lie ans?tze for the NS field

a

Ans?tze collected in table 1, of course, do not exhaust all possible ans?tze which

a a

reduce the NS equations. Here we consider several examples of ans?tze which do not

a

have the form (1.4). More complete consideration of this question will be given in our

next paper.

Because all ans?tze obtained within the framework of the Lie approach have form

a

(1.4), it is natural to call other ans?tze non-Lie. Our first example of this is the well

a

known ansatz

u = ??, (3.1)

where ? = ?(x) is a scalar function. If satisfies the Hamilton–Jacobi and Laplace

equations

?t + (??)2 + p = 0, (3.2)

?? = 0

then the function u (3.1) automatically satisfies the NS equations (1.1). It is an

example of non-local component reduction.

318 W.I. Fushchych, W.M. Shtelen, S.L. Slavutsky

Ansatz

u = a?(t, b · x, c · x), (3.3)

where a, b, c are constant vectors satisfying (2.29), reduces (1.1) to the two-dimen-

sional heat equation

?2 ?2

?t ? ?2 ? = 0, ?2 ? ?1 = b · x, ?2 = c · x, (3.4)

2 + ?? 2 ,

??1 2

Ansatz

(3.5)

u = x?(x), p = p(x)

reduces equations (1.1) to the system of pde for two scalar functions ? and p

x(?t + ??) + ?(? + p) = 0, ? + (x · ?)? = 0. (3.6)

New ans?tze and solutions of the NS equations (1.1) obtained within the framework

a

of conditional symmetry will be given in our next paper. The concept and the term

conditional invariance was firstly introduced by Fushchych [5] (see also Fushchych

and Nikitin [7]). Further development and applications of this concept are contained

in Fushchych et at [9], Fushchych and Serov [8], Levi and Winternitz [10].

Let us make some concluding remarks. It will be noted that the question of what

spin is carried by the NS field has a rather strange answer (Fushchych [4]): the NS

field carries not only spin 1 but all possible integer spins s = 0, 1, 2, . . .. It is due to

the fact that the space of solutions of the ns equations can be decomposed into an

infinite direct sum of subspaces invariant under operators Sab = ua ?ub ? ub ?ua from

algebra AO(3), and these subspaces are not invariant under operators Ga from (1.2)

because of the unboundedness of operators ?ua .

In hydrodynamics the linearized NS equations are sometimes used

ut ? ?u = 0, (3.7)

div u = 0.

The maximal invariance algebra of (3.7) is the seven-dimensional Lie algebra with

basis elements

?t , ?a , D = 2t?t + xa ?a , I = ua ?ua ,

(3.8)

Jab = xa ?b ? xb ?a + ua ?ub ? ub ?ua .

It should be pointed out that (3.7) are not Galilei invariant and therefore they fail in

adequately describing real hydrodynamics processes.

Acknowledgments. We would like to express our gratitude to the referees for

their useful suggestions.

1. Barannik L.F., Fushchych W.I., J. Math. Phys., 1989, 30, 280–290.

2. Bateman H., Erdelyi A., Higher transcendental functions, Vol. 1, New York, McGraw-Hill, 1953.

3. Birkhoff G., Hydrodynamics — A study in logic, fact and similitude, Princeton, Princeton University

Press, 1950.

4. Fushchych W.I., in Theoretic-Algebraic Methods in Problems of Mathematical Physics, Kiev, Insti-

tute of Mathematics, 1983, 4–23.

Reduction and exact solutions of the Navier–Stokes equations 319

5. Fushchych W.I., How to extend the symmetry of differential equations, in Symmetry and Solutions

of Nonlinear Equations of Mathematical Physics, Kiev, Institute of Mathematics, 1987, 4–16.

6. Fushchych W.I., Barannik A.F., Barannik L.F., Continuous subgroups of the generalized Galilei

group, Preprint N 85.19, Kiev, Institute of Mathematics, 1985.

7. Fushchych W.I., Nikitin A.G., Symmetries of Maxwell’s equations, Dordrecht, Reidel, 1987.

8. Fushchych W.I., Serov N., in Symmetry and Solutions of Equations of Mathematical Physics, Kiev,

Institute of Mathematics, 1989, 95–102.

9. Fushchych W.I., Shtelen W.M., Serov N.I., Symmetry analysis and exact solutions of nonlinear

equations of mathematical physics, Kiev, Naukova Dumka, 1989 (Engl. transl. Dordrecht, Kluwer,

in press).

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