ñòð. 57 |

solutions of (7.10) are functions

w1 = C (?(t))2 z2 ? 4z2 ?(t) ?(t)(?(t) ? 3)dt ,

4 2

1

w2 = f (z1 , z2 ) exp 1/2 2

2 (?C) ?(t)z2 + ?(t) ,

where ?(t) = ?(?C)1/2 ?(t)(?(t) ? 3)dt + 4?C ?(t)(?(t) ? 3)dt dt and f is

?(t)

an arbitrary solution of the following equation

?1

f1 ? f22 + (? ? 2)z2 ? ( 1 ?t ? ?1 + 2(?C)1/2 )z2 f2 = 0. (7.14)

2

After the change of the independent variables

|?(t)| exp 4(?C)1/2 z = |?(t)|1/2 exp 2(?C)1/2

?= ?(t)dt dt, ?(t)dt z2

in (7.14), we obtain equation (5.22) with ? (? ) = ?(t) ? 2 again.

?

Let us continue to system (7.8). The last equation of (7.8) integrates with respect to

z2 to the following expression: w3 = ??t ? ?1 z2 + ?. Here ? = ?(t) is an differentiable

function of z1 = y3 = t. Let us make the transformation from the symmetry group

of (7.2):

??? ? ?? ?

v (t, y ) = v (t, y ? ?(t)) + ?t (t), q (t, y ) = q(t, y ? ?(t)) ? ?tt (t) · y ,

?? v3 = v3 ,

? ? ?? ?

? ? ??

where ?tt · ? ? ? · ?tt = 0 and

?t · ? + ? + ?t ? ?1 (? · ?) ? |?|?2 (? · ?)(?t · ?) + |?|?2 (? · ?)(?t · ?) = 0.

?? ?? ? ??? ? ? ??? ?

Symmetry reduction and exact solutions of the Navier–Stokes equations 227

? ??

Hereafter |?|2 = ? · ?. This transformation does not modify ansatz (7.6), but it makes

the function ? vanish, i.e., w3 = ??t ? ?1 z2 . Therefore, without loss of generality we

?

may assume, at once, that w3 = ??t ? ?1 z2 .

Substituting the expression for w3 in the other equations of (7.8), we obtain that

s = z2 |?|?2 · ? ? ?t · ?t ? (?t · ?)?t ? ?1 |?|?2 + 1 ?tt ? ?1 ? (?t )2 ? ?2 ?

2? 1? ??? ?? ?

2 ?tt 2

? 2(?t · ?)|?|?2

? ?? w1 (z1 , z2 )dz2 ,

w1 ? ?1 ? ?1 z2 w2 ? |?|2 w22 = 0,

?1

1 1

(7.15)

w1 ? ?1 ? ?1 z2 w2 ? |?|2 w22 + ?|?|?2 2(?t · ?)|?|?2 z2 + w1 w2 = 0.

?2 ? ? ??

2 2

The change of the independent variables

?

? = (?(t)|?|)2 dt, z = ?(t)z2

reduces system (7.15) to the following form:

w? ? wzz = 0,

1 1

(7.16)

? ??

w? ? wzz + ?|?|?4 ? ?2 2(? t · ?)?z + w1 w2 = 0,

?? ? ??

2 2

? ?

? ? ? ?

where ?(? ) = ?(t), ?(? ) = ?(t), ? (? ) = ?(t).

?

Particular solutions of (7.15) are the functions

?

w1 = C1 + C2 ?(t)z2 + C3 1 (?(t)z2 )2 + (?(t)|?|)2 dt ,

2

w2 = f (t, z2 ) exp ? 2 (t)z2 + ? 1 (t)z2 + ? 0 (t) ,

2

where (? 2 (t), ? 1 (t), ? 0 (t)) is a particular solution of the system of ODEs:

?t ? 2?t ? ?1 ? 2 ? 4|?|2 (? 2 )2 + 1 C3 ?? 2 |?|?2 = 0,

? ?

2

2

?t ? ?t ? ?1 ? 1 ? 4|?|2 ? 2 ? 1 + 2?(?t · ?)|?|?4 + C2 ??|?|?2 = 0,

? ? ?? ?

1

?t ? 2|?|2 ? 2 ? |?|2 (? 1 )2 + ? C1 + C3 (?(t)|?|)2 dt |?|?2 = 0,

? ? ? ?

0

and f is an arbitrary solution of the following equation

f1 ? |?|2 f22 + (?t ? ?1 + 4|?|2 ? 2 )z2 + 2|?|2 ? 1 f2 = 0.

? ? ? (7.17)

Equation (7.17) is reduced by means of a local transformation of the independent

variables to the heat equation.

Consider the Lie reductions of system (7.2) to systems of ODEs. The second basis

2

operator of the each algebra Bk , k = 1, 5 induces, for the reduced system obtained

from system (7.2) by means of the first basis operator, either a Lie symmetry operator

from Table 2 or a operator giving a ansatz of form (6.4). Therefore, the Lie reduction

of system (7.2) with the algebras B1 ?B5 gives only solutions that can be constructed

2 2

1 1

for system (7.2) by means of reducing with the algebras B1 and B2 to system (6.1).

2

With the algebra B6 we obtain an ansatz and a reduced system of the following

forms:

v = ? + ??1 (?i · y )?t , v 3 = ?3 exp ??(?1 · y ) ,

?? ? ? ?i ??

(7.18)

s = h ? 1 ??1 (?tt · y )(?i · y ),

?i ? ? ?

2

228 W.I. Fushchych, R.O. Popovych

?? ??

where ?a = ?a (?), h = h(?), ? = t, ? = ? 11 ? 22 ? ? 12 ? 21 = ? 1 · ?1 = ? 2 · ?2 ,

? ?

?1 = (? 22 , ?? 21 ), ?2 = (?? 12 , ? 11 ), and

?t + ??1 (?i · ?)?t = 0, ?3 + ???1 (?1 · ?) ? ? 2 ??2 (?1 · ?1 ) ?3 = 0,

? ? ? ?i ?? ??

t

(7.19)

??1 (?i · ? i ) + ?t ? ?1 = 0.

??

t

Let us make the transformation from the symmetry group of system (7.2):

??? ? ?? ?? ??

v (t, y ) = v (t, y ? ?) + ?t , v 3 (t, y ) = v 3 (t, y ? ?), s(t, y ) = s(t, y ? ?) ? ?tt · y ,

??

? ? ? ??

where

?t + ??1 (?i · ?)?t + ? = 0.

? ? ? ?i ? (7.20)

It follows from (7.20) that ?tt = ??1 (?i · ?)?tt , i.e., ?tt · ? ? ?i · ?tt = 0. Therefore, this

? ? ? ?i ?i ? ? ?

trasformation does not modify ansatz (7.18), but it makes the functions ?i vanish.

And without loss of generality we may assume, at once, that ?i ? 0. Then

2

???1 |?| dt ,

?3 = C exp C = const.

The last equation of system (7.19) is the compatibility condition of system (7.2) and

ansatz (7.18).

8 Conclusion

In this article we reduced the NSEs to systems of PDEs in three and two independent

variables and systems of ODEs by means of the Lie method. Then, we investigated

symmetry properties of the reduced systems of PDEs and made Lie reductions of

systems which admitted non-trivial symmetry operators, i.e., operators that are not

induced by operators from A(N S). Some of the systems in two independent variables

were reduced to linear systems of either two one-dimensional heat equations or two

translational equations. We also managed to find exact solutions for most of the

reduced systems of ODEs.

Now, let us give some remaining problems. Firstly, we failed, for the present, to

describe the non-Lie ansatzes of form (1.6) that reduce the NSEs. (These ansatzes

include, for example, the well-known ansatzes for the Karman swirling flows (see

bibliography in [16]). One can also consider non-local ansatzes for the Navier–Stokes

field, i.e., ansatzes containing derivatives of new unknown functions.

Second problem is to study non-Lie (i.e., non-local, conditional, and Q-conditional)

symmetries of the NSEs [13].

And finally, it would be interesting to investigate compatability and to construct

exact solutions of overdetermined systems that are obtained from the NSEs by means

of different additional conditions. Usually one uses the condition where the nonli-

nearity has a simple form, for example, the potential form (see review [36]), i.e.,

rot((u · ?)u) = 0 (the NS fields satisfying this condition is called the generalized

Beltrami flows). We managed to describe the general solution of the NSEs with the

additional condition where the convective terms vanish [29, 30]. But one can give

other conditions, for example,

ut + (u · ?)u = 0,

u = 0,

and so on.

We will consider the problems above elsewhere.

Symmetry reduction and exact solutions of the Navier–Stokes equations 229

Appendix

A Inequivalent one-, two-, and three-dimensional

subalgebras of A(N S)

To find complete sets of inequivalent subalgebras of A(N S), we use the method given,

for example, in [27, 28]. Let us describe it briefly.

1. We find the commutation relations between the basis elements of A(N S).

2. For arbitrary basis elements V , W 0 of A(N S) and each ? ? R we calculate the

adjoint action

W (?) = Ad(?V )W 0 = Ad(exp(?V ))W 0

of the element exp(?V ) from the one-parameter group generated by the operator V

on W 0 . This calculation can be made in two ways: either by means of summing the

Lie series

?

?n n ?2

?

{V , W 0 } = W 0 + [V, W 0 ] + [V, [V, W 0 ]] + · · · , (A.1)

W (?) =

n! 1! 2!

n=0

where {V 0 , W 0 } = W 0 , {V n , W 0 } = [V, {V n?1 , W 0 }], or directly by means of solving

the initial value problem

dW (?)

W (0) = W 0 . (A.2)

= [V, W (?)],

d?

3. We take a subalgebra of a general form with a fixed dimension. Taking into

account that the subalgebra is closed under the Lie bracket, we try to simplify it by

means of adjoint actions as much as possible.

A.1 The commutation relations and the adjoint representation

of the algebra A(N S)

Basis elements (1.2) of A(N S) satisfy the following commutation relations:

[J12 , J23 ] = ?J31 , [J23 , J31 ] = ?J12 , [J31 , J12 ] = ?J23 ,

[?t , Jab ] = [D, Jab ] = 0, [?t , D] = 2?t ,

[?t , R(m)] = R(mt ), [D, R(m)] = R(2tmt ? m),

(A.3)

[?t , Z(?)] = Z(?t ), [D, Z(?)] = Z(2t?t + 2?),

[R(m), R(n)] = Z(mtt · n ? m · ntt ), [Jab , R(m)] = R(m),

?

[Jab , Z(?)] = [Z(?), R(m)] = [Z(?), Z(?)] = 0,

mb = ?ma , mc = 0, a = b = c = a.

where ma = mb ,

? ? ?

Note A.1 Relations (A.3) imply that the set of operators (1.2) generates an algebra

when, for example, the parameter-functions ma and ? belong to C ? ((t0 , t1 ), R)

?

(C0 ((t0 , t1 ), R), A((t0 , t1 ), R)), i.e., the set of infinite-differentiable (infinite-differen-

tiable finite, real analytic) functions from (t0 , t1 ) in R, where ?? ? t0 < t1 ? +?.

230 W.I. Fushchych, R.O. Popovych

But the NSEs (1.1) admit operators (1.3) and (1.4) with parameter-functions of a less

degree of smoothness. Moreover, the minimal degree of their smoothness depends

on the smoothness that is demanded for the solutions of the NSEs (1.1). Thus, if

ua ? C 2 ((t0 , t1 ) ? ?, R) and p ? C 1 ((t0 , t1 ) ? ?, R), where ? is a domain in R3 , then

it is sufficient that ma ? C 3 ((t0 , t1 ), R) and ? ? C 1 ((t0 , t1 ), R). Therefore, one can

consider the “pseudoalgebra” generated by operators (1.2). The prefix “pseudo-” means

that in this set of operators the commutation operation is not determined for all pairs

of its elements, and the algebra axioms are satisfied only by elements, where they are

defined. It is better to indicate the functional classes that are sets of values for the

parameters ma and ? in the notation of the algebra A(N S). But below, for simplicity,

we fix these classes, taking ma , ? ? C ? ((t0 , t1 ), R), and keep the notation of the

algebra generated by operators (1.2) in the form A(N S). However, all calculations

will be made in such a way that they can be translated for the case of a less degree

of smoothness.

Most of the adjoint actions are calculated simply as sums of their Lie series. Thus,

Ad(??t )D = D + 2??t , Ad(?D)?t = e?2? ?t ,

Ad(?Z(?))?t = ?t ? ?Z(?t ), Ad(?Z(?))D = D ? ?Z(2t?t + 2?),

Ad(?R(m))?t = ?t ? ?R(mt ) ? 1 ?2 Z(mt · mtt ? m · mttt ),

2

Ad(?R(m))D = D ? ?R(2tmt ? m) ?

(A.4)

? 1 ?2 Z(2tmt · mtt ? 2tm · mttt ? 4m · mtt ),

2

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