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obtained by means of reducing the NSEs with two- and three-dimensional subalgebras
of A(N S).
Here we consider system (2.9) with ?i vanishing. As mentioned in Note 2.5, in
this case the vector-function m has the form m = ?(t)e, where e = const, |e| = 1, and
? = ?(t) = |m(t)| = 0. Without loss of generality we can assume that e = (0, 0, 1),
i.e.,
m = (0, 0, ?(t)).
For such vector m, conditions (2.5) are satisfied by the following vector ni :
n1 = (1, 0, 0), n2 = (0, 1, 0).
Therefore, ansatz (2.4) and system (2.9) can be written, respectively, in the forms:
?1
u1 = v 1 , u2 = v 2 , u3 = ?(t) v 3 + ?t (t)x3 ,
(7.1)
?1 2
p = q ? 1 ?tt (t) ?(t) x3 ,
2
where v = v(y1 , y2 , y3 ), q = q(y1 , y2 , y3 ), yi = xi , y3 = t, and
vt + v j vj ? vjj + qi = 0,
i i i

vt + v j vj ? vjj = 0,
3 3 3
(7.2)
vi + ?3 = 0,
i

where ?3 = ?3 (t) = ?t /?.
It was shown in Note 2.8 that there exists a local transformation which make ?3
vanish. Therefore, we can consider system (7.2) only with ?3 vanishing and extend
the obtained results in the case ?3 = 0 by means of transformation (2.12). However it
will be sometimes convenient to investigate, at once, system (7.2) with an arbitrary
function ?3 .
The MIA of (7.2) with ?3 = 0 is given by the algebra
?
B = R3 (?), Z 1 (?), D1 , ?t , J 1 , ?v3 , v 3 ?v3
3 12

(see notations in Subsection 2.1). We construct complete sets of inequivalent one-
dimensional subalgebras of B and choose such algebras, among these subalgebras,
that can be used to reduce system (7.2) and do not lie in the linear span of the
?
operators R3 (?), Z 1 (?), J12 , i.e., the operators which are induced by operators from
1

A(N S) for arbitrary ?3 . As a result we obtain the following algebras (more exactly,
the following classes of algebras):
The one-dimentional subalgebras:
B1 = D3 + 2?J12 + 2?v 3 ?v3 + 2??v3 , where ?? = 0.
1 1 1
1.
B2 = ?t + ?J12 + ?v 3 ?v3 + ??v3 , where ?? = 0, ? ? {0; 1}.
1 1
2.
B3 = J12 + ?v 3 ?v3 + Z 1 (?(t)) , where ? = 0, ? ? C ? ((t0 , t1 ), R).
1 1
3.
?
B4 = R3 (?(t)) + ?v 3 ?v3 , where ? = 0,
1
4.
?(t) = (? 1 (t), ? 2 (t)) = (0, 0) ? t ? (t0 , t1 ), ? i ? C ? ((t0 , t1 ), R).
?
224 W.I. Fushchych, R.O. Popovych

The two-dimentional subalgebras:
?t + ?2 ?v3 , D3 + ?J12 + ?v 3 ?v3 + ?1 ?v3 ,
2 1 1
1. B1 =
??1 = 0, (? ? 2)?2 = 0.
where
D3 + 2?1 v 3 ?v3 + 2?1 ?v3 , J12 + ?2 v 3 ?v3 + ?2 ?v3 + Z 1 (?|t|?1 ) ,
2 1 1
2. B2 =
?1 ?1 = 0, ?2 ?2 = 0, ?1 ?2 ? ?2 ?1 = 0.
where
B3 = D3 + 2?J12 + 2?1 v 3 ?v3 + 2?1 ?v3 , R3 (|t|?+1/2 cos ?, |t|?+1/2 sin ? ) +
2 1 1
3.
+ ?2 v 3 ?v3 + ?2 ?v3 + Z 1 (?|t|??1 ) , where ? = ? ln |t|,
(?1 + ?)?1 ? ?2 ?1 = 0, ??2 = 0, ?? = 0.
B4 = ?t + ?1 v 3 ?v3 + ?1 ?v3 , J12 + ?2 v 3 ?v3 + ?2 ?v3 + Z 1 (?) ,
2 1
4.
where ?1 ?1 = 0, ?2 ?2 = 0, ?1 ?2 ? ?2 ?1 = 0.
B5 = ?t + ?J12 + ?1 v 3 ?v3 + ?1 ?v3 , R3 (e?t cos ?t, e?t sin ?t) +
2 1
5.
+ Z 1 (?e?t ) + ?2 v 3 ?v3 + ?2 ?v3 , where (?1 + ?)?1 ? ?2 ?1 = 0,
??2 = 0, ?? = 0.
? ? ?
B6 = R3 (? 1 ) + ?v 3 ?v3 , R3 (? 2 ) , where ? i = (? i1 (t), ? i2 (t)) = (0, 0)
2
6.
? t ? (t0 , t1 ), ? ij ? C ? ((t0 , t1 ), R), ?tt · ? 2 ? ? 1 · ?tt = 0, ? = 0.
?1 ? ? ?2
??
Hereafter ? 1 · ? 2 := ? 1i ? 2i .
Let us reduce system (7.2) to systems of PDEs in two independent variables. With
1 1
the algebras B1 –B4 we can construct the following complete set of Lie ansatzes of
codimension 1 for system (7.2) with ?3 = 0:
v 1 = |t|?1/2 (w1 cos ? ? w2 sin ? ) + 1 y1 t?1 ? ?y2 t?1 ,
1. 2
v 2 = |t|?1/2 (w1 sin ? + w2 cos ? ) + 1 y2 t?1 + ?y1 t?1 ,
2
(7.3)
v = |t| w + ? ln |t|,
3 ?3

q = |t|?1 s + 1 (? 2 + 1 )t?2 r2 ,
2 4

where ? = ? ln |t|, ?? = 0,
z1 = |t|?1/2 (y1 cos ? + y2 sin ? ), z2 = |t|?1/2 (?y1 sin ? + y2 cos ? ).
Here and below wa = wa (z1 , z2 ), s = s(z1 , z2 ), r = (y1 + y2 )1/2 .
2 2

v 1 = w1 cos ?t ? w2 sin ?t ? ?y2 ,
2.
v 2 = w1 sin ?t + w2 cos ?t + ?y1 ,
(7.4)
v 3 = w3 e?t + ?t,
q = s + 1 ? 2 r2 ,
2

where ? ? {0; 1}, ?? = 0,
z1 = y1 cos ?t + y2 sin ?t, z2 = ?y1 sin ?t + y2 cos ?t.
v 1 = y1 r?1 w3 ? y2 r?2 w1 ? ?y2 r?2 ,
3.
v 2 = y2 r?1 w3 + y1 r?2 w1 + ?y1 r?2 ,
(7.5)
v 3 = w2 e? arctan y2 /y1 ,
q = s + ?(t) arctan y2 /y1 ,
Symmetry reduction and exact solutions of the Navier–Stokes equations 225

= t, z2 = r, ? = 0, ? ? C ? ((t0 , t1 ), R).
where 1

v = (? · ?)?1 (w1 + ?)? + w3 ? + (? · y )?t ? z2 ?t
?? ? ? ? ?? ?
4. ?
v 3 = w2 exp ?(? · ?)?1 (? · y ) (7.6)
?? ??
q = s ? (? · ?)?1 (?tt · y )(? · y ) + 1 (? · ?)?2 (?tt · ?)(? · y )2 ,
?? ? ??? ?? ? ???
2

where z1 = t, z2 = (? · y ), ? = 0, v = (v 1 , v 2 ), y = (y1 , y2 ), ? i ? C ? ((t0 , t1 ), R),
?? ? ?
? = (?? 2 , ? 1 ).
?
Substituting ansatzes (7.3) and (7.4) into system (7.2) with ?3 = 0, we obtain a
reduced system of the form (6.1), where

?2 = ?1, ?3 = ?2?, ?4 = ?, ?5 = ? if and
?1 = 0, t>0
?2 = 1, ?3 = 2?, ?4 = ??, ?5 = ?? if
?1 = 0, t<0

for ansatz (7.3) and

?3 = ?2?,
?1 = 0, ?2 = 0, ?4 = ?, ?5 = ?

for ansatz (7.4). System (6.1) is investigated in Section 6 in detail.
Because the form of ansatzes (7.3) is not changed after transformation (2.12), it is
convinient to substitute their into a system of form (7.2) with an arbitrary function ?3 .
As a result of substituting, we obtain the following reduced systems:
?3 ?1 3 ?2
w1 + w3 w2 ? z2 (w1 + ?)2 ? (w22 + z2 w2 ? z2 w3 ) + s2 = 0,
3 3 3
3.
?1 1
w1 + w3 w2 ? w22 + z2 w2 + ? = 0,
1 1 1
(7.7)
?1 2 ?2
w1 + w3 w2 ? w22 ? z2 w2 + ?z2 w1 w2 = 0,
2 2 2

?1
w2 + z2 w3 = ??1 /?.
3

??1
w1 + w3 w2 ? (? · ?)w22 = 0,
1 1
4.
w1 + w3 w2 ? (? · ?)w22 + (? · ?)s2 + 2(w1 + ?)(? · ?)(? · ?)?1 ?
??3 ?? ????
3 3

? 2(?t · ?)(? · ?)?1 w3 + (2?t · ?t ? ?tt · ?)(? · ?)?1 z2 = 0,
? ??? ?? ? ??? (7.8)
w1 + w3 w2 ? (? · ?)w22 + ?(? · ?)?1 w1 + (?t · ?)(? · ?)?1 z2 w2 = 0,
??2 ?? ????
2 2

3
w2 + ?t /? = 0.

Unlike systems 8 and 9 from Subsection 3.2, systems (7.7) and (7.8) are not reduced
to linear systems of PDEs.
Let us investigate system (7.7). The last equation of (7.7) immediately gives
?1 ?1 3 ?2
(w2 + z2 w3 )2 = w22 + z2 w2 ? z2 w3 = 0,
3 3
(7.9)
?1
w3 = (? ? 1)z2 ? 1 ?t ? ?1 z2 ,
2

where ? = ?(t) is an arbitrary differentiable function of t = z2 . Then it follows from
the first equation of (7.7) that

?3 ?2
z2 (w1 + ?)2 dz2 ? 1 (? ? 1)2 z2 + 1 z2 (?t /?)t ? 1 (?t /?)2 ? ?t ln |z2 |.
2
s= 2 4 2
226 W.I. Fushchych, R.O. Popovych

Substituting (7.9) into the remaining equations of (7.7), we get
?1
w1 ? w22 + ?z2 ? 1 ?t ? ?1 z2 w2 + ? = 0,
1 1 1
2
(7.10)
?1 ?2
w1 ? w22 + (? ? 2)z2 ? 1 ?t ? ?1 z2 w2 + ?z2 w1 w2 = 0.
2 2 2
2

By means of changing the independent variables

|?(t)|dt, z = |?(t)|1/2 z2 , (7.11)
?=

system (7.10) can be transformed to a system of a simpler form:

w? ? wzz + ?z ?1 wz + ?|?|?1 = 0,
??
1 1 2
?
(7.12)
w? ? wzz + (? ? 2)z ?1 wz + ?z ?2 w1 w2 = 0,
2 2 2
?

?
where ? (? ) = ?(t), ?(? ) = ?(t), and ?(? ) = ?(t).
? ?
If ?(t) = ?2C?(t)(?(t) ? 1) for some fixed constant C, particular solutions of
(7.10) are functions

w1 = C?(t)z2 ,
2
w2 = f (z1 , z2 ) exp ?C ?(t)dt ,

where f is an arbitrary solution of the following equation
?1
f1 ? f22 + (? ? 2)z2 ? 1 ?t ? ?1 z2 f2 = 0. (7.13)
2

In the variables from (7.11), equation (7.13) has form (5.22) with ? (? ) = ?(t) ? 2.
?
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