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in all other cases.
Here ? i = ? i (t), ? = ?(t) are arbitrary smooth function of t = y3 .

Note 2.5 If functions ?b are determined by (2.10), then e?(t) = C|m(t)|, where C =
const, and the condition ?i = 0 implies that m = |m(t)|e, where e = const and |e| = 1.

Note 2.6 The vector-functions ni from Note 2.2 are determined up to the transfor-
mation

n1 = n1 cos ? ? n2 sin ?, n2 = n1 sin ? + n2 cos ?,

where ? = const. Therefore, ? can be chosen such that C2 = 0 (then C1 = 0).

Note 2.7 The operators R3 (? 1 , ? 2 ) + ?S and Z 1 (?) are induced by R(l) + Z(?) and
Z(?), respectively. Here l = ? i ni + ? 3 m, ?t (m · m) + 2? i (ni · m) = ?,
3
t

? ? 3 (m · m)?1 ((mt · ni )? i )2 ? 1 (mtt · ni )? 3 ? i + 1 (ltt · ni )? i = 0.
2 2 2

If m = |m|e, where e = const and |e| = 1, the operator J12 is induced by e1 J23 +
1

e2 J31 + e3 J12 .
For

m = ?3 e?t (?2 cos ?, ?2 sin ?, ?1 )T

with ? = ?t + ? and ?a = const, where ?1 + ?2 = 1, the operator ?t + ?J12 induces
2 2

the operator ?y3 ? ?1 ?J12 + ?v 3 ?v3 if the following vector-functions ni are chosen:
1

n2 = ?k 1 sin ?1 ? + k 2 cos ?1 ?,
n1 = k 1 cos ?1 ? + k 2 sin ?1 ?, (2.11)

where k 1 = (? sin ?, cos ?, 0)T and k 2 = (?1 cos ?, ?1 sin ?, ??2 )T .
For

m = ?3 |t + ?4 |?+1/2 (?2 cos ?, ?2 sin ?, ?1 )T
188 W.I. Fushchych, R.O. Popovych

with ? = ? ln |t + ?4 | + ? and ?a , ?4 = const, where ?1 + ?2 = 1, the operator
2 2

D + 2?4 ?t + 2?J12 induces the operator

D3 + 2?4 ?y3 ? 2?1 ?J12 + 2?v 3 ?v3 ,
1 1

where D3 = yi ?yi + 2y3 ?y3 ? v i ?vi ? 2q?q , if the vector-functions ni are chosen in
1

form (2.11). In all other cases the basis elements of the MIA of (2.9) are not induced
by operators from A(N S).

Note 2.8 The invariance algebras of systems of form (2.9) with different parameter-
functions ?3 = ?3 (t) and ?3 = ?3 (t) are similar . It suggests that there exists a local
? ?
transformation of variables which make ?3 vanish. So, let us transform variables in
the following way:
1
yi = yi e 2 ?(t) , e?(t) dt,
? y3 =
?
1
v i = v i + 1 yi ?3 (t) e? 2 ?(t) , v3 = v3 , (2.12)
? ?
2

q = qe??(t) + 1 yi yi (?3 (t)2 ) ? 2?3 (t) e??(t) .
? t
8

As a result, we obtain the system

v3 + v j vj ? vjj + qi + ?i (?3 )?3 = 0,
?i ? ?i ?i ? ?yv
v3 + v j vj ? vjj = 0,
?3 ? 3 ?3
?i
vi = 0

for the functions v a = v a (?1 , y2 , y3 ) and q = q (?1 , y2 , y3 ). Here subscripts 1, 2, and
? ?y?? ? ?y ? ?
3 denote differentiation with respect to y1 , y2 , and y3 , accordingly. Also ?i (?3 ) =
?? ? ?y
3
?i (t)e? 2 ?(t) .

3 Reduction of the Navier–Stokes equations
to systems of PDEs in two independent variables
3.1 Ansatzes of codimension two
In this subsection we give ansatzes that reduce the NSEs to systems of PDEs in two
independent variables. The ansatzes are constructed with the subalgebrical analysis of
A(N S) (see Subsection A.3) by means of the method discribed in Section B.

u1 = (rR)?1 ((x1 ? ?x2 )w1 ? x2 w2 + x1 x3 r?1 w3 ),
1.
u2 = (rR)?1 ((x2 + ?x1 )w1 + x1 w2 + x2 x3 r?1 w3 ),
(3.1)
u3 = x3 (rR)?1 w1 ? R?1 w3 ,
p = R?2 s,

where z1 = arctan x2 /x1 ? ? ln R, z2 = arctan r/x3 , ? ? 0.
Symmetry reduction and exact solutions of the Navier–Stokes equations 189

Here and below wa = wa (z1 , z2 ), s = s(z1 , z2 ), r = (x2 + x2 )1/2 , R = (x2 + x2 +
1 2 1 2
x3 ) , ?, ?, ?, µ, and ? are real constants.
2 1/2

u1 = |t|?1/2 r?1 (x1 w1 ? x2 w2 ) + 1 t?1 x1 + x1 r?2 ,
2. 2

u2 = |t|?1/2 r?1 (x2 w1 + x1 w2 ) + 1 t?1 x2 + x2 r?2 ,
2
(3.2)
u3 = |t|?1/2 w3 + ?r?1 w2 + 1 t?1 x3 ,
2

p = |t|?1 s ? 1 r?2 + 1 t?2 R2 + ?|t|?1 arctan x2 /x1 ,
2 8

where z1 = |t|?1/2 r, z2 = |t|?1/2 x3 ? ? arctan x2 /x1 , ? ? 0, ? ? 0.

u1 = r?1 (x1 w1 ? x2 w2 ) + x1 r?2 ,
3.
u2 = r?1 (x2 w1 + x1 w2 ) + x2 r?2 ,
(3.3)
u3 = w3 + ?r?1 w2 ,
p = s ? 1 r?2 + ? arctan x2 /x1 ,
2

where z1 = r, z2 = x3 ? ? arctan x2 /x1 , ? ? {0; 1}, ? ? 0 if ? = 1 and ? ? {0; 1} if
? = 0.

u1 = |t|?1/2 (µw1 + ?w3 ) cos ? ? |t|?1/2 w2 sin ? +
4.
+ ??t?1 cos ? + 1 t?1 x1 ? ?t?1 x2 ,
2

u2 = |t|?1/2 (µw1 + ?w3 ) sin ? + |t|?1/2 w2 cos ? +
+ ??t?1 sin ? + 1 t?1 x2 + ?t?1 x1 , (3.4)
2
u3 = |t|?1/2 (??w1 + µw3 ) + µ?t?1 + 1 t?1 x3 ,
2

p = |t|?1 s ? 1 t?2 ? 2 + 1 t?2 R2 + 1 ? 2 t?2 r2 +
2 8 2
+ ?|t|?3/2 (?x1 cos ? + ?x2 sin ? + µx3 ),

where

z1 = |t|?1/2 (µx1 cos ? + µx2 sin ? ? ?x3 ),
z2 = |t|?1/2 (x2 cos ? ? x1 sin ? ),
? = ?(?x1 cos ? + ?x2 sin ? + µx3 ) + 2??(x2 cos ? ? x1 sin ? ),
? = ? ln |t|, ? > 0, µ ? 0, ? ? 0, µ2 + ? 2 = 1, ?? = 0, ? ? 0.

u1 = |t|?1/2 w1 + 1 t?1 x1 ,
5. 2

u2 = |t|?1/2 w2 + 1 t?1 x2 ,
2
(3.5)
u3 = |t|?1/2 w3 + (? + 1 )t?1 x3 ,
2

p = |t|?1 s ? 1 ? 2 t?2 x2 + 1 t?2 R2 + ?|t|?3/2 x3 ,
3
2 8

where

z1 = |t|?1/2 x1 , z2 = |t|?1/2 x2 , ? ? 0.
?? = 0,
190 W.I. Fushchych, R.O. Popovych

u1 = (µw1 + ?w3 ) cos t ? w2 sin t + ?? cos t ? x2 ,
6.
u2 = (µw1 + ?w3 ) sin t + w2 cos t + ?? sin t + x1 ,
(3.6)
u3 = (??w1 + µw3 ) + µ?,
p = s ? 1 ? 2 + 1 r2 + ?(?x1 cos t + ?x2 sin t + µx3 ),
2 2

where
z1 = (µx1 cos t + µx2 sin t ? ?x3 ),
z2 = (x2 cos t ? x1 sin t),
? = ?(?x1 cos t + ?x2 sin t + µx3 ) + 2?(x2 cos t ? x1 sin t),
µ ? 0, ? ? 0, µ2 + ? 2 = 1, ?? = 0, ? ? 0.

u1 = w 1 , u2 = w 2 , u3 = w3 + ?x3 ,
7.
(3.7)
p = s ? 1 ? 2 x2 + ?x3 ,
3
2

where

? ? {0; 1}.
z1 = x1 , z2 = x2 , ?? = 0,

u1 = x1 w1 ? x2 r?2 (w2 ? ?(t)),
8.
u2 = x2 w1 + x1 r?2 (w2 ? ?(t)),
(3.8)
u3 = (?(t))?1 (w3 + ?t (t)x3 + ? arctan x2 /x1 ),
p = s ? 1 ?tt (t)(?(t))?1 x2 + ?t (t) arctan x2 /x1 ,
3
2

where

?, ? ? C ? ((t0 , t1 ), R).
? ? {0; 1},
z1 = t, z2 = r,

u = w + ??1 (ni · x)mi ? ??1 (k · x)kt ,
9. t
(3.9)
p = s ? 1 ??1 (mi · x)(ni · x) ? 1 ??2 (mi · k)(ni · x)(k · x),
tt tt
2 2

where
mi ? C ? ((t0 , t1 ), R3 ),
z2 = (k · x),
z1 = t,
m1 · m2 ? m1 · m2 = 0, k = m 1 ? m2 , n1 = m2 ? k,
tt tt
n2 = k ? m1 , ? = ?(t) = k · k = 0 ? t ? (t0 , t1 ).

3.2 Reduced systems
Substituting ansatzes (3.1)–(3.9) into the NSEs (1.1), we obtain the following systems
of reduced equations:
w2 w1 + w3 w2 ? w1 w3 cot z2 ? (w1 )2 ? (w2 + ?w1 )2 sin2 z2 ?
1 1
1.
? (w3 )2 ? (? 2 + sin?2 z2 )w11 + w22 ? ?w1 ? 2w2 ? 2w1 ?
1 1 1 3 2

? 2w1 sin z2 + w2 cos z2 ? w1 sin?1 z2 ? (2s + ?s1 ) sin2 z2 = 0,
1
Symmetry reduction and exact solutions of the Navier–Stokes equations 191

w2 w1 + w3 w2 + w3 (w2 + 2?w1 ) cot z2 ?
2 2

? ? (w1 )2 + (w3 )2 + (w2 + ?w1 )2 sin2 z2 ?
? (? 2 + sin?2 z2 )w11 + w22 + 3?w1 + 2?(w2 + ?w1 + w1 ) sin z2 +
2 2 2 3 1

+ (2w1 + 2w1 cot z2 ? w2 ? 2?w1 ) sin?1 z2 ?
1 3

? (w2 + 2?w2 ) cos z2 + 2?s sin2 z2 + (1 + ? 2 sin2 z2 )s1 = 0,
2 1 (3.10)
w2 w1 + w3 w2 ? (w3 )2 cot z2 ? (w2 + ?w1 )2 sin z2 cos z2 ?
3 3

? (? 2 + sin?2 z2 )w11 + w22 + ?w1 + 2w2 sin z2 +
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