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equal to 1. Some similarity solutions of these equations were constructed in [22]. The
particular case of system (6.1) for ?1 = ?2 = ?4 = ?5 = 0 and ?3 = 1 was considered
in [31].
In this section we study symmetry properties of system (6.1) and construct large
sets of its exact solutions.
Theorem 6.1 The MIA of (6.1) is the algebra
E1 = ?1 , ?2 , ?s if ?1 = 0, ?4 = 0.
1.
E2 = ?1 , ?2 , ?s , ?w3 ? ?1 z2 ?s if ?1 = 0, ?4 = 0, (?1 , ?2 , ?5 ) = (0, 0, 0).
2.
?
E3 = ?1 , ?2 , ?s , ?w3 ? ?1 z2 ?s , D ? 3w3 ?w3 if ?1 = 0, ?k = 0, k = 2, 5.
3.
E4 = ?1 , ?2 , ?s , J, (w3 + ?5 /?4 )?w3 if ?1 = 0, ?4 = 0.
4.
E5 = ?1 , ?2 , ?s , J, ?w3 if ?1 = ?4 = 0, (?2 , ?3 ) = (0, 0), ?5 = 0.
5.
E6 = ?1 , ?2 , ?s , J, ?w3 , w3 ?w3 if ?1 = ?4 = ?5 = 0, (?2 , ?3 ) = (0, 0).
6.
?
E7 = ?1 , ?2 , ?s , J, ?w3 , D + 2w3 ?w3 if ?5 = 0, ?l = 0, l = 1, 4.
7.
?
E8 = ?1 , ?2 , ?s , J, ?w3 , D, w3 ?w3 if ?n = 0, n = 1, 5.
8.
?
Here D = zi ?i ? wi ?wi ? 2s?s , J = z1 ?2 ? z2 ?1 + w1 ?w2 ? w2 ?w1 , ?i = ?zi .
Note 6.1 The bases of the algebras E6 and E8 contain the operator w3 ?w3 that is not
induced by elements of A(N S).
Note 6.2 If ?4 = 0, the constant ?5 can be made to vanish by means of local
transformation
?1
s = s ? ?1 ?5 ?4 z2 ,
w3 = w3 + ?5 /?4 , (6.2)
? ?
where the independent variables and the functions wi are not transformed. Therefore,
we consider below that ?5 = 0 if ?4 = 0.
Note 6.3 Making the non-local transformation
(6.3)
s = s + ?2 ?,
?
where ?1 = w2 , ?2 = ?w1 (such a function ? exists in view of the last equation of
(6.1)), in system (6.1) with ?3 = 0, we obtain a system of form (6.1) with ?3 = ?2 = 0.
? ?
In some cases (?1 = 0, ?3 = ?4 = ?5 = 0, ?2 = 0; ?1 = ?3 = ?4 = 0, ?2 = 0)
transformation (6.3) allows the symmetry of (6.1) to be extended and non-Lie solutions
to be constructed. Moreover, it means that in the cases listed above system (6.1) is
invariant under the non-local transformation
wi = e?? wi , s = e?2? s + ?2 (e?2? ? 1)?,
zi = e? zi , w3 = e?? w3 ,
? ? ? ?
where
? = ?3 if ?3 = ?4 = ?5 = 0, ?1 , ?2 = 0;
? = 2 if ?1 = ?3 = ?4 = 0, ?2 , ?5 = 0;
? = 0 if ?1 = ?3 = ?4 = ?5 = 0, ?2 = 0.
Symmetry reduction and exact solutions of the Navier–Stokes equations 213

Let us consider an ansatz of the form:
w1 = a1 ?1 ? a2 ?3 + b1 ?2 ,
w2 = a2 ?1 + a1 ?3 + b2 ?2 ,
(6.4)
w3 = ?2 + b3 ?2 ,
s = h + d1 ? 2 + d 2 ? 1 ? 2 + 1 d3 ? 2 ,
2
2

where a2 + a2 = 1, ? = ?1 = a1 z2 ? a2 z1 , ?2 = a1 z1 + a2 z2 , B, ba , da = const,
1 2

bi = Bai , b3 (B + ?4 ) = 0,
(6.5)
d2 = ?2 B ? ?1 b3 a1 , d3 = ?B 2 ? ?1 b3 a2 ,

Here and below ?a = ?a (?) and h = h(?). Indeed, formulas (6.4) and (6.5) determine
a whole set of ansatzes for system (6.1). This set contains both Lie ansatzes, construc-
ted by means of subalgebras of the form

a1 ?1 + a2 ?2 + a3 (?w3 ? ?1 z2 ?s ) + a4 ?s , (6.6)

and non-Lie ansatzes. Equation (6.5) is the necessary and sufficient condition to
reduce (6.1) by means of an ansatz of form (6.3). As a result of reduction we obtain
the following system of ODEs:
?3 ?1 ? ?1 + µ1j ?j + d1 + d2 ? + ?2 ?3 = 0,
? ??
? ?? ? ?2 + µ2j ?j + ?5 = 0,
32
??
(6.7)
? ?? ? ?3 + h? ? ?2 ?1 + ?1 a1 ?2 = 0,
33
??
3
?? = ?,
where µ11 = ?B, µ12 = ??1 a2 , µ21 = ?b3 , µ22 = ??4 , ? = ?3 ? B. If ? = 0, system
(6.7) implies that
?3 = C0 = const,
h = ?2 ?1 (?)d? ? ?1 a1 ?2 (?)d?,

and the functions ?i satisfy system (4.23), where ?11 = d1 +?2 C0 , ?21 = d2 , ?12 = ?5 ,
?22 = 0. If ? = 0, then ?3 = ?? (translating ?, the integration constant can be made
to vanish),
h = ? 1 ? 2 ? 2 + ?2 ?1 (?)d? ? ?1 a1 ?2 (?)d?,
2

and the functions satisfy system (4.29), where ?11 = d1 , ?21 = d2 + ?2 ?, ?12 = ?5 ,
?22 = 0.

Note 6.4 Step-by-step reduction of the NSEs (1.1) by means of ansatzes (3.4)–(3.7)
and (6.4) is equivalent to a particular case of immediate reduction of the NSEs (1.1)
to ODEs by means of ansatzes 5 and 6 from Subsection 4.1.
214 W.I. Fushchych, R.O. Popovych

Table 1. Complete sets of inequivalent one-dimensional subalgebras of the algebras
E1 ? E8 (a and al (l = 1, 4) are real constants)
Values of
Algebra Subalgebras
parameters

a2 + a2 = 1
E1 a1 ?1 + a2 ?2 + a3 ?s , ?s 1 2

a2 + a2 = 1,
a1 ?1 + a2 ?2 + a3 (?w3 ? ?1 z2 ?s ) , 1 2
E2
?1 + a4 ?s , ?w3 ? ?1 z2 ?s , ?s a4 = 0

a2 + a2 = 1,
a1 ?1 + a2 ?2 + a3 (?w3 ? ?1 z2 ?s ) , ?1 + a4 ?s , 1 2
a3 ? {?1; 0; 1},
E3
?
D ? 3w3 ?w3 , ?w3 ? ?1 z2 ?s , ?s a4 ? {?1; 1}

J + a1 ?s + a2 w3 ?w3 , ?2 + a1 ?s + a2 w3 ?w3 ,
E4
w3 ?w3 + a1 ?s , ?s

J + a1 ?s + a2 ?w3 , ?2 + a1 ?s + a2 ?w3 ,
E5
?w3 + a1 ?s >, < ?s

J + a1 ?s + a2 w3 ?w3 , ?2 + a1 ?s + a2 w3 ?w3 ,
a2 = 0,
E6 J + a1 ?s + a3 ?w3 , ?2 + a1 ?s + a3 ?w3 , a3 ? {?1; 0; 1}
w3 ?w3 + a1 ?s , ?w3 + a1 ?s , ?s

a2 ? {?1; 0; 1},
?
D + aJ + 2w3 ?w3 , J + a1 ?s + a2 ?w3 ,
a1 ? {?1; 0; 1}
E7
?2 + a1 ?s + a2 ?w3 , ?w3 + a2 ?s , ?s if a2 = 0

? ?
D + aJ + a3 w3 ?w3 , D + aJ + a3 ?w3 ,
ai ? {?1; 0; 1},
J + a1 ?s + a4 w3 ?w3 , ?2 + a1 ?s + a4 w3 ?w3 ,
E8
a4 = 0
J + a1 ?s + a2 ?w3 , ?2 + a1 ?s + a2 ?w3 ,
w3 ?w3 + a1 ?s , ?w3 + a1 ?s , ?s
Symmetry reduction and exact solutions of the Navier–Stokes equations 215

Now let us choose such algebras, among the algebras from Table 1, that can be
used to reduce system (6.1) and do not belong to the set of algebras (6.6). By means
of the chosen algebras we construct ansatzes that are tabulated in the form of Table 2.

Table 2. Ansatzes reducing system (6.1) (r = (z1 + z2 )1/2 )
2 2

Values Invariant
N Algebra Ansatz
of ?n variable

w1 = r?2 (z1 ?1 ? z2 ?2 ),
?1 = 0,
?
D ? 3w3 ?w3 ? = arctan z2 w2 = r?2 (z2 ?1 + z1 ?2 ),
1 ?k = 0, z1
w3 = r?3 ?3 , s = r?2 h
k = 2, 5

w1 = ?1 , w2 = ?2 ,
?1 = 0, ?2 + a1 ?s + a2 w3 ?w3 ,
2 w3 = ?3 ea2 z2 ,
? = z1
?5 = 0 a2 = 0
s = h + a1 z2

w1 = z1 ?1 ? z2 r?2 ?2 ,
w2 = z2 ?1 + z1 r?2 ?2 ,
?1 = 0,
3 J + a1 ?s + a2 ?w3 ?=r
w3 = ?3 + a2 arctan z2 ,
?4 = 0 z1
s = h + a1 arctan z2
z1

w1 = z1 ?1 ? z2 r?2 ?2 ,
w2 = z2 ?1 + z1 r?2 ?2 ,
?1 = 0, J + a1 ?s + a2 w3 ?w3
4 ?=r z2
a2 = 0 if ?4 = 0 w3 = ?3 ea2 arctan z1 ,
?5 = 0
s = h + a1 arctan z2z1

?2
? = arctan z2 ? w2 = r?2 (z1 ?1 ? z2 ?2 ),
1 1 2
?5 = 0,
?
D + aJ + 2w3 ?w3 z1
5 ?l = 0, w = r (z2 ? + z1 ? ),
?a ln r
w3 = r2 ?3 , s = r?2 h
l = 1, 4

w1 = r?2 (z1 ?1 ? z2 ?2 ),
? = arctan z2 ? w2 = r?2 (z2 ?1 + z1 ?2 ),
?n = 0, ? z1
6 D + aJ + a1 ?w3
?a ln r w3 = ?3 + a1 ln r,
n = 1, 5
s = r?2 h

?2
? = arctan z2 ? w2 = r?2 (z1 ?1 ? z2 ?2 ),
1 1 2
?
D + aJ + a1 w3 ?w3 ,
?n = 0, z1
7 w = r (z2 ? + z1 ? ),
?a ln r
n = 1, 5 a1 = 0
w3 = ra1 ?3 , s = r?2 h
216 W.I. Fushchych, R.O. Popovych

Substituting the ansatzes from Table 2 into system (6.1), we obtain the reduced
systems of ODEs in the functions ?a and h:

?2 ?1 ? ?1 ? ?1 ?1 ? ?2 ?2 ? 2h + ?1 ?3 sin ? + 2?2 = 0,
1. ? ?? ?
? ?? ? ??? + h? ? 2?? + ?1 ? cos ? = 0,
22 2 1 3
(6.8)
?2 ?3 ? ?3 ? 3?1 ?3 ? 9?3 = 0,
? ??
2
?? = 0.

?1 ?1 ? ?1 + ?2 ?2 + h? = 0,
2. ? ??
? ?? ? ?2 ? ?2 ?1 + a1 = 0,
12
??
(6.9)
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