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u3 = v 3 + ?(t)r?1 v 2 + ?t (t) arctan x2 /x1 ,
p = q ? 1 ?tt (t)(?(t))?1 x2 ? 1 r?2 + ?(t) arctan x2 /x1 ,
3
2 2

where
?, ? ? C ? ((t0 , t1 ), R).
y3 = x3 ? ?(t) arctan x2 /x1 ,
y1 = t, y2 = r,
Note 2.1 The expression for the pressure p from ansatz (2.3) is indeterminate in the
points t ? (t0 , t1 ) where ?(t) = 0. If there are such points t, we will consider ansatz
(2.3) on the intervals (tn , tn ) that are contained in the interval (t0 , t1 ) and that satisfy
01
one of the conditions:
a) ?(t) = 0 ? t ? (tn , tn );
01

? t ? (tn , tn ).
b) ?(t) = 0 01

In the last case we consider ?tt /? := 0.
184 W.I. Fushchych, R.O. Popovych

u = v i ni + (m · m)?1 v 3 m + (m · m)?1 (m · x)mt ? yi ni ,
4. t

p = q ? 3 (m · m)?1 ((mt · ni )yi )2 ? (m · m)?1 (mtt · x)(m · x) + (2.4)
2

+ 1 (mtt · m)(m · m)?2 (m · x)2 ,
2

where
m, ni ? C ? ((t0 , t1 ), R3 ).
yi = ni · x, y3 = t,

ni · m = n1 · n2 = n1 · n2 = 0, |ni | = 1. (2.5)
t

Note 2.2 There exist vector-functions ni which satisfy conditions (2.5). They can be
constructed in the following way: let us fix the vector-functions k i = k i (t) such that
k i · m = k 1 · k 2 = 0, |k i | = 1, and set

n1 = k 1 cos ?(t) ? k 2 sin ?(t),
(2.6)
n2 = k 1 sin ?(t) + k 2 cos ?(t).

n1 · n2 = kt · k 2 ? ?t = 0 if ? = (kt · k 2 )dt.
1 1
Then t

2.2 Reduced systems
1–2. Substituting ansatzes (2.1) and (2.2) into the NSEs (1.1), we obtain reduced
systems of PDEs with the same general form
v a va ? vaa + q1 + ?1 v 2 = 0,
1 1

v a va ? vaa + q2 ? ?1 v 1 = 0,
2 2
(2.7)
v a va ? vaa + q3 = 0,
3 3

a
va = ?2 .
Hereafter subscripts 1, 2, and 3 of functions denote differentiation with respect to y1 ,
y2 , and y3 , accordingly. The constants ?i take the values
?1 = ?2?, ?2 = ? 3 3
1. if t > 0, if
?1 = 2?, ?2 = t < 0.
2 2
?1 = ?2?,
2. ?2 = 0.
For ansatzes (2.3) and (2.4) the reduced equations have the form
?1 ?2 1 ?2 2
v1 + v 1 v2 + v 3 v3 ? y2 v 2 v 2 ? v22 + (1 + ? 2 y2 )v33 ? 2?y2 v3 + q2 = 0,
1 1 1 1
3.
?1 ?2 2
v1 + v 1 v2 + v 3 v3 + y2 v 1 v 2 ? v22 + (1 + ? 2 y2 )v33 +
2 2 2 2

?2 1 ?2 ?1 ?1
+ 2?y2 v3 + 2y2 v 2 ? ?y2 q3 + ?y2 = 0,
(2.8)
?2 3 ?3 1 ?1
v1 + v 1 v2 + v 3 v3 ? v22 + (1 + ? 2 y2 )v33 ? 2? 2 y2 v3 + 2?1 y2 v 2 +
3 3 3 3

?1 ?1 ?2 ?2
+ 2?y2 (y2 v 2 )2 + (1 + ? 2 y2 )q3 ? ?11 ? ?1 y3 ? ??y2 = 0,
?1
y2 v 1 + v2 + v3 = 0.
1 3

v3 + v j vj ? vjj + qi + ?i (y3 )v 3 = 0,
i i i
4.
v3 + v j vj ? vjj = 0,
3 3 3
(2.9)
vi + ?3 (y3 ) = 0,
i
Symmetry reduction and exact solutions of the Navier–Stokes equations 185

where
?i = ?i (y3 ) = 2(m · m)?1 (mt · ni ),
(2.10)
?3 = ?3 (y3 ) = (m · m)?1 (mt · m).

2.3 Symmetry of reduced systems
Let us study symmetry properties of systems (2.7), (2.8), and (2.9). All results of this
subsection are obtained by means of the standard Lie algorithm [28, 27]. First, let us
consider system (2.7).

Theorem 2.1 The MIA of system (2.7) is the algebra
1
?a , ?q , J12 if ?1 = 0;
a)
1
?a , ?q , Jab if ?1 = 0, ?2 = 0;
b)
1 1
?a , ?q , Jab , D1 if ?1 = ?2 = 0.
c)

Here Jab = ya ?b ? yb ?a + v a ?vb ? v b ?va , D1 = ya ?a ? v a ?va ? 2q?q .
1 1

Note 2.3 All Lie symmetry operators of (2.7) are induced by operators from A(N S):
1 1
The operators Jab and D1 are induced by Jab and D. The operators ca ?a (ca = const)
and ?q are induced by either

Z(|t|?1 ),
R(|t|1/2 (c1 cos ? ? c2 sin ?, c1 sin ? + c2 cos ?, c3 )),

where ? = ? ln |t|, for ansats (2.1) or

R(c1 cos ?t ? c2 sin ?t, c1 sin ?t + c2 cos ?t, c3 ), Z(1)

for ansatz (2.2), respectively. Therefore, Lie reductions of system (2.7) give only
solutions that can be obtained by reducing the NSEs with two- and three-dimensional
subalgebras of A(N S).

Let us continue to system (2.8). We denote Amax as the MIA of (2.8). Studying
symmetry properties of (2.8), one has to consider the following cases:
A. ?, ? ? 0. Then

Amax = ? 1 , D2 , R1 (?(y1 )), Z 1 (?(y1 )) ,
1

where
D2 = 2y1 ?1 + y2 ?2 + y3 ?3 ? v a ?va ? 2q?q ,
1

R1 (?(y1 )) = ??3 + ?1 ?v3 ? ?11 y3 ?q , Z 1 (?(y1 )) = ?(y1 )?q .

Here and below ? = ?(y1 ) and ? = ?(y1 ) are arbitrary smooth functions of y1 = t.
B. ? ? 0, ? ? 0. In this case an extension of Amax exists for ? = (C1 y1 + C2 )?1 ,
where C1 , C2 = const. Let C1 = 0. We can make C2 vanish by means of equivalence
?1
transformation (A.6), i.e., ? = Cy1 , where C = const. Then

Amax = D2 , R1 (?(y1 )), Z 1 (?(y1 )) .
1
186 W.I. Fushchych, R.O. Popovych

If C1 = 0, ? = C = const and

Amax = ?1 , R1 (?(y1 )), Z 1 (?(y1 )) .

For other values of ?, i.e., when ?11 ? = ?1 ?1 ,

Amax = R1 (?(y1 )), Z 1 (?(y1 )) .

C. ? = 0. By means of equivalence transformation (A.6) we make ? = 0. In this
case an extension of Amax exists for ? = ±|C1 y1 + C2 |1/2 , where C1 , C2 = const. Let
C1 = 0. We can make C2 vanish by means of equivalence transformation (A.6), i.e.,
? = C|y1 |1/2 , where C = const. Then

Amax = D2 , R2 (|y1 |1/2 ), R2 (|y1 |1/2 ln |y1 |), Z 1 (?(y1 )) ,
1

where R2 (?(y1 )) = ??3 + ?1 ?v3 . If C1 = 0, i.e., ? = C = const,

Amax = ? 1 , ?3 , y1 ?3 + ?v3 Z 1 (?(y1 )) .

For other values of ?, i.e., when (? 2 )11 = 0,

Amax = R2 (?(y1 )), R2 (?(y1 ) (?(y1 ))?2 dy1 ), Z 1 (?(y1 )) .

Note 2.4 In all cases considered above the Lie symmetry operators of (2.8) are
induced by operators from A(N S): The operators ?1 , D2 , and Z 1 (?(y1 )) are induced
1

by ?t , D, and Z(?(t)), respectively. The operator R(0, 0, ?(t)) induces the operator
R1 (?(y1 )) for ? ? 0 and the operator R2 (?(y1 )) (if ?11 ? ? ??11 = 0) for ? = 0.
Therefore, the Lie reduction of system (2.8) gives only solutions that can be obtained
by reducing the NSEs with two- and three-dimentional subalgebras of A(N S).

When ? = ? = 0, system (2.8) describes axially symmetric motion of a fluid and
can be transformed into a system of two equations for a stream function ?1 and a
function ?2 that are determined by

?1 = ?y2 v 3 ,
?1 = y2 v 1 , ?2 = y2 v 2 .
3 2

The transformed system was studied by L.V. Kapitanskiy [20, 21].
Consider system (2.9). Let us introduce the notations
t = y3 , ? = ?(t) = ?3 (t)dt,
R3 (? 1 (t), ? 2 (t)) = ? i ?yi + ?t ?vi ? ?tt yi ?q ,
i i

Z 1 (?(t)) = ?(t)?q , S = ?v3 ? ?i (t)yi ?q ,
E(?(t)) = 2??t + ?t yi ?yi + (?tt yi ? ?t v i )?vi ? (2?t q + 1 ?ttt yj yj )?q ,
2
J12 = y1 ?2 ? y2 ?1 + v 1 ?v2 ? v 2 ?v1 .
1

Theorem 2.2 The MIA of(2.9) is the algebra

1) R3 (? 1 (t), ? 2 (t)), Z 1 (?(t)), S, E(?1 (t)), E(?2 (t)), v 3 ?v3 , J12 ,
1

where ?1 = e??(t) e?(t) dt and ?2 = e??(t) , if ?i = 0;

2) R3 (? 1 (t), ? 2 (t)), Z 1 (?(t)), S, E(?(t)) + 2a1 v 3 ?v3 + 2a2 J12 ,
1
Symmetry reduction and exact solutions of the Navier–Stokes equations 187

where a1 , a2 , and a3 are fixed constants, ? = e??(t) e?(t) dt + a3 , if
3 3
?1 = e 2 ? ?? 2 ?a1 C1 cos(a2 ln ?) ? C2 sin(a2 ln ?) ,
? ? ?
3 3
?2 = e 2 ? ?? 2 ?a1 C1 sin(a2 ln ?) + C2 cos(a2 ln ?)
? ? ?

with ? = ?(t) = | e?(t) dt + a3 |, C1 , C2 = const, (C1 , C2 ) = (0, 0);
??

3) R3 (? 1 (t), ? 2 (t)), Z 1 (?(t)), S, E(?(t)) + 2a1 v 3 ?v3 + 2a2 J12 ,
1

where a1 and a2 are fixed constants, ? = e??(t) , if
3
?1 = e 2 ??a1 ? C1 cos(a2 ?) ? C2 sin(a2 ?) ,
?
? ?
3
?2 = e 2 ??a1 ? C1 sin(a2 ?) + C2 cos(a2 ?)
?
? ?

e?(t) dt, C1 , C2 = const, (C1 , C2 ) = (0, 0);
with ? = ?(t) =
??

4) R3 (? 1 (t), ? 2 (t)), Z 1 (?(t)), S

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