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i
J0a = i?a + xa ?t + (?1 + i?2 )?a .
2?
Оператори (25) породжують перетворення Лоренца для t i x, а функцiя ? пере-
творюється як
?·?
1 i
(?0 + ?3 ) ch ? + ?0 ? ?3 ? (?1 + i?2 ) sh ?
? (x ) = ?,
?
2 ?
де ?0 — одинична матриця 2 ? 2, ? = {?1 , ?2 , ?3 } — довiльнi постiйнi, ? = (?1 +
2

?2 + ?3 )1/2 .
2 2



1. Фущич В.И., Никитин А.Г., Симметрия уравнений Максвелла, Киев, Наук. думка, 1983, 200 с.
2. Fushchych W.I., Shtelen W.M., On nonlocal transformations, Lett. Nuovo Cim., 1985, 44, № 1,
40–42.
3. Штелень В.М., Нелиевская симметрия й нелокальные преобразования, Препринт № 87.6, Киев,
1987, 28 с.
4. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелиненных
уравнений математической физики, Киев, Наук. думка, 1989, 336 с.
5. Fushchych W.I., On additional invariance of Dirac and Maxwell equations, Lett. Nuovo Cim., 1974,
11, № 10, 508–511.
6. Фущич В.И., О дополнительной инвариантности уравнений Клейна–Гордона–Фока, Докл. АН
СССР, 1976, 230, № 3, 570–573.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 306–319.

Reduction and exact solutions
of the Navier–Stokes equations
W.I. FUSHCHYCH, W.M. SHTELEN, S.L. SLAVUTSKY
?
We construct a complete set of G(1, 3)-inequivalent ans?tze of codimension 1 for the
a
Navier–Stokes (NS) field which reduce the ns equations to systems of ordinary di-
fferential equations (ODE). Having solved these ODEs we thereby obtain solutions of
the NS equations. Formulae of group multiplication of solutions are given. Several non-
Lie ans?tze are discussed.
a

1. Introduction
The NS equations
?u
+ (u · ?)u ? ?u + ?p = 0, (1.1)
div u = 0,
?t
where u = u(x) = {u1 , u2 , u3 } is the velocity field of a fluid, p = p(x) is the
pressure, x = {t, x} ? R(4), ? = {?/?xa }, a = 1, 2, 3, ? is Laplacian, are basic
equations of hydrodynamics which describe motion of an incompressible viscous fluid.
The problem of finding exact solutions of nonlinear equations (1.1) is an important but
rather complicated one. Considerable progress in solving this problem can be achieved
by making use of a symmetry approach. Equations (1.1) have non-trivial symmetry
properties; it is well known (see, e.g. Birkhoff [3]) that they are invariant under the
?
extended Galilei group G(1, 3) generated by operators
? ?
?t ? , ?a ? , Ga = t?a + ?ua ,
?t ?xa (1.2)
= xa ?b ? xb ?a + ua ?ub ? ub ?ua , D = 2t?t + xa ?a ? ua ?ua ? 2p?p ,
Jab
where ?ua ? ?/?ua , ?p ? ?/?p. Recently it was shown (Ovsyannikov [12], Lloyd [11])
that the maximal, in the sense of Lie invariance algebra, of the NS equations (1.1)
?
is the direct sum of eleven-dimensional AG(1, 3) (1.2) and infinite-dimensional algeb-
ra A? with basis elements
Q = f a ?a + f?a ?ua ? xa f a ?p ,
? (1.3)
R = g?p ,
where f a = f a (t) and g = g(t) are arbitrary differentiable functions of t; dot means
differentiation with respect to t.
In this paper we systematically use symmetry properties of (1.1) to find their exact
?
solutions. In section 2 we describe the complete set of G(1, 3)-inequivalent ans?tze of
a
codimension 1
ua (t, x) = f ab (x)?b (?) + g a (x), (1.4)
p(x) = F (x)?(?),
where the functions f ab , g a and F , and new variable ? = ?(x) are determined by
?
means of operators of three-dimensional subalgebras of AG(1, 3) (1.2). We consider
J. Phys. A: Math. Gen., 1991, 24, P. 971–984.
Reduction and exact solutions of the Navier–Stokes equations 307

?
three-dimensional subalgebras of AG(1, 3) because an ansatz of the form (1.4), inva-
riant under such a subalgebra, reduces (1.1) to a system of ODE immediately. As
a rule reduced systems of ode can be solved by a standard method. (In most cases we
find the general solutions of these reduced systems of ODE). Ans?tze of the type (1.4),
a
which are obtained by means of Lie symmetry operators, we shall call Lie ans?tze. a
The method of finding exact solutions of PDE used here is based on Lie’s ideas of
invariant solutions and it is described in full detail in Fushchych et al [9].
Starting from solutions of the reduced systems of ODE (which are, of course,
solutions of the NS equations) one can construct multiparameter families of solutions
for the NS equations. To do this one has to use formulae of group multiplication of
solutions which are given at the end of section 2.
In section 3 we consider some non-Lie ans?tze for the NS field. These ans?tze
a a
cannot be obtained within the framework of the local Lie approach used in section 2.
?
2. G(1, 3)-inequivalent ans?tze of codimension 1 for the NS field
a
and exact solutions of the NS equations (1.1)
?
Let Qj ? Q1 , Q2 , Q3 be a three-dimensional subalgebra of AG(1, 3) (1.2). It
follows from (1.2) that the general form of operator Qj is
? a
(2.1)
Qj = ?j (x)?? + ?j (u)?ua + ?j (p)?p ,
?
where ? = 0, 3, ?0 ? ?/?t; ?j , ?j , ?j are linear functions of x, u, p. The explicit form
? a
?
of an ansatz (1.4) is determined as the solution of the following equations
?
?j (x)?? ?(x) = 0,
Qj [ua ? f ab (x)?b (?) ? g a (x)] = 0, (2.2)
Qj [p ? F (x)?(?)] = 0.
?
Equations (2.2) can be solved rather easily. All three-dimensional G(1, 3)-inequivalent
?
subalgebras of AG(1, 3) are found in Fushchych et al [6] and Barannik and Fush-
chych [1] with the help of the method developed by Patera et al [13]. In table 1 we
list these three-dimensional subalgebras and give corresponding invariant ans?tze of
a
the form (1.4) obtained as solutions of equations (2.2).
In this table f , g, h, ? are differentiable functions of corresponding invariant
variable ?; ? = 0 is an arbitrary constant.
Let us substitute ans?tze from table 1 into the ns equations (1.1). As a result we
a
obtain the following systems of ODE:
1? . f? = 0, g = 0, h = 0.
?
?
2? . hf? ? f = 0, hg ? g = 0, hh ? h + ? = 0, h = 0.
? ?? ?
?? ?
3? . g + hf? ? f = 0, hg ? g = 0, hh ? h + ? = 0, h = 0.
? ?? ?
?? ?
4? . f?h + 2f = 0, gh + 2? = 0, 1 ? 2hh ? 4h ? 2? = 0, h = 0.
? ? ? ?
? g ?
5? . 1 + hf? ? f = 0, g h ? g = 0, hh ? h + ? = 0, h = 0.
? ?? ?? ?
?
6? . g ? 2hf? ? 4f = 0, hg + 2? = 0, 1 ? 2hh ? 4h ? 2? = 0, h = 0.
? ? ? ?
? g ?
7? . (?f ? h)f? ? 2(?2 + 1)f + ?? = 0, (?f ? h)g ? 2(?2 + 1)? = 0,
? ? ? g
(?f ? h)h ? 2(?2 + 1)h ? ? + 1 = 0, ?f? ? h = 0.
? ? ?
? 2
?
8 . ?f?(h ? ?g) + g ? (?2 + 1)f = 0, ?g(h ? ?g) + ?? ? (?2 + 1)? = 0,
? ? ? g
? ? ?
1 ? h(h ? ?g) ? ? ? (?2 + 1)h = 0, h ? ?g = 0.
? ?
308 W.I. Fushchych, W.M. Shtelen, S.L. Slavutsky

?
Table 1. G(1, 3)-inequivalent ans?tze of codimension 1 for the NS field
a
Invariant
N Algebra Anzatz
variable ?
u1 = f (?), u2 = g(?), u3 = h(?), p = ?(?)
1 ?1 , ?2 , ?3 t
u1 = f (?), u2 = g(?), u3 = h(?), p = ?(?)
2 ?t , ?1 , ?2 x3
u1 = x2 + f (?), u2 = g(?), u3 = h(?),
3 ?t , ?1 , G1 + G2 x3
p = ?(?)
t2 ? 2x3 u1 = f (?), u2 = g(?), u3 = t + h(?), p = ?(?)
4 ?1 , ?2 , ?t + G3
u1 = t + f (?), u2 = g(?), u3 = h(?), p = ?(?)
5 ?1 , ?2 , ?t + G1 x3
t2 ? 2x3 u1 = x2 + f (?), u2 = g(?), u3 = t + h(?),
6 ?1 , ?2 + G1 ,
?t + G3 p = ?(?)
t2 + 2?x1 ? 2x3 u1 = f (?), u2 = g(?), u3 = t + h(?), p = ?(?)
7 ?1 + ??3 , ?2 ,
?t + G3
?x2 ? x3 + (t2 /2) u1 = x2 + f (?), u2 = g(?), u3 = t + h(?),
8 ?1 , ?t + G3 ,
G1 + ?2 + ??3 p = ?(?)
(x2 + x2 )1/2 u1 = x1 f (?) ? x2 g(?), u2 = x1 g(?) + x2 f (?),
9 ?t , ?3 , J12 1 2
u3 = h(?), p = ?(?)
(x2 + x2 )1/2 u1 = x1 f (?) ? x2 g(?), u2 = x1 g(?) + x2 f (?),
10 ?t + G3 , ?3 , J12 1 2
u3 = t + h(?), p = ?(?)
u1 = (1/x2 )f (?), u2 = (1/x2 )g(?),
11 ?t , ?3 , D x1 /x2
u3 = (1/x2 )h(?), p = (1/x2 )?(?)
2

ln(x2 + x2 )+ u1 = (x2 + x2 )?1 (x1 f (?) ? x2 g(?)),
12 ?t , ?3 , J12 + ?D 1 2 1 2
?1
u = (x1 + x2 )?1 (x1 g(?) + x2 f (?)),
2 2
2? tan (x1 /x2 ) 2
u = (x1 + x2 )?1/2 h(?), p = (x2 + x2 )?1 ?(?)
3 2
2 1 2

(x2 + x2 )1/2 /x3 u1 = (x2 + x2 )?1 (x1 f (?) ? x2 g(?)),
13 ?t , J12 , D 1 2 1 2
u2 = (x2 + x2 )?1 (x1 g(?) + x2 f (?)),
1 2
u3 = (x2 + x2 )?1/2 h(?), p = (x2 + x2 )?1 ?(?)
1 2 1 2

(x2 + x2 )1/2 /t u1 = (1/t)(x1 f (?) ? x2 g(?)),
14 ?3 , J12 , D 1 2
u2 = (1/t)(x1 g(?) + x2 f (?)),
v
u3 = (1/ t)h(?), p = (1/t)?(?)
(x2 + x2 )1/2 /t u1 = (1/t)(x1 f (?) ? x2 g(?)),
15 G3 , J12 , D 1 2
u2 = (1/t)(x1 g(?) + x2 f (?)),
v
u3 = (1/ t)h(?) + (x3 /t), p = (1/t)?(?)
v v v
u1 = (1/vt)f (?), u2 = (1/ t)g(?),
16 ?t , ?2 , D x3 / t
u3 = (1/ t)h(?), p = (1/t)?(?)
v v
u1 = (1/vt)f (?) + (?x2 /t),
17 ?t , D, G2 + ?G1 x3 / t
u2 = (1/vt)g(?) + (x2 /t),
u3 = (1/ t)h(?), p = (1/t)?(?)
v v
u1 = (1/vt)f (?) + (x1 /t),
18 G1 , G2 , D x3 / t
u2 = (1/vt)g(?) + (x2 /t),
u3 = (1/ t)h(?), p = (1/t)?(?)
v v v
u1 = (1/vt)f (?), u2 = (1/ t)g(?) + (x2 /t),
19 ?1 , G2 , D x3 / t
u3 = (1/ t)h(?), p = (1/t)?(?)
Reduction and exact solutions of the Navier–Stokes equations 309

1 3 3
9? . f 2 ? g 2 + ?f f? + ? = f? + f , 2f g + ?f g = g + g ,
?
? ? ??
? ? ?
? 1?
?f h = h + h, 2f + ? f? = 0.
?
?
1 3 3
10? . f 2 ? g 2 + ?f f? + ? = f? + f , 2f g + ?f g = g + g ,
?
? ? ??
? ? ?
? 1?
1 + ?f h = h + h, 2f + ? f? = 0.
?
?
11? . ? ? g(f + ? f?) + ? = 2(1 + ?)f + ?(2f? + ? f ),
?

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