стр. 44 |

+ 2 3 ?1 ?2 ?22 + 3 (?2 ? 1)2 ?2 ?3 ?33 ? 3 (?2 ? 1) ?

2

?1 ?3 ?1 ?3

1 2 1 1

? ?2 + 2 ?2 ?12 ? 3 ?2 (?2 ? 1) ?2 + 2 ?23 ? 3 3 (?2 ? 1)2 ?2 .

1

?1 2 3 2

?1 ?3 ?1 ?3

1 x3 x0

= ?(?1 , ?2 , ?3 ) ? , ?1 = (x2 + x2 )1/2 ,

6.8 arcsin 1 2

c ?3 ?

x0 x2

? arcsin , ?3 = (x2 + u2 )1/2 ,

?2 = 3

? 2 + x2

x1 2

1 1

0 < c < 1, ? ? R, ? > 0, A = ?3 + 2 ?1 ,

?1 c ?3

1 1 2

B1 = 2 3 ?2 + 2 ?1 ?2 ?3 ? 2 ?1 ?2 ?3 ? ?3 ?3 ,

2 2 1

c ?1 ?3 ?1 ?1

1 2

B2 = ?(?2 ? 1)2 ?1 ?3 , B3 = 2 3 ? c2 ? 2 ?1 ?3 ? ?1 ?3 ,

3

c ?1 ?3 3

12 1

? 2 ? 2 ? 2 + ? 2 ? 2 ? ? 2 ?3 ,

B4 = 1 1

2

?1 ?1

1 1 1

B5 = ??2 ?2 ? 2 2 ?2 ? 2 ?2 ?2 ? 2 2 2 ?2 ,

13 1 3

c ?3 ?1 c ?1 ?3

1 1 1

?2 ? 2 2 + ?2 ?2 ? ?2 ?1 , P = 2 3 (?2 ? 1)2 ?1 ?11 +

B6 = 3 3

2

c2 ?3 c ?3 c ?3

1 22 2 1 2

?2 ?3 + 2 3 2 ?2 ?3 + 2 3 ?3 + 2 2 3 ?1 ?2 ?22 +

+

c ?1 ?3 2 c ?3 1 c ?1 ?3

3

?1

1 2 1

+ 3 (?2 ? 1)2 ?2 ?3 ?33 ? 2 3 (?2 ? 1) ?2 + 2 ?2 ?12 ?

2 1

?1 c ?3 ?1

2 1 1

? 3 ?2 (?2 ? 1) ?2 + 2 2 ?23 ? 2 3 3 (?2 ? 1)2 ?2 .

2 3 2

?1 c ?3 c ?1 ?3

Анзаци (6.1)–(6.8) можна записати у виглядi (3), де ? = ?(x, u) = (?1 (x, u),

?2 (x, u), ?3 (x, u) — iнварiанти пiдгруп групи P (1, 4).

1. Фущич В.И., Серов Н.И., Симметрия и некоторые точные решения многомерного уравнения

Монжа–Ампера, Докл. АН СССР, 1983, 273, № 3, 543–546.

2. Fushchych W., Shteten W., Serov N., Symmetry analysis and exact solutions of equations of nonli-

near mathematics physics, Dordrecht, Kluwer Academic Publishers, 1993, 435 p.

3. Федорчук В.М., Непрерывные подгруппы неоднородной группы де Ситтера P (1, 4), Препринт

№ 78.18, Киев, Ин-т математики АН УССР, 1978, 36 c.

4. Федорчук В.М., Расщепляющиеся подалгебры алгебры Ли обобщенной группы Пуанкаре

P (1, 4), 1979, Укр. мат. журн., 31, № 6, 717–722.

5. Федорчук В.М., Нерасщепляющиеся подалгебры алгебры Ли обобщенной группы Пуанкаре

P (1, 4), Укр. мат. журн., 1981, 33, № 5, 696–700.

6. Федорчук В.М., Фущич В.И., О подгруппах обобщенной группы Пуанкаре, в кн. Теоретико-

групповые методы в физике, Т. 1, М., Наука, 1980, 61–66.

7. Fushchych W.I., Barannik A.F., Barannik L.F., Fedorchuk V.M., Continious subgroups of the Poin-

car? group P (1, 4), J. Phys. A: Math. Gen., 1985, 18, № 14, 2893–2899.

e

W.I. Fushchych, Scientific Works 2003, Vol. 5, 180–239.

Symmetry reduction and exact solutions

of the Navier–Stokes equations

W.I. FUSHCHYCH, R.O. POPOVYCH

Ansatzes for the Navier–Stokes field are described. These ansatzes reduce the Navier–

Stokes equations to system of differential equations in three, two, and one independent

variables. The large sets of exact solutions of the Navier–Stokes equations are

constructed.

1 Introduction

The Navier–Stokes equations (NSEs)

ut + (u · ?)u ? u + ?p = 0,

(1.1)

div u = 0

which describe the motion of an incompressible viscous fluid are the basic equations

of modern hydrodynamics. In (1.1) and below u = {ua (t, x)} denotes the velocity

field of a fluid, p = p(t, x) denotes the pressure, x = {xa }, ?t = ?/?t, ?a = ?/?xa ,

? = {?a }, = ? · ? is the Laplacian, the kinematic coefficient of viscosity and fluid

density are set equal to unity. Repead indices denote summation whereby we consider

the indices a, b to take on values in {1, 2, 3} and the indices i, j to take on values in

{1, 2}.

The problem of finding exact solutions of non-linear equations (1.1) is an important

but rather complicated one. There are some ways to solve it. Considerable progress in

this field can be achieved by means of making use of a symmetry approach. Equations

(1.1) have non-trivial symmetry properties. It was known long ago [37, 2] that they

are invariant under the eleven-parametric extended Galilei group. Let us denote it by

G1 (1, 3). This group includes the Galilei group and scale transformations. The Lie

algebra AG1 (1, 3) of G1 (1, 3) is generated by the operators

P0 , Jab , D, Pa , Ga ,

where

D = 2t?t + xa ?a ? ua ?ua ? 2p?p ,

P 0 = ?t ,

Jab = xa ?b ? xb ?a + ua ?ub ? ub ?ua , a = b,

Ga = t?a + ?ua , Pa = ?a .

Relatively recently it was found by means of the Lie method [8, 5, 26] that the

maximal Lie invariance algebra (MIA) of the NSEs (1.1) is the infinite-dimensional

algebra A(N S) with the basis elements

(1.2)

?t , D, Jab , R(m), Z(?),

J. Nonlinear Math. Phys., 1994, 1, № 1, P. 75–113; № 2, P. 156–188.

Symmetry reduction and exact solutions of the Navier–Stokes equations 181

where

R(m) = R(m(t)) = ma (t)?a + ma (t)?ua ? ma (t)xa ?p , (1.3)

t tt

(1.4)

Z(?) = Z(?(t)) = ?(t)?p ,

ma = ma (t) and ? = ?(t) are arbitrary smooth functions of t (degree of their

smoothness is discussed in Note A.1).

The algebra AG1 (1, 3) is a subalgebra of A(N S). Indeed, setting ma = ?ab , where

b is fixed, we obtain R(m) = ?b , and if ma = ?ab t then R(m) = Gb . Here ?ab is the

Kronecker symbol (?ab = 1 if a = b, ?ab = 0 if a = b).

Operators (1.2) generate the following invariance transformations of system (1.1):

?t : u(t, x) = u(t + ?, x),

? p(t, x) = p(t + ?, x)

?

(translations with respect to t),

u(t, x) = Bu(t, B T x), p(t, x) = p(t, B T x)

Jab : ? ?

(space rotations),

u(t, x) = e? u(e2? t, e? x), p(t, x) = e2? p(e2? t, e? x)

D: ? ?

(1.5)

(scale transformations),

R(m) : u(t, x) = u(t, x ? m(t)) + mt (t),

?

p(t, x) = p(t, x ? m(t)) ? mtt · x ? 1 m · mtt

? 2

(these transformations include the space translations

and the Galilei transformations),

Z(?) : u(t, x) = u(t, x),

? p(t, x) = p(t, x) + ?(t).

?

Here ? ? R, B = {?ab } ? O(3), i.e. BB T = {?ab }, B T is the transposed matrix.

Besides continuous transformations (1.5) the NSEs admit discrete transformations

of the form

t = t, xa = xa , a = b, xb = ?xb ,

? ? ?

(1.6)

p = p, ua = ua , a = b, ub = ?ub ,

? ? ?

where b is fixed. Invariance under transformations (1.5) and (1.6) means that (u, p) is

??

a solution of (1.1) if (u, p) is a solution of (1.1).

A complete review of exact solutions found for the NSEs before 1963 is contained

in [1]. We should like also to mark more modern reviews [16, 7, 36] despite their

subjects slightly differ from subjects of our investigations. To find exact solutions

of (1.1), symmetry approach in explicit form was used in [2, 31, 32, 6, 20, 21, 4,

17, 15, 12, 10, 11, 30]. This article is a continuation and a extention of our works

[15, 12, 10, 11, 30]. In it we make symmetry reduction of the NSEs to systems

of PDEs in three and two independent variables and to systems of ODEs, using

subalgebraic structure of A(N S). We investigate symmetry properties of the reduced

systems of PDEs and construct exact solutions of the reduced systems of ODEs when

it is possible. As a result, large classes of exact solutions of the NSEs are obtained.

182 W.I. Fushchych, R.O. Popovych

The reduction problem for the NSEs is to describe ansatzes of the form [9]:

ua = f ab (t, x)v b (?) + g a (t, x), p = f 0 (t, x)q(?) + g 0 (t, x) (1.7)

that reduce system (1.1) in four independent variables to systems of differential equati-

ons in the functions v a and q depending on the variables ? = {?n } (n = 1, N ), where

N takes on a fixed value from the set {1, 2, 3}. In formulas (1.7) f ab , g a , f 0 , g 0 , and

?n are smooth functions to be described. In such a general formulation the reducti-

on problem is too complex to solve. But using Lie symmetry, some ansatzes (1.7)

reducing the NSEs can be obtained. According to the Lie method, first a complete

set of A(N S)-inequivalent subalgebras of dimension M = 4 ? N is to be constructed.

For N = 3, N = 2, and N = 1 such sets are given in Subsections A.2, A.3, and

A.4, correspondingly. Knowing subalgebraic structure of A(N S), one can find explicit

forms for the functions f ab , g a , f 0 , g 0 , and ?n and obtain reduced systems in the

functions v k and q. This is made in Section 2 (N = 3), Section 3 (N = 2) and Secti-

on 4 (N = 1). Moreover, in Subsection 2.3 symmetry properties of all reduced systems

of PDEs in three independent variables are investigated, and in Subsection 4.3 exact

solutions of the reduced systems of ODEs are constructed. Symmetry properties and

exact solutions of some reduced systems of PDEs in two independent variables are

discussed in Sections 5 and 6. In Section 7 we make symmetry reduction of a some

reduced system of PDEs in three independent variables.

In conclusion of the section, for convenience, we give some abbreviations, notati-

ons, and default rules used in this article.

Abbreviations:

the NSEs: the Navier–Stokes equations

the MIA: the maximal Lie invariance algebra (of either a some equation or a some

system of equations)

a ODE: a ordinary differential equation

a PDE: a partial differential equation

Notations:

C ? ((t0 , t1 ), R): the set of infinite-differentiable functions from (t0 , t1 ) into R, where

?? ? t0 < t1 ? +?

C ? ((t0 , t1 ), R3 ): the set of infinite-differentiable vector-functions from (t0 , t1 ) into

R3 , where ?? ? t0 < t1 ? +?

?t = ?/?t , ?a = ?/?xa , ?ua = ?/?ua , . . .

Default rules:

Repead indices denote summation whereby we consider the indices a, b to take on

values in {1, 2, 3} and the indices i, j to take on values in {1, 2}.

All theorems on the MIAs of PDEs are proved by means of the standard Lie

algorithm.

Subscripts of functions denote differentiation.

Symmetry reduction and exact solutions of the Navier–Stokes equations 183

2 Reduction of the Navier–Stokes equations

to systems of PDEs in three independent variables

2.1 Ansatzes of codimension one

In this subsection we give ansatzes that reduce the NSEs to systems of PDEs in three

independent variables. The ansatzes are constructed with the subalgebraic analysis of

A(N S) (see Subsection A.2) by means of the method discribed in Section B.

u1 = |t|?1/2 (v 1 cos ? ? v 2 sin ? ) + 1 x1 t?1 ? ?x2 t?1 ,

1. 2

u2 = |t|?1/2 (v 1 sin ? + v 2 cos ? ) + 1 x2 t?1 + ?x1 t?1 ,

2

(2.1)

u3 = |t|?1/2 v 3 + 1 x3 t?1 ,

2

p = |t|?1 q + 1 ? 2 t?2 r2 + 1 t?2 xa xa ,

2 8

where

y1 = |t|?1/2 (x1 cos ? + x2 sin ? ), y2 = |t|?1/2 (?x1 sin ? + x2 cos ? ),

y3 = |t|?1/2 x3 , ? ? 0, ? = ? ln |t|.

v a = v a (y1 , y2 , y3 ), q = q(y1 , y2 , y3 ), r = (x2 + x2 )1/2 .

Here and below 1 2

u1 = v 1 cos ?t ? v 2 sin ?t ? ?x2 ,

2.

u2 = v 1 sin ?t + v 2 cos ?t + ?x1 ,

(2.2)

u3 = v 3 ,

p = q + 1 ? 2 r2 ,

2

where

y1 = x1 cos ?t + x2 sin ?t, y2 = ?x1 sin ?t + x2 cos ?t,

? ? {0; 1}.

y3 = x3 ,

u1 = x1 r?1 v 1 ? x2 r?1 v 2 + x1 r?2 ,

3.

u2 = x2 r?1 v 1 + x1 r?1 v 2 + x2 r?2 ,

(2.3)

стр. 44 |