On Lie Reduction of the Navier-Stokes Equations

Lie reduction of the Navier-Stokes equations to systems of partial differential equations in three and two independent variables and to ordinary differential equations is described. The Navier-Stokes equations (NSEs) ~ut + (~u · ~ ∇)~u−4~u+ ~ ∇p = ~0, div ~u = 0 (1) which describe the motion of an incompressible viscous fluid are the basic equations of hydrodynamics. In (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}, 4 = ~ ∇· ~ ∇ is the Laplacian. The kinematic coefficient of viscosity and fluid density are set equal to unity. The problem of finding exact solutions of the nonlinear equations (1) is an important and rather complicated one. There are some ways to solve it. Considerable progress in this field can be achieved by means of a symmetry approach. Equations (1) have nontrivial symmetry properties. Relatively recently, it was found by means of the Lie method [2, 1] that the maximal Lie invariance algebra of the NSEs (1) is the infinite-dimensional algebra A(NS) with the basis elements D = 2t∂t + xa∂a − u∂ua − 2p∂p, Jab = xa∂b − xb∂a + u∂ub − u∂ua , a < b, ∂t, R(~ m) = R(~ m(t)) = m∂a +mt ∂ua −mttxa∂p, Z(χ) = Z(χ(t)) = χ∂p, (2) where ma = ma(t) and χ = χ(t) are arbitrary smooth functions of t (for example, from C((t0, t1),R) ). Hereafter, repeated 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}. Algebra (2) contains, as a subalgebra, the eleven-dimensional extended Galilei algebra < ∂t, D, Jab, ∂a, Ga = t∂a + ∂ua > . To find exact solutions of (1), the symmetry approach in explicit form was used in a number of papers (see references in [4]). In our works [4, 7, 5, 6, 12], we made a complete symmetry reduction of the NSEs to systems of PDEs in three and two independent variables and to systems of ODEs, using the subalgebraic structure of A(NS). Copyright c © 1995 by Mathematical Ukraina Publisher. All rights of reproduction in any form reserved.