Beyond partially invariant solutions

Abstract:
Group analysis of differential equations founded by S. Lie in the end of 19th century was given a new life in the middle of the 20th century. One of the major achievements of a new theory was an introduced by L.V. Ovsyannikov notion of a partially invariant solution of a system of differential equations. Let the system of equations admits a local Lie group. If some solution of this system doesn't change under the action of any transformation from the given group, then it’s called an invariant solution. Examples of invariant solutions are well known: for instance, these are motions of continuous media with plane, cylindrical and spherical waves. Algorithm of invariant solution construction is built long time ago. Also, it may turn out that the initial solution is not invariant, however its orbit under the action of the given group fills out an invariant manifold in the space of independent variables and sought functions. Such solution is called a partially invariant one, since only some of sought functions in it are invariant. For classification of partially invariant solutions two integer numbers are introduced: rank (the number of independent variables in the system for the invariant part of sought functions) and defect (the number of non-invariant sought functions). Invariant solutions have a zero defect.

L.V. Ovsyannikov has established that functionally-invariant solutions of V.I. Smirnov – S.L. Sobolev and double waves in gas dynamics are partially invariant solutions to the corresponding systems of equations. This has stimulated the development of partial invariance theory, which has lead to construction of new classes of exact solutions to the equations of gas dynamics, Navier-Stokes, magnetohydrodynamics, elasticity theory, plasticity theory and other models, which cannot be obtained by any intuitive reasons. In the current paper there are proposed a number of heuristic approaches for broadening the class of exact solutions to equations of continuous media mechanics on the base on their partially invariant solutions.

Solution to the Navier-Stokes equations, known as von Karman’s vortex is neither invariant nor partially invariant solution to the mentioned system. However it can be obtained with the help of two-step procedure. This solution is an invariant solution to some partially invariant sub-model of the Navier - Stokes equations; this sub-model inherits a part of group which is admitted by the initial system of three-dimensional non-stationary Navier-Stokes equations (S.V. Meleshko, V.V. Pukhnachev, 1999).

Another approach is illustrated on an example of a problem of deformation of visco-elastic strip with free boundaries. The corresponding partially invariant solution of the equations of motion of Maxwell’s incompressible visco-elastic media has a rank one and turns to be very poor. It was shown that refusal from invariance of the part of sought functions leads to significant broadening of class of exact solutions to the problem (V.V. Pukhnachev, 2009).

One more opportunity is given by the application of method of invariant manifolds (V.A. Galaktionov, 1996) for the construction of exact solutions of resolving system, to which the invariant part of partially invariant solution satisfies. In particular, it was possible to build a solution to problem about non-stationary flow of viscous liquid with a critical point if the solid boundary in the flow-out regime, while there are no stationary analogues to this solution (V.V. Kuznetsov and V.V. Pukhnachev, 2009).

A number of systems of equations of continuous media mechanics have a specific structure. One of the sought functions satisfies the equation which has a trivial solution for sure. In gas dynamics it’s entropy, in the theory of heat gravitational convection it’s temperature, in the equations of rotationally symmetric motion of an ideal incompressible liquid it’s circular component of velocity, in the system of equations of concentration boundary layer it’s concentration of admixture. Using this specific character it is possible to build new solutions to the above mentioned systems on the base of some partially invariant solutions to these systems of equations (V.V. Pukhnachev, S.V. Meleshko, E.Yu. Meshcheryakova, 2002-2009).

The work is supported by RFBR grant (No. 07-01-00168) and program “Leading Scientific Schools of Russian Federation” grant (NSh-2260.2008.1).