Symmetries in the Navier-Stokes equations

There are considered the Navier-Stokes equations (NSE) with a zero (or potential) external force vector. The group of transformations admitted by these equations is infinite-dimensional: coefficients of operators of the corresponding Lie algebra contain four arbitrary functions of time. Rich properties of the NSE symmetry produce a lot of their exact solutions. Classical examples of invariant solutions of the Navier-Stokes equations are the solutions by Poiseuille, Couette and Hamel-Jeffery. Notion of a partially invariant solution of a system of differential equations, introduced by Ovsiannikov, allowed us to extend a class of exact solutions of the NSE essentially. However this class is not exhausted by invariant and partially invariant solutions. Its further extension could be obtained with a help of a two-step procedure: first we reduce the initial NSE model to a partially invariant sub-model of less dimension, which inherits a part of the group, admitted by the NSE, and then we construct invariant solutions of the reduced sub-model. Following this procedure, there was revealed the group-theoretical nature of the famous von Karman solution.

Symmetry approach turned to be especially useful in application to the free boundary problems for the NSE. Local theory of specified problems is close to be completed nowadays. At the same time, the global solvability of non-stationary problems in time is proved only under the condition that the initial velocity vector is small. Solvability of stationary problems is established under restrictions of smallness of Reynolds number or capillary number. Employment of properties of invariance of free boundary conditions allowed us to obtain non-local results in problems with a free boundary, which effective dimension does not exceed 2. Non-stationary problems on a rotating ring and on the viscous layer spread over a rotating plane as well as analogues of stationary flows of Hamel-Jeffery and Poiseuille are among this kind of problems. Moreover, with a help of partially invariant solutions there are constructed examples of blowing up and shrinking of the flow domain phenomena in free boundary problems.

One more method to obtain new exact solutions of the NSE is the "raising" of the known solutions by reduction of their properties of invariance. As an example, there is considered a non-stationary analogue of the classical Hamel-Jeffery flow, which describes stationary self-similar flow in a flat diffusor. If we depart from the property of stationarity, but keep the self-similarity, we'll come to a non-standard boundary-valued problem in a stripe for a quasi-linear elliptical equation of fourth order, which is satisfied by a stream function of flow in self-similar variables. Its specificity consists in the fact that on the "right" infinity limit values of a stream function in dependence on the polar angle could be set arbitrarily, meanwhile on the "left" infinity they are determined by preserved in time value of flux of liquid through the cross-section of sector as a solution of Hamel-Jeffery problem. Case of zero flux represents a special interest for investigation. In this case the velocity field of flow possess a finite Dirichlet integral at any arbitrary positive time value.