Some Remarks on the Non-uniqueness of the Stationary Solutions of Navier-stokes Equations

We formulate some conditions when non-uniqueness of approximate solutions of the stationary Navier-Stokes equations occurs. Introduction. In this paper we will be concerned with the stationary Navier-Stokes equations for incompressible liquid in a bounded domain in Rn (n ∈ {2, 3}). Since a long time, it has been known that there exists a weak solution of this problem. For example, general existence results are proved in [8], Section 7 and in [11], Chapter II.1. As far as the problem of uniqueness or non-uniqueness is concerned, the situation is different. There are no general results. It is known that the solution is unique in a rather restrictive case when viscosity is large compared with external forces (see [8], Chapter I, Section 7, [11], Chapter II.1, [12], Part II, Section 9). On the other hand, there are some specific results that assert non-uniqueness. For example, W. Velte proved a non-uniqueness result for the Taylor problem, i.e. the problem describing the flow of a viscous incompressible liquid in a domain of R3 bounded by two infinite cylinders with the same vertical axis (see [13], [14], [11]). Let us also mention the Quette-Taylor experiment which indicates that if the Reynolds number of the flow increases, then the flow loses its stability and new steady states appear (see [12], Section 9). There are some abstract results which assert that generically with respect to various parameters of the flow, the number of solutions of the Navier-Stokes problem is finite and odd (see [2], [3], [4], [7], [10]). In the paper [5] some abstract criterion of non-uniqueness of the equation of the Navier-Stokes type is proved. This criterion is applied to the finitedimensional Galerkin equations corresponding to the homogeneous boundary value problem for the stationary Navier-Stokes equations.