ON A CRITERION FOR LOCATlNG STABLE STATIONARY SOLUTIONS TO THE NAVIER-STOKES EQUATIONS

Kq wvrds cd phr-crc.5: Navier-Stokes equations. stahlc ationarj solutions. long time Cialcrkin approximations. resoJvcnt of Navier-Stokes operator<. IN7'RODUCTION 1.r IS KNOL!'N that for some flows. even if the driving forces are independent of time, the flow that actually occurs is time-dependent. This means that the " permanent regime ". which takes place after a transient period, can become time-dependent. Understanding this regime is directly related to a better understanding of the behaviour for t+ + % of the corresponding solution of the Navier-Stokes equations (N.S.E.) and of the functional invariant set (attractor) that represents it. Since the global existence of strong solutions to the 3-dimensional N.S.E. is still an open problem. most of the studies concerning the large time behaviour of the N.S.E. have been restricted to the '-dimensional case. Assuming that the driving forces of the flow are time independent, then stationary solutions to the N.S.E. always exist. Moreover, if these driving forces are " small enough " , then there is only one stationary solution. and this unique solution is stable (i.e. any time-dependent solution to the N.S.E. converges, as t-f + %. to this unique stationary solution in the L' norm) (see e.g. [l]). C. Foias and R. Temam showed in [2] and [3] that for an open dense set of generic forces (in the appropriate functional space) the number of stationary solutions to the N.S.E. is finite. A remarkable property of any 2-dimensional solution to the N.S.E., which was established by C. Foias and G. Prodi [4], is that, as t-+ + 33, the solution is completely determined by a finite number of modes. A similar result was established by C. Foias and R. Temam in [5] for the Galerkin approximation. An upper estimate for the number of determining modes was given by C. [7] for oth,er aspects of the large time behaviour of Galerkin approximations to the N.S.E.) A major problem in this direction is: how to relate the large time behaviour of the exact solutions of the N.S.E. to the large time behaviour of their Galerkin approximations? In [8]. P. Constantin. C. Foias. and R. Temam gave sufficient conditions for inferring the existence of a nearby " stable " stationary solution of the N.S.E. from the apparent '*stability " of one of the time-dependent Galerkin approximate solutions. Namely. if m is sufficiently large (W b MZ where M, is described in section 3), if …