ABSTRACT
We consider the Stokes system in a three-dimensional cylinder Ω = G × (0, L) of R3, G being a bounded smooth domain of R2. We study the following uniqueness property: If u is a solution of the Stokes system in Ω × (0, T) with Dirichlet boundary conditions, T being a positive time, and its third component vanishes, i.e. u 3 ≡ 0, then can we ensure that u ≡ 0? We prove that this property does hold for “almost every” crosssection G. By using the Fourier expansion of solutions the problem is reduced to show that, generically with respect to the cross-section G, there is no eigenfunction of the Stokes system with third component identically zero. We also show how this uniqueness result can be applied to obtain approximate controllability properties of the Stokes system with scalar controls oriented in the direction (0,0,1) of R3. Actually, it was while working on the approximate controllability problem that we were led to the problem of generic uniqueness studied in the present paper. We also prove that the results above fail when G is a ball of R2.
