A generic uniqueness result for the Stokes system and its control theoretical consequences. (English) Zbl 0852.35112

Lions, Jacques-Louis; Zuazua, Enrique

Marcellini, Paolo (ed.) et al., Partial differential equations and applications. Collected papers in honor of Carlo Pucci on the occasion of his 70th birthday. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 177, 221-235 (1996).

Given a three-dimensional cylinder \(\Omega= G\times (0, L)\), where \(G\) is a bounded smooth domain of the plane, \(T> 0\) and \(\omega\) a non-empty and open subset of \(\Omega\), we can study the following uniqueness property: If \(u\) is a solution of the Stokes system in \(\Omega\times (0, T)\) with homogeneous boundary condition, and its third component vanishes in \(\omega\times (0, T)\), then can be ensured that \(u\equiv 0\) in \(\Omega\times (0, T)\)?. The authors show that this property holds for “almost every” cross section \(G\) (in particular, it holds if all the eigenvalues of the Laplacian in \(H^1_0(G)\) are simple), but it fails when \(G\) is a ball of \(\mathbb{R}^2\). Moreover, they apply this uniquness result to obtain approximate controllability properties of the Stokes system with scalar controls oriented in the direction \((0, 0, 1)\) of \(\mathbb{R}^3\).
For the entire collection see [Zbl 0834.00042].

Reviewer: L.A.Fernandez (Santander)

MSC:

35Q30	Navier-Stokes equations
93B05	Controllability

Keywords:

Stokes system; homogeneous boundary condition; approximate controllability properties of the Stokes system