A few remarks on the transport-Stokes system

We consider the so-called transport-Stokes system which describes sedimentation of inertialess suspensions in a viscous flow and couples a transport equation and the steady Stokes equations in the full three-dimensional space. First we present a global existence and uniqueness result for L 1 ∩L p initial densities where p≥3. Secondly, we prove that, in the case where p>3, the flow map which describes the trajectories of these solutions is analytic with respect to time. Finally we establish the small-time global exact controllability of the transport-Stokes system. These results extend to the transport-Stokes system some results obtained for the incompressible Euler system respectively by Yudovich in Yud63, by Chemin in Che92, Che95 and by Coron, and Glass, in Cor96, Gla00.