Stopping a viscous fluid by a feedback dissipative external field: I. The stationary Stokes problem

Abstract

In this work we consider a planar stationary flow of an incompressible viscous fluid in a semi-infinite strip governed by the standard Stokes system. We show how this fluid can be stopped at a finite distance of the entrance of the semi-infinite strip by means of a feedback source depending in a sub-linear way on the velocity field. This localization effect is proved reducing the problem to a non-linear bi-harmonic type one for which the localization of solutions is obtained by means of the application of an energy method, in the spirit of the monograph by Antontsev, Díaz and Shmarev [5]. Since the presence of the non-linear terms defined by the source is not standard in the fluid mechanics literature, we establish also some results about the existence and uniqueness of weak solutions for this problem.