Local feedback regularization of three-dimensional Navier-Stokes equations on bounded domains
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The specific problem we consider here is inspired by recent advances in the control of nonlinear distributed parameter systems and its possible applications to hydrodynamics. The main objective is to investigate the extent to which the 3-dimensional Navier-Stokes system can be regularized using a particular, physically motivated, feedback control law. The specific choice of feedback mechanism is motivated by a work of O.A. Ladyzhenskaya  in which she introduces a modification of the Navier-Stokes equation on a three dimensional bounded domain and shows that the resulting perturbed system possesses global dynamics and, furthermore, this dynamics is stable. It is in this sense that we understand the system to be regularized.