On the controllability of the Navier-Stokes equation in spite of boundary layers

TL;DR

This paper demonstrates the controllability of the incompressible Navier-Stokes equations in bounded domains with partial boundary control, showing that solutions can be driven to rest at any positive time.

Contribution

The authors establish a controllability result for Navier-Stokes equations with slip boundary conditions, even with boundary under-determination, extending previous control theory results.

Findings

Existence of weak Leray solutions that vanish at a specified positive time.

Controllability achieved despite boundary under-determination.

Application to 2D and 3D bounded domains with slip boundary conditions.

Abstract

In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connected bounded domain, either in 2D or in 3D, with a Navier slip-with-friction boundary condition except on a part of the boundary. This under-determination encodes that one has control over the remaining part of the boundary. We prove that for any initial data, for any positive time, there exists a weak Leray solution which vanishes at this given time.