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Controllability of the Navier-Stokes equation in a rectangle with a little help of a distributed phantom force
We consider the 2D incompressible Navier-Stokes equation in a rectangle with the usual no-slip boundary condition prescribed on the upper and lower boundaries. We prove that for any positive time, for any finite energy initial data, there exist controls on the left and right boundaries and a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space, such that the corresponding solution is at rest at the given final time. Our work improves earlier results in [17, 18]fatcat:sbkuzuo6i5h27bcrj3qp7s3z3q