Uniform local null control of the Leray-α model^∗,∗∗

¹ Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa PB, Brasil
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² Departamento EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
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Received: 22 November 2013
Revised: 8 February 2014

Abstract

This paper deals with the distributed and boundary controllability of the so called Leray-α model. This is a regularized variant of the Navier−Stokes system (α is a small positive parameter) that can also be viewed as a model for turbulent flows. We prove that the Leray-α equations are locally null controllable, with controls bounded independently of α. We also prove that, if the initial data are sufficiently small, the controls converge as α → 0⁺ to a null control of the Navier−Stokes equations. We also discuss some other related questions, such as global null controllability, local and global exact controllability to the trajectories, etc.