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

Published online by Cambridge University Press:  08 August 2014

Fágner D. Araruna
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa PB, Brasil. fagner@mat.ufpb.br
Enrique Fernández-Cara
Affiliation:
Departamento EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; cara@us.es; desouza@us.es
Diego A. Souza
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa PB, Brasil. fagner@mat.ufpb.br Departamento EDAN, University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; cara@us.es; desouza@us.es
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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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2014

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