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.
Mots-clés : null controllability, Carleman inequalities, Leray-αmodel, Navier−Stokes equations
@article{COCV_2014__20_4_1181_0,
author = {Araruna, F\'agner D. and Fern\'andez-Cara, Enrique and Souza, Diego A.},
title = {Uniform local null control of the {Leray-}$\alpha $ model},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1181--1202},
publisher = {EDP-Sciences},
volume = {20},
number = {4},
year = {2014},
doi = {10.1051/cocv/2014011},
zbl = {1297.93031},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2014011/}
}
TY - JOUR AU - Araruna, Fágner D. AU - Fernández-Cara, Enrique AU - Souza, Diego A. TI - Uniform local null control of the Leray-$\alpha $ model JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 1181 EP - 1202 VL - 20 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014011/ DO - 10.1051/cocv/2014011 LA - en ID - COCV_2014__20_4_1181_0 ER -
%0 Journal Article %A Araruna, Fágner D. %A Fernández-Cara, Enrique %A Souza, Diego A. %T Uniform local null control of the Leray-$\alpha $ model %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1181-1202 %V 20 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014011/ %R 10.1051/cocv/2014011 %G en %F COCV_2014__20_4_1181_0
Araruna, Fágner D.; Fernández-Cara, Enrique; Souza, Diego A. Uniform local null control of the Leray-$\alpha $ model. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 4, pp. 1181-1202. doi : 10.1051/cocv/2014011. http://www.numdam.org/articles/10.1051/cocv/2014011/
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