We consider the controllability of a viscous incompressible fluid modeled by the Navier–Stokes system with a nonlinear viscosity. To prove the controllability to trajectories, we linearize around a trajectory and the corresponding linear system includes a nonlocal spatial term. Our main result is a Carleman estimate for the adjoint of this linear system. This estimate yields in a standard way the null controllability of the linear system and the local controllability to trajectories. Our method to obtain the Carleman estimate is completely general and can be adapted to other parabolic systems when a Carleman estimate is available.
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@article{CRMATH_2021__359_6_719_0,
author = {Guerrero, Sergio and Takahashi, Tak\'eo},
title = {Controllability to trajectories of a {Ladyzhenskaya} model for a viscous incompressible fluid},
journal = {Comptes Rendus. Math\'ematique},
pages = {719--732},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {6},
year = {2021},
doi = {10.5802/crmath.202},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.202/}
}
TY - JOUR AU - Guerrero, Sergio AU - Takahashi, Takéo TI - Controllability to trajectories of a Ladyzhenskaya model for a viscous incompressible fluid JO - Comptes Rendus. Mathématique PY - 2021 SP - 719 EP - 732 VL - 359 IS - 6 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.202/ DO - 10.5802/crmath.202 LA - en ID - CRMATH_2021__359_6_719_0 ER -
%0 Journal Article %A Guerrero, Sergio %A Takahashi, Takéo %T Controllability to trajectories of a Ladyzhenskaya model for a viscous incompressible fluid %J Comptes Rendus. Mathématique %D 2021 %P 719-732 %V 359 %N 6 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.202/ %R 10.5802/crmath.202 %G en %F CRMATH_2021__359_6_719_0
Guerrero, Sergio; Takahashi, Takéo. Controllability to trajectories of a Ladyzhenskaya model for a viscous incompressible fluid. Comptes Rendus. Mathématique, Tome 359 (2021) no. 6, pp. 719-732. doi : 10.5802/crmath.202. http://www.numdam.org/articles/10.5802/crmath.202/
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