In this paper we deal with the local exact controllability of the Navier-Stokes system with nonlinear Navier-slip boundary conditions and distributed controls supported in small sets. In a first step, we prove a Carleman inequality for the linearized Navier-Stokes system, which leads to null controllability of this system at any time . Then, fixed point arguments lead to the deduction of a local result concerning the exact controllability to the trajectories of the Navier-Stokes system.
Keywords: Navier-Stokes system, controllability, slip
Guerrero, Sergio 1
@article{COCV_2006__12_3_484_0,
author = {Guerrero, Sergio},
title = {Local exact controllability to the trajectories of the {Navier-Stokes} system with nonlinear {Navier-slip} boundary conditions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {484--544},
year = {2006},
publisher = {EDP Sciences},
volume = {12},
number = {3},
doi = {10.1051/cocv:2006006},
mrnumber = {2224824},
zbl = {1106.93011},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv:2006006/}
}
TY - JOUR AU - Guerrero, Sergio TI - Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 484 EP - 544 VL - 12 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2006006/ DO - 10.1051/cocv:2006006 LA - en ID - COCV_2006__12_3_484_0 ER -
%0 Journal Article %A Guerrero, Sergio %T Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 484-544 %V 12 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv:2006006/ %R 10.1051/cocv:2006006 %G en %F COCV_2006__12_3_484_0
Guerrero, Sergio. Local exact controllability to the trajectories of the Navier-Stokes system with nonlinear Navier-slip boundary conditions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 484-544. doi: 10.1051/cocv:2006006
[1] , Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975. | Zbl | MR
[2] , L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984). | Zbl
[3] and, Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. | Zbl | Numdam
[4] , Thesis, University of Seville (1993).
[5] and, Analysis of turbulent boundary layers. Applied Mathematics and Mechanics, No. 15. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1974). | Zbl | MR
[6] , On the controllability of the -D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1995/96) 35-75. | Zbl | Numdam
[7] , and, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31-61. | Zbl
[8] ,, and, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83/12 (2004) 1501-1542.
[9] and, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Analyse non Lin. 17 (2000) 583-616. | Zbl | Numdam
[10] and, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996). | Zbl | MR
[11] , An introduction to the Mathematical Theory of the Navier-Stokes equations, Vol. I. Springer-Verlag, New York (1994). | Zbl | MR
[12] , Local exact controllability for the 2-D Navier-Stokes equations with the Navier slip boundary conditions, in Turbulence Modelling and Vortex Dynamics, Istanbul, Springuer Berlin, 1996. Lect. Notes . Phys. 491 (1997) 148-168 | Zbl
[13] , Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Zbl | Numdam
[14] and, Global Carleman estimates for weak elliptic non homogeneous Dirichlet problem. Int. Math. Research Notices 16 (2003) 883-913.
[15] and, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications. Lect. Notes Pure Appl. Math. 218 (2001) | Zbl | MR
[16] and, Problèmes aux limites non homogènes et applications (3 volumes). Dunod, Gauthiers-Villars, Paris (1968). | Zbl
[17] , Intégration et probabilités. Analyse de Fourier et analyse spectrale. Masson (1982). | Zbl | MR
[18] , Incompressible flow. Wiley-Interscience, New York (1984). | Zbl | MR
[19] , Boundary-Layer Theory. McGraw-Hill, New York (1968). | Zbl | MR
[20] and, On a boundary value problem for a stationnary system of Navier-Stokes equations. Trudy Mat. Inst. Steklov 125 (1973) 196-210. | Zbl
[21] , An introduction to Sobolev spaces and interpolation spaces. Course (2000), URL: http://www.math.cmu.edu/cna/publications/SOB+Int.pdf.
[22] , Navier-Stokes equations. Theory and numerical analysis. Studies in Mathematics and its applications, 2. North Holland Publishing Co., Amsterdam-New York-Oxford (1977). | Zbl | MR
[23] , Exact boundary controllability for the semilinear wave equation, H. Brezis and J.L. Lions Eds., Pitman, New York in Nonlinear Partial Differential Equations Appl. X (1991) 357-391. | Zbl
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