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
@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}, publisher = {EDP-Sciences}, volume = {12}, number = {3}, year = {2006}, doi = {10.1051/cocv:2006006}, mrnumber = {2224824}, zbl = {1106.93011}, language = {en}, url = {http://archive.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 - http://archive.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 http://archive.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, Volume 12 (2006) no. 3, pp. 484-544. doi : 10.1051/cocv:2006006. http://archive.numdam.org/articles/10.1051/cocv:2006006/
[1] Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975. | MR | Zbl
,[2] L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984). | Zbl
,[3] Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. | Numdam | Zbl
and ,[4] Thesis, University of Seville (1993).
,[5] Analysis of turbulent boundary layers. Applied Mathematics and Mechanics, No. 15. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1974). | MR | Zbl
and ,[6] On the controllability of the -D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1995/96) 35-75. | Numdam | Zbl
,[7] Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31-61. | Zbl
, and ,[8] Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83/12 (2004) 1501-1542.
, , and ,[9] Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Analyse non Lin. 17 (2000) 583-616. | Numdam | Zbl
and ,[10] Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996). | MR | Zbl
and ,[11] An introduction to the Mathematical Theory of the Navier-Stokes equations, Vol. I. Springer-Verlag, New York (1994). | MR | Zbl
,[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. | Numdam | Zbl
,[14] Global Carleman estimates for weak elliptic non homogeneous Dirichlet problem. Int. Math. Research Notices 16 (2003) 883-913.
and ,[15] Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications. Lect. Notes Pure Appl. Math. 218 (2001) | MR | Zbl
and ,[16] Problèmes aux limites non homogènes et applications (3 volumes). Dunod, Gauthiers-Villars, Paris (1968). | Zbl
and ,[17] Intégration et probabilités. Analyse de Fourier et analyse spectrale. Masson (1982). | MR | Zbl
,[18] Incompressible flow. Wiley-Interscience, New York (1984). | MR | Zbl
,[19] Boundary-Layer Theory. McGraw-Hill, New York (1968). | MR | Zbl
,[20] On a boundary value problem for a stationnary system of Navier-Stokes equations. Trudy Mat. Inst. Steklov 125 (1973) 196-210. | Zbl
and ,[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). | MR | Zbl
,[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
,Cited by Sources: