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.

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 T>0. Then, fixed point arguments lead to the deduction of a local result concerning the exact controllability to the trajectories of the Navier-Stokes system.

DOI: 10.1051/cocv:2006006
Classification: 34B15, 35Q30, 76D03, 93B05, 93C10
Mots-clés : Navier-Stokes system, controllability, slip
Guerrero, Sergio 1

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, boîte courrier 187, 75035 Cedex 05, Paris, France;
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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://www.numdam.org/articles/10.1051/cocv:2006006/

[1] R.A. Adams, Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975. | MR | Zbl

[2] J.-P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984). | Zbl

[3] S. Anita and V. Barbu, Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. | Numdam | Zbl

[4] J.A. Bello, Thesis, University of Seville (1993).

[5] T. Cebeci and A.M. Smith, Analysis of turbulent boundary layers. Applied Mathematics and Mechanics, No. 15. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1974). | MR | Zbl

[6] J.-M. Coron, On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM: COCV 1 (1995/96) 35-75. | Numdam | Zbl

[7] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31-61. | Zbl

[8] E. Fernández-Cara, S. Guerrero, O.Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83/12 (2004) 1501-1542.

[9] E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré, Analyse non Lin. 17 (2000) 583-616. | Numdam | Zbl

[10] A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996). | MR | Zbl

[11] G.P. Galdi, An introduction to the Mathematical Theory of the Navier-Stokes equations, Vol. I. Springer-Verlag, New York (1994). | MR | Zbl

[12] O.Yu. Imanuvilov, 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] O.Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Numdam | Zbl

[14] O.Yu. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak elliptic non homogeneous Dirichlet problem. Int. Math. Research Notices 16 (2003) 883-913.

[15] O.Yu. Imanuvilov and M. Yamamoto, 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

[16] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications (3 volumes). Dunod, Gauthiers-Villars, Paris (1968). | Zbl

[17] P. Malliavin, Intégration et probabilités. Analyse de Fourier et analyse spectrale. Masson (1982). | MR | Zbl

[18] R.L. Panton, Incompressible flow. Wiley-Interscience, New York (1984). | MR | Zbl

[19] H. Schlichting, Boundary-Layer Theory. McGraw-Hill, New York (1968). | MR | Zbl

[20] V.A. Solonnikov and V.E. Schadilov, On a boundary value problem for a stationnary system of Navier-Stokes equations. Trudy Mat. Inst. Steklov 125 (1973) 196-210. | Zbl

[21] L. Tartar, An introduction to Sobolev spaces and interpolation spaces. Course (2000), URL: http://www.math.cmu.edu/cna/publications/SOB+Int.pdf.

[22] R. Temam, 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] E. Zuazua, 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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