Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T09:07:19.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Local exact controllability to the trajectories of the Navier-Stokes system withnonlinear Navier-slip boundary conditions

Published online by Cambridge University Press:  20 June 2006

Sergio Guerrero
Sergio Guerrero*
Affiliation:
Dpto. E.D.A.N., University of Sevilla, Aptdo. 1160, 41080 Sevilla, Spain; sguerrero@us.es.
Get access

Abstract

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.

Keywords

Navier-Stokes systemcontrollabilityslip.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 3 , July 2006 , pp. 484 - 544
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R.A. Adams, Sobolev spaces. Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
J.-P. Aubin, L'analyse non linéaire et ses motivations économiques. Masson, Paris (1984).
Anita, S. and Barbu, V., Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157–173. CrossRef
J.A. Bello, Thesis, University of Seville (1993).
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).
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.
Fabre, C., Puel, J.-P. and Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh 125A (1995) 31–61. CrossRef
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.
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.
A. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations. Lecture Notes #34, Seoul National University, Korea (1996).
G.P. Galdi, An introduction to the Mathematical Theory of the Navier-Stokes equations, Vol. I. Springer-Verlag, New York (1994).
Imanuvilov, O.Yu., 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 CrossRef
Imanuvilov, O.Yu., Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39–72. CrossRef
Imanuvilov, O.Yu. and Puel, J.-P., Global Carleman estimates for weak elliptic non homogeneous Dirichlet problem. Int. Math. Research Notices 16 (2003) 883–913. CrossRef
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)
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications (3 volumes). Dunod, Gauthiers-Villars, Paris (1968).
P. Malliavin, Intégration et probabilités. Analyse de Fourier et analyse spectrale. Masson (1982).
R.L. Panton, Incompressible flow. Wiley-Interscience, New York (1984).
H. Schlichting, Boundary-Layer Theory. McGraw-Hill, New York (1968).
Solonnikov, V.A. and Schadilov, V.E., On a boundary value problem for a stationnary system of Navier-Stokes equations. Trudy Mat. Inst. Steklov 125 (1973) 196–210.
L. Tartar, An introduction to Sobolev spaces and interpolation spaces. Course (2000), URL: http://www.math.cmu.edu/cna/publications/SOB+Int.pdf.
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).
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.