On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions

Jean-Michel Coron

ESAIM: Control, Optimisation and Calculus of Variations (1996)

  • Volume: 1, page 35-75
  • ISSN: 1292-8119

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Coron, Jean-Michel. "On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions." ESAIM: Control, Optimisation and Calculus of Variations 1 (1996): 35-75. <http://eudml.org/doc/90500>.

@article{Coron1996,
author = {Coron, Jean-Michel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Navier-Stokes equations; controllability; Navier slip boundary conditions; slip boundary conditions; approximate controllability},
language = {eng},
pages = {35-75},
publisher = {EDP Sciences},
title = {On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions},
url = {http://eudml.org/doc/90500},
volume = {1},
year = {1996},
}

TY - JOUR
AU - Coron, Jean-Michel
TI - On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 1996
PB - EDP Sciences
VL - 1
SP - 35
EP - 75
LA - eng
KW - Navier-Stokes equations; controllability; Navier slip boundary conditions; slip boundary conditions; approximate controllability
UR - http://eudml.org/doc/90500
ER -

References

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  1. [1] R.A. Adams: Sobolev spaces, Academic Press, San Diego, London, 1978. Zbl0314.46030MR450957
  2. [2] C. Bardos, F. Golse, and D. Levermore: Fluid dynamic limits of kinetic equations I: formal derivations, J. Statistical Physics, 63, 1991, 323-344. MR1115587
  3. [3] F. Coron: Derivation of slip boundary conditions for the Navier-Stokes System from the Boltzmann equation, J. Statistical Physics, 54, 1989, 829-857. Zbl0666.76103MR988561
  4. [4] J.-M. Coron: Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5, 1992, 295-312. Zbl0760.93067MR1164379
  5. [5] J.-M. Coron: Stabilization of controllable systems, preprint, 1993, to appear in Nonholonomic geometry, A. Bellaïche and J.-J. Risler ed., Progress in Math., Birkhäuser. Zbl0858.93059MR1421826
  6. [6] J.-M. Coron: Relations entre commandabilité et stabilisations non linéaires, in Nonlinear partial differential equations and their applications, Collège de France seminars, Paris 1989-1991, Vol.11, H. Brezis and J.-L. Lions eds., Pitman Res. Notes Math. Ser., London, 299, 1994, 68-86. Zbl0813.93014MR1268900
  7. [7] J.-M. Coron: Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris, 317, 1993, 271-276. Zbl0781.76013MR1233425
  8. [8] J.-M. Coron: On the controllability of 2-D incompressible perfect fluids, J. Math. Pures et Appliquées, 75, 1996, 155-188. Zbl0848.76013MR1380673
  9. [9] C. Fabre: Uniqueness result for Stokes equations and their consequences in linear and nonlinear control problems, in Contrôlabilité approchée des solutions de quelques équations d'évolution, Habilitation à diriger des recherches, Université Pierre et Marie Curie, January 1996. MR1396664
  10. [10] E. Fernández-Cara and M. González-Burgos: A result concerning approximate controllability for the Navier-Stokes Equations, SIAM J. Control, to appear. Zbl0833.93009
  11. [11] E. Fernández-Cara and J. Real: On a conjecture due to J.-L. Lions, Nonlinear Analysis, Theory, Methods et Appl., 21, 1993, 835-847. Zbl0844.35082MR1249663
  12. [12] A.V. Fursikov: Exact boundary zero controllability of three-dimensional Navier-Stokes equations, J. Dynamical Control et Systems, 1, 1995, 325-350. Zbl0951.93005MR1354539
  13. [13] A.V. Fursikov and O. Yu. Imanuvilov: On controllability of certain Systems simulating a fluid flow, in Flow Control, IMA vol. in Math. and its Appl. , M.D. Gunzburger ed., Springer Verlag, New York, 68, 1995, 149-184. Zbl0922.93006MR1348646
  14. [14] A.V. Fursikov and O.Yu. Imanuvilov: On exact boundary zero controllability of the two-dimensional Navier-Stokes equation, Acta Appl. Math., 36, 1994, 1-10. Zbl0809.93006MR1308746
  15. [15] A.V. Fursikov and O.Yu. Imanuvilov: Local exact controllability of the Navier-Stokes equations, RIM-GARC preprint series 95-92, Seoul National University, February 1996. MR1404773
  16. [16] G. Geymonat and E. Sanchez-Palencia: On the vanishing viscosity limit for acoustic phenomena in a bounded region, Arch. Rat. Mechanics and Analysis, 75, 1981, 257-268. Zbl0475.76079MR605891
  17. [17] B.E. Launder and D.B. Spalding: Mathematical models of turbulence, Academic Press, 1972. Zbl0288.76027
  18. [18] J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, 1969. Zbl0189.40603MR259693
  19. [19] J.-L. Lions: Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles, Gauthier-Villars, Paris, 1968. Zbl0179.41801MR244606
  20. [20] J.-L. Lions: Are there connections between turbulence and controllability?, 9th IN-RIA International Conference, Antibes, June 12-25, 1990. 
  21. [21] J.-L. Lions: Exact controllability for distributed Systems. Some trends and some problems, in: Applied and Industrial Mathematics, R. Spigler ed., Kluwer Academic Publishers, Dordrecht, Boston, London, 1991, 59-84. Zbl0735.93006MR1147191
  22. [22] J.-L. Lions and E. Magenes: Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, 1968. Zbl0165.10801MR247243
  23. [23] P. Maremonti: Some theorems of existence for solutions of the Navier-Stokes equations with slip boundary condition in half-space, Ricerche di Matematica, 40, 1991, 81-135. Zbl0754.35110MR1191888
  24. [24] C. L. M. H. Navier: Sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. France, 6, 1823, 389-440. 
  25. [25] G.G. Stokes: On the effect of internal friction of fluids on the motion of pendulums, Trans. Cambridge Philos. Soc., 9, 1851, 8-106. 

Citations in EuDML Documents

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  1. Jacques-Louis Lions, Enrique Zuazua, Exact boundary controllability of Galerkin's approximations of Navier-Stokes equations
  2. Caroline Fabre, Uniqueness results for stokes equations and their consequences in linear and nonlinear control problems
  3. O. Yu. Imanuvilov, On exact controllability for the Navier-Stokes equations
  4. T. Horsin, On the controllability of the burger equation
  5. Olivier Glass, Exact boundary controllability of 3-D Euler equation
  6. Viorel Barbu, Feedback stabilization of Navier–Stokes equations
  7. Viorel Barbu, Feedback stabilization of Navier–Stokes equations
  8. Olivier Glass, Contrôlabilité de l’équation d’Euler tridimensionnelle pour les fluides parfaits incompressibles
  9. Olivier Glass, Exact boundary controllability of 3-D Euler equation
  10. S. Guerrero, O. Yu. Imanuvilov, Remarks on global controllability for the Burgers equation with two control forces

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