no. 4
Stabilization of the 2D incompressible Euler system in an infinite strip
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 737-762

The paper is devoted to the study of a stabilization problem for the 2D incompressible Euler system in an infinite strip with boundary controls. We show that for any stationary solution (c,0) of the Euler system there is a control which is supported in a given bounded part of the boundary of the strip and stabilizes the system to (c,0).

@article{AIHPC_2013__30_4_737_0,
     author = {Nersisyan, Hayk},
     title = {Stabilization of the {2D} incompressible {Euler} system in an infinite strip},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {737--762},
     year = {2013},
     publisher = {Elsevier},
     volume = {30},
     number = {4},
     doi = {10.1016/j.anihpc.2012.12.002},
     mrnumber = {3082482},
     zbl = {1291.35202},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2012.12.002/}
}
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Nersisyan, Hayk. Stabilization of the 2D incompressible Euler system in an infinite strip. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 737-762. doi: 10.1016/j.anihpc.2012.12.002

[1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35-92 | MR | Zbl

[2] A. Agrachev, A. Sarychev, Navier–Stokes equations controllability by means of low modes forcing, J. Math. Fluid Mech. 7 (2005), 108-152 | MR | Zbl

[3] C. Bardos, U. Frisch, Finite-time regularity for bounded and unbounded ideal incompressible fluids using Hölder estimates, Lecture Notes in Math. 565 (1976), 1-13 | MR | Zbl

[4] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 no. 3 (1984), 61-66 | MR | Zbl

[5] J.L. Bona, R. Smith, The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 no. 1287 (1975), 555-601 | MR | Zbl

[6] M. Chapouly, On the global null controllability of a Navier–Stokes system with Navier slip boundary conditions, J. Differential Equations 247 no. 7 (2009), 2094-2123 | MR | Zbl

[7] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Syst. 5 (1992), 295-312 | MR | Zbl

[8] J.-M. Coron, On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. 75 no. 2 (1996), 155-188 | MR | Zbl

[9] J.-M. Coron, On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var. 1 (1996), 35-75 | MR | EuDML | Zbl | Numdam

[10] J.-M. Coron, On the null asymptotic stabilization of 2-D incompressible Euler equation in a simply connected domain, SIAM J. Control Optim. 37 no. 6 (1999), 1874-1896 | MR | Zbl

[11] J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs vol. 136 (2007) | MR

[12] G. Duvaut, J.-L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris (1972) | MR | Zbl

[13] A.V. Fursikov, O.Yu. Imanuvilov, Exact controllability of the Navier–Stokes and Boussinesq equations, Russian Math. Surv. 54 no. 3 (1999), 93-146 | MR | Zbl

[14] O. Glass, Exact boundary controllability of 3-D Euler equation, ESAIM Control Optim. Calc. Var. 5 (2000), 1-44 | MR | EuDML | Zbl | Numdam

[15] O. Glass, L. Rosier, On the control of the motion of a boat, preprint, 2011. | MR

[16] O. Glass, Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case, SIAM J. Control Optim. 44 no. 3 (2005), 1105-1147 | MR | Zbl

[17] T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188-200 | MR | Zbl

[18] T. Kato, G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 no. 7 (1988), 891-907 | MR | Zbl

[19] H. Nersisyan, Controllability of 3D incompressible Euler equations by a finite-dimensional external force, ESAIM Control Optim. Calc. Var. 16 no. 3 (2010), 677-694 | MR | EuDML | Zbl | Numdam

[20] A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Comm. Math. Phys. 266 no. 1 (2006), 123-151 | MR | Zbl

[21] R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, vol. 2, North-Holland Publishing Co. (1977) | MR

[22] W. Wolibner, Un théorème sur lʼexistence du mouvement plan dʼun fluide parfait, homogène, incompressible, pendant un temps infiniment long, Math. Z. 37 no. 1 (1933), 698-726 | MR | EuDML | Zbl

[23] V. Yudovich, The flow of a perfect, incompressible liquid through a given region, Dokl. Akad. Nauk SSSR 146 (1962), 561-564 | MR

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