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 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 .
@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},
volume = {30},
year = {2013},
pages = {737-762},
doi = {10.1016/j.anihpc.2012.12.002},
mrnumber = {3082482},
zbl = {1291.35202},
language = {en},
url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_4_737_0}
}
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) pp. 737-762. doi : 10.1016/j.anihpc.2012.12.002. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_4_737_0/
[1] , , , 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 162050 | Zbl 0123.28706
[2] , , Navier–Stokes equations controllability by means of low modes forcing, J. Math. Fluid Mech. 7 (2005), 108-152 | MR 2127744 | Zbl 1075.93014
[3] , , Finite-time regularity for bounded and unbounded ideal incompressible fluids using Hölder estimates, Lecture Notes in Math. 565 (1976), 1-13 | MR 467034 | Zbl 0355.76016
[4] , , , Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 no. 3 (1984), 61-66 | MR 763762 | Zbl 0573.76029
[5] , , The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 no. 1287 (1975), 555-601 | MR 385355 | Zbl 0306.35027
[6] , 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 2560050 | Zbl 1178.35285
[7] , Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Syst. 5 (1992), 295-312 | MR 1164379 | Zbl 0760.93067
[8] , On the controllability of 2-D incompressible perfect fluids, J. Math. Pures Appl. 75 no. 2 (1996), 155-188 | MR 1380673 | Zbl 0848.76013
[9] , 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 | Numdam | MR 1393067 | Zbl 0872.93040
[10] , 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 1720143 | Zbl 0954.76010
[11] , Control and Nonlinearity, Mathematical Surveys and Monographs vol. 136 (2007) | MR 2302744
[12] , , Les inéquations en mécanique et en physique, Dunod, Paris (1972) | MR 464857 | Zbl 0298.73001
[13] , , Exact controllability of the Navier–Stokes and Boussinesq equations, Russian Math. Surv. 54 no. 3 (1999), 93-146 | MR 1728643 | Zbl 0970.35116
[14] , Exact boundary controllability of 3-D Euler equation, ESAIM Control Optim. Calc. Var. 5 (2000), 1-44 | Numdam | MR 1745685 | Zbl 0940.93012
[15] O. Glass, L. Rosier, On the control of the motion of a boat, preprint, 2011. | MR 3021777
[16] , 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 2178059 | Zbl 1130.93403
[17] , On classical solutions of the two-dimensional non-stationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188-200 | MR 211057 | Zbl 0166.45302
[18] , , Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 no. 7 (1988), 891-907 | MR 951744 | Zbl 0671.35066
[19] , Controllability of 3D incompressible Euler equations by a finite-dimensional external force, ESAIM Control Optim. Calc. Var. 16 no. 3 (2010), 677-694 | Numdam | MR 2674632 | Zbl 1193.35141
[20] , Approximate controllability of three-dimensional Navier–Stokes equations, Comm. Math. Phys. 266 no. 1 (2006), 123-151 | MR 2231968 | Zbl 1105.93016
[21] , Navier–Stokes Equations. Theory and Numerical Analysis, vol. 2, North-Holland Publishing Co. (1977) | MR 609732
[22] , 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 1545430 | Zbl 0008.06901
[23] , The flow of a perfect, incompressible liquid through a given region, Dokl. Akad. Nauk SSSR 146 (1962), 561-564 | MR 163529