Geodesic
Stabilizing a solution of the 2D Navier-Stokes system in the exterior of a bounded domain by means of a control on the boundary
Sbornik. Mathematics, Tome 203 (2012) no. 9, pp. 1244-1268 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The problem of stabilizing a solution of the 2D Navier-Stokes system defined in the exterior of a bounded domain with smooth boundary is investigated. For a given initial velocity field a control on the boundary of the domain must be constructed such that the solution stabilizes to a prescribed vortex solution or trivial solution at the rate of $1/{t^k}$. On the way, related questions are investigated, concerning the behaviour of the spectrum of an operator under a relatively compact perturbation and the existence of attracting invariant manifolds. Bibliography: 21 titles.
Keywords: Navier-Stokes system, Lamb-Oseen vortex, stabilization, boundary control, invariant manifolds. 
@article{SM_2012_203_9_a1,
     author = {A. V. Gorshkov},
     title = {Stabilizing a~solution of the {2D} {Navier-Stokes} system in the exterior of a~bounded domain by means of a~control on the boundary},
     journal = {Sbornik. Mathematics},
     pages = {1244--1268},
     year = {2012},
     volume = {203},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2012_203_9_a1/}
}
TY  - JOUR
AU  - A. V. Gorshkov
TI  - Stabilizing a solution of the 2D Navier-Stokes system in the exterior of a bounded domain by means of a control on the boundary
JO  - Sbornik. Mathematics
PY  - 2012
SP  - 1244
EP  - 1268
VL  - 203
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2012_203_9_a1/
LA  - en
ID  - SM_2012_203_9_a1
ER  - 
%0 Journal Article
%A A. V. Gorshkov
%T Stabilizing a solution of the 2D Navier-Stokes system in the exterior of a bounded domain by means of a control on the boundary
%J Sbornik. Mathematics
%D 2012
%P 1244-1268
%V 203
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2012_203_9_a1/
%G en
%F SM_2012_203_9_a1
A. V. Gorshkov. Stabilizing a solution of the 2D Navier-Stokes system in the exterior of a bounded domain by means of a control on the boundary. Sbornik. Mathematics, Tome 203 (2012) no. 9, pp. 1244-1268. http://geodesic.mathdoc.fr/item/SM_2012_203_9_a1/

[1] A. Fursikov, “Real process corresponding to the 3D Navier–Stokes system, and its feedback stabilization from the boundary”, Partial differential equations, Amer. Math. Soc. Transl. Ser. 2, 206, Amer. Math. Soc., Providence, RI, 2002, 95–123 | MR | Zbl

[2] A. V. Fursikov, “Stabilizability of two-dimensional Navier–Stokes equations with help of a boundary feedback control”, J. Math. Fluid Mech., 3:3 (2001), 259–301 | DOI | MR | Zbl

[3] A. V. Fursikov, “Stabilizability of a quasi-linear parabolic equation by means of a boundary control with feedback”, Sb. Math., 192:4 (2001), 593–639 | DOI | MR | Zbl

[4] A. V. Gorshkov, “Stabilization of the one-dimensional heat equation on a semibounded rod”, Russian Math. Surveys, 56:2 (2001), 409–410 | DOI | MR | Zbl

[5] A. V. Gorshkov, “Stabilization of a semilinear parabolic equation in the exterior of a bounded domain by means of boundary controls”, Sb. Math., 194:10 (2003), 1475–1502 | DOI | MR | Zbl

[6] Th. Gallay, C. E. Wayne, “Invariant manifolds and the long-time asymptotics of the Navier–Stokes and vorticity equations on $\mathbb{R}^2$”, Arch. Ration. Mech. Anal., 163:3 (2002), 209–258 | DOI | MR | Zbl

[7] Th. Gallay, C. E. Wayne, “Global stability of vortex solutions of the two-dimensional Navier–Stokes equation”, Comm. Math. Phys., 255:1 (2005), 97–129 | DOI | MR | Zbl

[8] S. M. Belotserkovsky, I. K. Lifanov, Method of discrete vortices, CRC Press, Boca Raton, FL, 1993 | MR | MR | Zbl

[9] G.-H. Cottet, P. D. Koumoutsakos, Vortex methods. Theory and practice, Cambridge Univ. Press, Cambridge, 2000 | MR | Zbl

[10] A. V. Gorshkov, “Stabilization of linear parabolic equations defined in an exterior of a bounded domain by a boundary control”, J. Dynam. Control Systems, 10:1 (2004), 1–9 | DOI | MR | Zbl

[11] M. V. Keldysh, “On the completeness of the eigenfunctions of some classes of non-selfadjoint linear operators”, Russian Math. Surveys, 26:4 (1971), 15–44 | DOI | MR | Zbl | Zbl

[12] A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Transl. Math. Monogr., 71, Amer. Math. Soc., Providence, RI, 1988 | MR | MR | Zbl | Zbl

[13] A. S. Markus, V. I. Matsaev, “Comparison theorems for spectra of linear operators, and spectral asymptotics.”, Trans. Moscow Math. Soc., 1984, no. 1, 139–187 | MR | Zbl | Zbl

[14] A. A. Shkalikov, “On the basis property of root vectors of a perturbed self-adjoint operator”, Proc. Steklov Inst. Math., 269 (2010), 284–298 | DOI | MR | Zbl

[15] O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Appl. Math. Sci., 49, Springer-Verlag, New York, 1985 | MR | MR | Zbl | Zbl

[16] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin–New York, 1981 | DOI | MR | MR | Zbl | Zbl

[17] L. Hörmander, Linear partial differential operators, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1963 | MR | MR | Zbl | Zbl

[18] K. Yosida, Functional analysis, Grundlehren Math. Wiss., 123, Springer-Verlag, Berlin–Göttingen–Heidelberg; Academic Press, New York, 1965 | MR | MR | Zbl | Zbl

[19] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 ; I. C. Gohberg, M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, RI, 1969 | MR | MR | Zbl

[20] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin–Heidelberg–New York, 1966 | MR | MR | Zbl | Zbl

[21] A. V. Babin, M. I. Vishik, Attractors of evolution equations, Stud. Math. Appl., 25, North-Holland, Amsterdam, 1992 | MR | MR | Zbl | Zbl