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Long-time behaviour of a radially symmetric fluid-shell interaction system
1. | Kharkiv Karazin National University, 4 Svobody sq., 61077 Kharkiv, Ukraine |
2. | Kharkiv Automobile and Highway National University, 25 Yaroslava Mudrogo st., 61002 Kharkiv, Ukraine |
We study the long-time dynamics of a coupled system consisting of the 2D Navier-Stokes equations and full von Karman elasticity equations. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
![]() ![]() |
[2] |
V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions
for a nonlinear fluid-structure interaction model, Fluids and Waves, Contemp. Math., AMS,
Providence, RI, 440 (2007), 55-82. |
[3] |
A. Boutet de Monvel and I. Chueshov,
Uniqueness theorem for weak solutions of von Karman evolution equations, J. Math. Anal. Appl., 221 (1998), 419-429.
doi: 10.1006/jmaa.1997.5681. |
[4] |
A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont,
Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404.
doi: 10.1007/s00021-004-0121-y. |
[5] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015.
![]() ![]() |
[6] |
I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999.
![]() ![]() |
[7] |
I. Chueshov,
A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Meth. Appl. Sci., 34 (2011), 1801-1812.
|
[8] |
I. Chueshov and T. Fastovska,
On interaction of circular cylindrical shells with a Poiseuille type flow, Evolution Equations and Control Theory, 5 (2016), 605-629.
doi: 10.3934/eect.2016021. |
[9] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,
Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. |
[10] |
I. Chueshov and I. Ryzhkova,
A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656.
|
[11] |
I. Chueshov and I. Ryzhkova,
Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Diff. Eqs., 254 (2013), 1833-1862.
doi: 10.1016/j.jde.2012.11.006. |
[12] |
I. Chueshov and I. Ryzhkova,
On the interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Mathematical Journal, 65 (2013), 158-177.
doi: 10.1007/s11253-013-0771-0. |
[13] |
Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee,
Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9 (2003), 633-650.
doi: 10.3934/dcds.2003.9.633. |
[14] |
G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2 edition, Springer-Verlag, New York, 2011.
![]() ![]() |
[15] |
G. Galdi, C. Simader and H. Sohr,
A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q, q}$, Math. Annalen, 331 (2005), 41-74.
doi: 10.1007/s00208-004-0573-7. |
[16] |
C. Grandmont,
Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737.
doi: 10.1137/070699196. |
[17] |
M. Guidorzi, M. Padula and P. I. Plotnikov,
Hopf solutions to a fluid-elastic interaction model, MMAS, 18 (2008), 215-269.
doi: 10.1142/S0218202508002668. |
[18] |
H. Koch and I. Lasiecka,
Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Prog. Nonlinear Differ. Equ. Appl, 50 (2002), 197-216.
|
[19] |
O. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.
![]() ![]() |
[20] |
J. -L. Lions and E. Magenes,
Problémes aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968. |
[21] |
J. Simon,
Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
|
[22] |
R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988.
![]() ![]() |
[23] |
R. Temam,
Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
[24] |
H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.
![]() ![]() |
show all references
References:
[1] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.
![]() ![]() |
[2] |
V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions
for a nonlinear fluid-structure interaction model, Fluids and Waves, Contemp. Math., AMS,
Providence, RI, 440 (2007), 55-82. |
[3] |
A. Boutet de Monvel and I. Chueshov,
Uniqueness theorem for weak solutions of von Karman evolution equations, J. Math. Anal. Appl., 221 (1998), 419-429.
doi: 10.1006/jmaa.1997.5681. |
[4] |
A. Chambolle, B. Desjardins, M. Esteban and C. Grandmont,
Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005), 368-404.
doi: 10.1007/s00021-004-0121-y. |
[5] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015.
![]() ![]() |
[6] |
I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999.
![]() ![]() |
[7] |
I. Chueshov,
A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Meth. Appl. Sci., 34 (2011), 1801-1812.
|
[8] |
I. Chueshov and T. Fastovska,
On interaction of circular cylindrical shells with a Poiseuille type flow, Evolution Equations and Control Theory, 5 (2016), 605-629.
doi: 10.3934/eect.2016021. |
[9] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,
Mem. Amer. Math. Soc., 195 (2008), ⅷ+183 pp. |
[10] |
I. Chueshov and I. Ryzhkova,
A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656.
|
[11] |
I. Chueshov and I. Ryzhkova,
Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, J. Diff. Eqs., 254 (2013), 1833-1862.
doi: 10.1016/j.jde.2012.11.006. |
[12] |
I. Chueshov and I. Ryzhkova,
On the interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Mathematical Journal, 65 (2013), 158-177.
doi: 10.1007/s11253-013-0771-0. |
[13] |
Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee,
Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst., 9 (2003), 633-650.
doi: 10.3934/dcds.2003.9.633. |
[14] |
G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2 edition, Springer-Verlag, New York, 2011.
![]() ![]() |
[15] |
G. Galdi, C. Simader and H. Sohr,
A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q, q}$, Math. Annalen, 331 (2005), 41-74.
doi: 10.1007/s00208-004-0573-7. |
[16] |
C. Grandmont,
Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716-737.
doi: 10.1137/070699196. |
[17] |
M. Guidorzi, M. Padula and P. I. Plotnikov,
Hopf solutions to a fluid-elastic interaction model, MMAS, 18 (2008), 215-269.
doi: 10.1142/S0218202508002668. |
[18] |
H. Koch and I. Lasiecka,
Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Prog. Nonlinear Differ. Equ. Appl, 50 (2002), 197-216.
|
[19] |
O. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.
![]() ![]() |
[20] |
J. -L. Lions and E. Magenes,
Problémes aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968. |
[21] |
J. Simon,
Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
|
[22] |
R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988.
![]() ![]() |
[23] |
R. Temam,
Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
[24] |
H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.
![]() ![]() |
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