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March  2018, 38(3): 1315-1348. doi: 10.3934/dcds.2018054

Long-time behaviour of a radially symmetric fluid-shell interaction system

Tamara Fastovska 1,2,,

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

* Corresponding author: Tamara Fastovska

Received  April 2017 Revised  September 2017 Published  December 2017

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.

Keywords: Fluid-structure interaction, linearized 3D Navier-Stokes equations, Poiseulle flow, nonlinear circular shell, attractor.
Mathematics Subject Classification: Primary: 74F10, 35B41; Secondary: 35Q30, 74K25.
Citation: Tamara Fastovska. Long-time behaviour of a radially symmetric fluid-shell interaction system. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1315-1348. doi: 10.3934/dcds.2018054
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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