American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015.   Google Scholar
[6] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999.   Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2 edition, Springer-Verlag, New York, 2011.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19] O. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.   Google Scholar
[20]

J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.  Google Scholar

[21]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.   Google Scholar

[22] R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988.   Google Scholar
[23]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.  Google Scholar

[24] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.   Google Scholar

show all references

References:
[1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015.   Google Scholar
[6] I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999.   Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2 edition, Springer-Verlag, New York, 2011.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19] O. Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969.   Google Scholar
[20]

J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.  Google Scholar

[21]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.   Google Scholar

[22] R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988.   Google Scholar
[23]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.  Google Scholar

[24] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.   Google Scholar
[1]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[2]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[3]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[4]

Cung The Anh, Dang Thi Phuong Thanh, Nguyen Duong Toan. Uniform attractors of 3D Navier-Stokes-Voigt equations with memory and singularly oscillating external forces. Evolution Equations & Control Theory, 2021, 10 (1) : 1-23. doi: 10.3934/eect.2020039
[5]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110
[6]

Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021005
[7]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349
[8]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020408
[9]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074
[10]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001
[11]

Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062
[12]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[13]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[14]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141
[15]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[16]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[17]

Balázs Kósa, Karol Mikula, Markjoe Olunna Uba, Antonia Weberling, Neophytos Christodoulou, Magdalena Zernicka-Goetz. 3D image segmentation supported by a point cloud. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 971-985. doi: 10.3934/dcdss.2020351
[18]

Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143
[19]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128
[20]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (100)
  • HTML views (162)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences