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December  2016, 5(4): 605-629. doi: 10.3934/eect.2016021

On interaction of circular cylindrical shells with a Poiseuille type flow

Igor Chueshov 1, and  Tamara Fastovska 2,

1. 

Kharkov National Universit, Department of Mathematics and Mechanics, 4 Svobody sq, 61077 Kharkov
2. 

Department of Mathematics and Mechanics, Kharkov Karazin National University, 4, Svobody sq., Kharkov 61077

Received  July 2016 Revised  September 2016 Published  October 2016

We study dynamics of a coupled system consisting of the 3D Navier--Stokes equations which is linearized near a certain Poiseuille type flow between two unbounded circular cylinders and nonlinear elasticity equations for the transversal displacements of the bounding cylindrical shells. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.
Keywords: Fluid--structure interaction, nonlinear circular shell, attractor., Poiseulle flow, linearized 3D Navier--Stokes equations.
Mathematics Subject Classification: Primary: 74F10, 35B41; Secondary: 35Q30, 74K2.
Citation: Igor Chueshov, Tamara Fastovska. On interaction of circular cylindrical shells with a Poiseuille type flow. Evolution Equations & Control Theory, 2016, 5 (4) : 605-629. doi: 10.3934/eect.2016021
References:
[1]

M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-14648-5.  Google Scholar

[2]

G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z.  Google Scholar

[3]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, New prospects in direct, inverse and control problems for evolution equations, Springer INdAM Series, 10 (2014), 49-78. doi: 10.1007/978-3-319-11406-4_3.  Google Scholar

[4]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic- hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Ser.S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[5]

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

[6]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and Waves, Contemp. Math., vol. 440, AMS, Providence, RI, (2007), 55-82. doi: 10.1090/conm/440/08476.  Google Scholar

[7]

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

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar

[9]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 2002. http://www.emis.de/monographs/Chueshov/. Google Scholar

[10]

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. doi: 10.1002/mma.1496.  Google Scholar

[11]

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. doi: 10.1090/memo/0912.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[13]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635.  Google Scholar

[14]

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

[15]

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

[16]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7.  Google Scholar

[17]

E. H. Donnell, A new theory for the buckling of thin cylinders under axial compression and bending, Trans. ASME, 56 (1934), 795-806. Google Scholar

[18]

D. A. Evensen, Nonlinear Fexural Vibrations of Thin-Walled Circular Cylinders, NASA TN D-4090., 1967. Google Scholar

[19]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[20]

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

[21]

N. Große and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr., 286 (2013), 1586-1613. doi: 10.1002/mana.201300007.  Google Scholar

[22]

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

[23]

D. Fujiwara, Concrete characterizations of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 ( 1967), 82-86. doi: 10.3792/pja/1195521686.  Google Scholar

[24]

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

[25]

M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model, Applicable Analysis 88 (2009), 1053-1065. doi: 10.1080/00036810903114841.  Google Scholar

[26]

M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. doi: 10.1002/mma.1104.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 497-513. doi: 10.1051/cocv:1999119.  Google Scholar

[31]

M. E. Taylor, Partial Differential Equations 1. Basic theory, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[32]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[33]

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

[34]

H. Triebel, Theory of Function Spaces 2, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[35]

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

show all references

References:
[1]

M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-14648-5.  Google Scholar

[2]

G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z.  Google Scholar

[3]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, New prospects in direct, inverse and control problems for evolution equations, Springer INdAM Series, 10 (2014), 49-78. doi: 10.1007/978-3-319-11406-4_3.  Google Scholar

[4]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic- hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Ser.S, 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[5]

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

[6]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in Fluids and Waves, Contemp. Math., vol. 440, AMS, Providence, RI, (2007), 55-82. doi: 10.1090/conm/440/08476.  Google Scholar

[7]

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

[8]

I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015. doi: 10.1007/978-3-319-22903-4.  Google Scholar

[9]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 2002. http://www.emis.de/monographs/Chueshov/. Google Scholar

[10]

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. doi: 10.1002/mma.1496.  Google Scholar

[11]

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. doi: 10.1090/memo/0912.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[13]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656. doi: 10.3934/cpaa.2013.12.1635.  Google Scholar

[14]

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

[15]

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

[16]

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal., 176 (2005), 25-102. doi: 10.1007/s00205-004-0340-7.  Google Scholar

[17]

E. H. Donnell, A new theory for the buckling of thin cylinders under axial compression and bending, Trans. ASME, 56 (1934), 795-806. Google Scholar

[18]

D. A. Evensen, Nonlinear Fexural Vibrations of Thin-Walled Circular Cylinders, NASA TN D-4090., 1967. Google Scholar

[19]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, $2^{nd}$ edition, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[20]

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

[21]

N. Große and C. Schneider, Sobolev spaces on Riemannian manifolds with bounded geometry: general coordinates and traces, Math. Nachr., 286 (2013), 1586-1613. doi: 10.1002/mana.201300007.  Google Scholar

[22]

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

[23]

D. Fujiwara, Concrete characterizations of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 ( 1967), 82-86. doi: 10.3792/pja/1195521686.  Google Scholar

[24]

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

[25]

M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model, Applicable Analysis 88 (2009), 1053-1065. doi: 10.1080/00036810903114841.  Google Scholar

[26]

M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. doi: 10.1002/mma.1104.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction, ESAIM: Control, Optimisation and Calculus of Variations, 4 (1999), 497-513. doi: 10.1051/cocv:1999119.  Google Scholar

[31]

M. E. Taylor, Partial Differential Equations 1. Basic theory, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[32]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[33]

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

[34]

H. Triebel, Theory of Function Spaces 2, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[35]

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

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