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

On interaction of circular cylindrical shells with a Poiseuille type flow

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.
Citation: Igor Chueshov, Tamara Fastovska. On interaction of circular cylindrical shells with a Poiseuille type flow. Evolution Equations and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

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

[29]

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

[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.

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

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