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Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping
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 |
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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