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On Dirichlet, Poncelet and Abel problems
A global attractor for a fluid--plate interaction model
1. | Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody sq., 61077, Kharkov, Ukraine |
2. | Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody Sq., Kharkov, 61022, Ukraine |
References:
[1] |
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. |
[2] |
G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction I. Explicit semigroup generator and its spectral properties, in "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 15-54.
doi: 10.1090/conm/440/08475. |
[3] |
G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolc-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. |
[4] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. |
[5] |
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., 440, AMS, Providence, RI, (2007), 55-82.
doi: 10.1090/conm/440/08476. |
[6] |
V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1173-207.
doi: 10.1512/iumj.2008.57.3284. |
[7] |
H. Beirão da Veiga, On the existence of strong solution to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52.
doi: 10.1007/s00021-003-0082-5. |
[8] |
V. V. Bolotin, "Nonconservative Problems of Elastic Stability," Pergamon Press, Oxford, 1963. |
[9] |
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. |
[10] |
I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/. |
[11] |
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. |
[12] |
I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping, Comm. Pure Appl. Anal., 11 (2012), 659-674.
doi: 10.3934/cpaa.2012.11.659. |
[13] |
I. Chueshov and I. Lasiecka, Attractors for second order evolution equations, J. Dynam. Diff. Eqs., 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[14] |
I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping," Memoirs of AMS, vol.195, no. 912, AMS, Providence, RI, 2008. |
[15] |
I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations," Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[16] |
I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents, Preprint arXiv:1204.5864v1. |
[17] |
I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, Preprint arXiv:1112.6094v1. |
[18] |
I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in "System Modeling and Optimization: 25th IFIP TC7 Conference, Berlin, Germany, Sept. 2011," Springer, in press. |
[19] |
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. |
[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] |
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. |
[22] |
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. |
[23] |
M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech., 10 (2008), 388-401.
doi: 10.1007/s00021-006-0236-4. |
[24] |
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. |
[25] |
M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466.
doi: 10.1002/mma.1104. |
[26] |
M. Guidorzi, M. Padula and P. I. Plotnikov, Hopf solutions to a fluid-elastic interaction model, Math. Models Methods Appl. Sci., 18 (2008), 215-269.
doi: 10.1142/S0218202508002668. |
[27] |
N. Kopachevskii and Yu. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane, Russian J. Math. Phys., 5 (1998), 459-472. |
[28] |
O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow," GIFML, Moscow, 1961 (1st Russian edition); Nauka, Moscow, 1970 (2nd Russian edition); Gordon and Breach, New York, 1963 and 1969 (English translations of the 1st Russian edition). |
[29] |
J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM, Philadelphia, 1989.
doi: 10.1137/1.9781611970821. |
[30] |
J. Lagnese, Modeling and stabilization of nonlinear plates, Int. Ser. Num. Math., 100 (1991), 247-264. |
[31] |
J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates," Masson, Paris, 1988. |
[32] |
J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal. 43 (2011), 389-410.
doi: 10.1137/10078983X. |
[33] |
J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1, (French), Dunod, Paris, 1968. |
[34] |
J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod, Paris, 1969. |
[35] |
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. |
[36] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1986. |
[37] |
G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems," Elsevier Sciences, Amsterdam, 2 (2002), 885-992.
doi: 10.1016/S1874-575X(02)80038-8. |
[38] |
J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM Journal on Control and Optimization, 48 (2010), 5398-5443.
doi: 10.1137/080744761. |
[39] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, Ser. 4, 148 (1987), 65-96.
doi: 10.1007/BF01762360. |
[40] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[41] |
R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
[42] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 1978. |
show all references
References:
[1] |
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. |
[2] |
G. Avalos and R. Triggiani, The coupled PDE system arising in fluid-structure interaction I. Explicit semigroup generator and its spectral properties, in "Fluids and Waves," Contemp. Math., 440, AMS, Providence, RI, (2007), 15-54.
doi: 10.1090/conm/440/08475. |
[3] |
G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolc-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. |
[4] |
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992. |
[5] |
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., 440, AMS, Providence, RI, (2007), 55-82.
doi: 10.1090/conm/440/08476. |
[6] |
V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J., 57 (2008), 1173-207.
doi: 10.1512/iumj.2008.57.3284. |
[7] |
H. Beirão da Veiga, On the existence of strong solution to a coupled fluid-structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21-52.
doi: 10.1007/s00021-003-0082-5. |
[8] |
V. V. Bolotin, "Nonconservative Problems of Elastic Stability," Pergamon Press, Oxford, 1963. |
[9] |
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. |
[10] |
I. Chueshov, "Introduction to the Theory of Infinite-Dimensional Dissipative Systems," Acta, Kharkov, 1999 (in Russian); English translation: Acta, Kharkov, 2002. Available from: http://www.emis.de/monographs/Chueshov/. |
[11] |
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. |
[12] |
I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping, Comm. Pure Appl. Anal., 11 (2012), 659-674.
doi: 10.3934/cpaa.2012.11.659. |
[13] |
I. Chueshov and I. Lasiecka, Attractors for second order evolution equations, J. Dynam. Diff. Eqs., 16 (2004), 469-512.
doi: 10.1007/s10884-004-4289-x. |
[14] |
I. Chueshov and I. Lasiecka, "Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping," Memoirs of AMS, vol.195, no. 912, AMS, Providence, RI, 2008. |
[15] |
I. Chueshov and I. Lasiecka, "Von Karman Evolution Equations," Springer, New York, 2010.
doi: 10.1007/978-0-387-87712-9. |
[16] |
I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents, Preprint arXiv:1204.5864v1. |
[17] |
I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, Preprint arXiv:1112.6094v1. |
[18] |
I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, in "System Modeling and Optimization: 25th IFIP TC7 Conference, Berlin, Germany, Sept. 2011," Springer, in press. |
[19] |
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. |
[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] |
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. |
[22] |
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. |
[23] |
M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech., 10 (2008), 388-401.
doi: 10.1007/s00021-006-0236-4. |
[24] |
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. |
[25] |
M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466.
doi: 10.1002/mma.1104. |
[26] |
M. Guidorzi, M. Padula and P. I. Plotnikov, Hopf solutions to a fluid-elastic interaction model, Math. Models Methods Appl. Sci., 18 (2008), 215-269.
doi: 10.1142/S0218202508002668. |
[27] |
N. Kopachevskii and Yu. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane, Russian J. Math. Phys., 5 (1998), 459-472. |
[28] |
O. Ladyzhenskaya, "Mathematical Theory of Viscous Incompressible Flow," GIFML, Moscow, 1961 (1st Russian edition); Nauka, Moscow, 1970 (2nd Russian edition); Gordon and Breach, New York, 1963 and 1969 (English translations of the 1st Russian edition). |
[29] |
J. Lagnese, "Boundary Stabilization of Thin Plates," SIAM, Philadelphia, 1989.
doi: 10.1137/1.9781611970821. |
[30] |
J. Lagnese, Modeling and stabilization of nonlinear plates, Int. Ser. Num. Math., 100 (1991), 247-264. |
[31] |
J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates," Masson, Paris, 1988. |
[32] |
J. Lequeurre, Existence of strong solutions to a fluid-structure system, SIAM J. Math. Anal. 43 (2011), 389-410.
doi: 10.1137/10078983X. |
[33] |
J.-L. Lions and E. Magenes, "Problémes aux Limites non Homogénes et Applications," Vol. 1, (French), Dunod, Paris, 1968. |
[34] |
J. L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French), Dunod, Paris, 1969. |
[35] |
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. |
[36] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer, New York, 1986. |
[37] |
G. Raugel, Global attractors in partial differential equations, in "Handbook of Dynamical Systems," Elsevier Sciences, Amsterdam, 2 (2002), 885-992.
doi: 10.1016/S1874-575X(02)80038-8. |
[38] |
J.-P. Raymond, Feedback stabilization of a fluid-structure model, SIAM Journal on Control and Optimization, 48 (2010), 5398-5443.
doi: 10.1137/080744761. |
[39] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed Applicata, Ser. 4, 148 (1987), 65-96.
doi: 10.1007/BF01762360. |
[40] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer, New York, 1988.
doi: 10.1007/978-1-4684-0313-8. |
[41] |
R. Temam, "Navier-Stokes Equations: Theory and Numerical Analysis," Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. |
[42] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North Holland, Amsterdam, 1978. |
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