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Stability analysis of non-linear plates coupled with Darcy flows
Rational decay rates for a PDE heat--structure interaction: A frequency domain approach
1. | Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, United States |
2. | Department of Mathematics, University of Virginia, Charlottesville, VA 22903 |
References:
[1] |
F. Abdullah, D. Mercier, and S. Nicaise, Spectral analysis and exponential or polynomial stability and exponential or polynomial stability of some indefinite sign damped problems,, preprint, (2012). Google Scholar |
[2] |
W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837.
doi: 10.2307/2000826. |
[3] |
G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optimiz., 55 (2007), 163.
doi: 10.1007/s00245-006-0884-z. |
[4] |
G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method,, Applicationes Mathematicae, 35 (2008), 259.
doi: 10.4064/am35-3-2. |
[5] |
G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system,, special issue of Georgian Math. J., 15 (2008), 403.
|
[6] |
G. Avalos, I. Lasiecka and R. Triggiani, Optimal rational decay of a parabolic-hyperbolic system with boundary interface,, (2012)., (2012). Google Scholar |
[7] |
G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: Explicit semigroup generator and its spectral properties,, in, 440 (2007), 15.
doi: 10.1090/conm/440/08475. |
[8] |
G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface,, Discr. Cont. Dynam. Sys., 22 (2008), 817.
doi: 10.3934/dcds.2008.22.817. |
[9] |
G. Avalos and R. Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction,, J. Diff. Eqns., 245 (2008), 737.
doi: 10.1016/j.jde.2007.10.036. |
[10] |
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. Cont. Dynam. Sys., 2 (2009), 417.
doi: 10.3934/dcdss.2009.2.417. |
[11] |
G. Avalos and R. Triggiani, Coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behavior of the resolvent operator on the imaginary axis,, Applicable Analysis, 88 (2009), 1357.
doi: 10.1080/00036810903278513. |
[12] |
G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Eqns., 9 (2009), 341.
doi: 10.1007/s00028-009-0015-9. |
[13] |
G. Avalos and R. Triggiani, Rational decay rates for a fluid-structure interaction model via a resolvent-based approach,, (2013)., (2013). Google Scholar |
[14] |
V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions,, in, 440 (2007), 55. Google Scholar |
[15] |
A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.
doi: 10.1007/s00208-009-0439-0. |
[16] |
K. N. Boyadzhiev and N. Levan, Strong stability of Hilbert space contraction semigroups,, Stud. Sci. Math. Hung., 30 (1995), 162.
|
[17] |
H. Cohen and S. I. Rubinow, "Some Mathematical Topics in Biology,", Proc. Symp. on System Theory, (1965), 321. Google Scholar |
[18] |
P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988).
|
[19] |
Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discr. Contin. Dynam. Sys., 9 (2003), 633.
doi: 10.3934/dcds.2003.9.633. |
[20] |
T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface,, Asymptotic Analysis, 51 (2007), 17.
|
[21] |
L. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, IMA J. Appl. Math., 75 (2010), 881.
doi: 10.1093/imamat/hxq038. |
[22] |
B. Kellogg, Properties of solutions of elliptic boundary value problems,, in, (1972), 47. Google Scholar |
[23] |
J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Diff. Eqns., 50 (1983), 163.
doi: 10.1016/0022-0396(83)90073-6. |
[24] |
I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators,, J. Math. Pures et Appl., 65 (1986), 149.
|
[25] |
I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control,, Appl. Math. Optimiz., 19 (1986), 243. Google Scholar |
[26] |
I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions,, Appl. Math. Optim., 25 (1992), 189.
doi: 10.1007/BF01182480. |
[27] |
I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. I,", Cambridge University Press, (2000).
|
[28] |
I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. II,", Cambridge University Press, (2000).
|
[29] |
I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equations of Neumann type, Part I: The $ L_2 $ boundary case ,, Annali Matem. Pura e Applicata, (1990), 285.
|
[30] |
I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler Bernoulli equations ,, Applied Math. Optimization, (1993), 277.
|
[31] |
I. Lasiecka and R. Triggiani, A sharp trace regularity result of Kirchhoff and thermoelastic plate equations with free boundary conditions ,, Rocky Mountain. J.Math., (2000), 981.
|
[32] |
N. Levan, The stabilizability problem: A Hilbert space operator decomposition approach,, Special issue on the mathematical foundations of system theory, 25 (1978), 721.
doi: 10.1109/TCS.1978.1084539. |
[33] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).
|
[34] |
J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications, Vol. I,", Die Grundlehren der mathematischen Wissenschaften, (1972).
|
[35] |
Y. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach space,, Stud. Math., 88 (1988), 37.
|
[36] |
J. P. Quinn and D. L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping,, Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 97.
|
[37] |
, J. M. Rivera,, private communication, (2012). Google Scholar |
[38] |
J. E. Muñoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory,, JMAA, 326 (2007), 691.
doi: 10.1016/j.jmaa.2006.03.022. |
[39] |
J. E. Muñoz Rivera, M. G. Naso and F. Vagni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory,, JMAA, 286 (2003), 692.
doi: 10.1016/S0022-247X(03)00511-0. |
[40] |
D. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Review, 20 (1978), 639.
doi: 10.1137/1020095. |
[41] |
H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2001).
doi: 10.1007/978-3-0348-8255-2. |
[42] |
R. Triggiani, A cosine operator approach to modeling boundary input problems for hyperbolic systems,, in, 6 (1978), 380.
|
[43] |
R. Triggiani, Exact boundary controllability of $L_2(\Omega) \times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems,, Appl. Math. Optimiz., 18 (1988), 241.
doi: 10.1007/BF01443625. |
[44] |
R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach,, J. Math. Anal. Appl., 137 (1989), 438.
doi: 10.1016/0022-247X(89)90255-2. |
[45] |
X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system in fluid-structure interaction,, Arch. Rat. Mech. Anal., 184 (2007), 49.
doi: 10.1007/s00205-006-0020-x. |
show all references
References:
[1] |
F. Abdullah, D. Mercier, and S. Nicaise, Spectral analysis and exponential or polynomial stability and exponential or polynomial stability of some indefinite sign damped problems,, preprint, (2012). Google Scholar |
[2] |
W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups,, Trans. Amer. Math. Soc., 306 (1988), 837.
doi: 10.2307/2000826. |
[3] |
G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optimiz., 55 (2007), 163.
doi: 10.1007/s00245-006-0884-z. |
[4] |
G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method,, Applicationes Mathematicae, 35 (2008), 259.
doi: 10.4064/am35-3-2. |
[5] |
G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system,, special issue of Georgian Math. J., 15 (2008), 403.
|
[6] |
G. Avalos, I. Lasiecka and R. Triggiani, Optimal rational decay of a parabolic-hyperbolic system with boundary interface,, (2012)., (2012). Google Scholar |
[7] |
G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: Explicit semigroup generator and its spectral properties,, in, 440 (2007), 15.
doi: 10.1090/conm/440/08475. |
[8] |
G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface,, Discr. Cont. Dynam. Sys., 22 (2008), 817.
doi: 10.3934/dcds.2008.22.817. |
[9] |
G. Avalos and R. Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction,, J. Diff. Eqns., 245 (2008), 737.
doi: 10.1016/j.jde.2007.10.036. |
[10] |
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. Cont. Dynam. Sys., 2 (2009), 417.
doi: 10.3934/dcdss.2009.2.417. |
[11] |
G. Avalos and R. Triggiani, Coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behavior of the resolvent operator on the imaginary axis,, Applicable Analysis, 88 (2009), 1357.
doi: 10.1080/00036810903278513. |
[12] |
G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, J. Evol. Eqns., 9 (2009), 341.
doi: 10.1007/s00028-009-0015-9. |
[13] |
G. Avalos and R. Triggiani, Rational decay rates for a fluid-structure interaction model via a resolvent-based approach,, (2013)., (2013). Google Scholar |
[14] |
V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions,, in, 440 (2007), 55. Google Scholar |
[15] |
A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455.
doi: 10.1007/s00208-009-0439-0. |
[16] |
K. N. Boyadzhiev and N. Levan, Strong stability of Hilbert space contraction semigroups,, Stud. Sci. Math. Hung., 30 (1995), 162.
|
[17] |
H. Cohen and S. I. Rubinow, "Some Mathematical Topics in Biology,", Proc. Symp. on System Theory, (1965), 321. Google Scholar |
[18] |
P. Constantin and C. Foias, "Navier-Stokes Equations,", Chicago Lectures in Mathematics, (1988).
|
[19] |
Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, Discr. Contin. Dynam. Sys., 9 (2003), 633.
doi: 10.3934/dcds.2003.9.633. |
[20] |
T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface,, Asymptotic Analysis, 51 (2007), 17.
|
[21] |
L. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system,, IMA J. Appl. Math., 75 (2010), 881.
doi: 10.1093/imamat/hxq038. |
[22] |
B. Kellogg, Properties of solutions of elliptic boundary value problems,, in, (1972), 47. Google Scholar |
[23] |
J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Diff. Eqns., 50 (1983), 163.
doi: 10.1016/0022-0396(83)90073-6. |
[24] |
I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators,, J. Math. Pures et Appl., 65 (1986), 149.
|
[25] |
I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control,, Appl. Math. Optimiz., 19 (1986), 243. Google Scholar |
[26] |
I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions,, Appl. Math. Optim., 25 (1992), 189.
doi: 10.1007/BF01182480. |
[27] |
I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. I,", Cambridge University Press, (2000).
|
[28] |
I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. II,", Cambridge University Press, (2000).
|
[29] |
I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equations of Neumann type, Part I: The $ L_2 $ boundary case ,, Annali Matem. Pura e Applicata, (1990), 285.
|
[30] |
I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler Bernoulli equations ,, Applied Math. Optimization, (1993), 277.
|
[31] |
I. Lasiecka and R. Triggiani, A sharp trace regularity result of Kirchhoff and thermoelastic plate equations with free boundary conditions ,, Rocky Mountain. J.Math., (2000), 981.
|
[32] |
N. Levan, The stabilizability problem: A Hilbert space operator decomposition approach,, Special issue on the mathematical foundations of system theory, 25 (1978), 721.
doi: 10.1109/TCS.1978.1084539. |
[33] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires,", Dunod; Gauthier-Villars, (1969).
|
[34] |
J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications, Vol. I,", Die Grundlehren der mathematischen Wissenschaften, (1972).
|
[35] |
Y. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach space,, Stud. Math., 88 (1988), 37.
|
[36] |
J. P. Quinn and D. L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping,, Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 97.
|
[37] |
, J. M. Rivera,, private communication, (2012). Google Scholar |
[38] |
J. E. Muñoz Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory,, JMAA, 326 (2007), 691.
doi: 10.1016/j.jmaa.2006.03.022. |
[39] |
J. E. Muñoz Rivera, M. G. Naso and F. Vagni, Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory,, JMAA, 286 (2003), 692.
doi: 10.1016/S0022-247X(03)00511-0. |
[40] |
D. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Review, 20 (1978), 639.
doi: 10.1137/1020095. |
[41] |
H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2001).
doi: 10.1007/978-3-0348-8255-2. |
[42] |
R. Triggiani, A cosine operator approach to modeling boundary input problems for hyperbolic systems,, in, 6 (1978), 380.
|
[43] |
R. Triggiani, Exact boundary controllability of $L_2(\Omega) \times H^{-1}(\Omega)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems,, Appl. Math. Optimiz., 18 (1988), 241.
doi: 10.1007/BF01443625. |
[44] |
R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach,, J. Math. Anal. Appl., 137 (1989), 438.
doi: 10.1016/0022-247X(89)90255-2. |
[45] |
X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system in fluid-structure interaction,, Arch. Rat. Mech. Anal., 184 (2007), 49.
doi: 10.1007/s00205-006-0020-x. |
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