# American Institute of Mathematical Sciences

December  2013, 2(4): 563-598. doi: 10.3934/eect.2013.2.563

## Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588 2 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152-3240

Received  May 2013 Revised  September 2013 Published  November 2013

We consider a coupled parabolic--hyperbolic PDE system arising in fluid--structure interaction, where the coupling is exercised at the interface between the two media. This paper is a study in contrast on stability properties of the overall coupled system under two scenarios: the case with interior dissipation of the structure, and the case without. In the first case, uniform stabilization is achieved (by a $\lambda$-domain analysis) without geometrical conditions on the structure, but only on an explicitly identified space $Ĥ$ of codimension one with respect to the original energy state space $H$ where semigroup well-posedness holds. In the second case, only rational (a fortiori strong) stability is possible, again only on the space $Ĥ$, however, under geometrical conditions of the structure, which e.g., exclude a sphere. Many classes of good geometries are identified. Recent papers [6,9] show uniform stabilization on all of $H$, and without geometrical conditions; however, with dissipation at the boundary interface.
Citation: George Avalos, Roberto Triggiani. Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability. Evolution Equations and Control Theory, 2013, 2 (4) : 563-598. doi: 10.3934/eect.2013.2.563
##### References:
 [1] G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. & Optimiz., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z. [2] 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), 259-280. doi: 10.4064/am35-3-2. [3] G. Avalos, I. Lasiecka, and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, invited paper, special issue of Georgian Math. J., 15 (2008), 403-437; dedicated to the memory of J. L. Lions. [4] G. Avalos, I. Lasiecka and R. Triggiani, Optimal rational decay of a parabolic-hyperbolic system with boundary interface,, 2012., (). [5] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: Explicit semigroup generator and its spectral properties, Fluids and Waves, AMS Contemp. Math., 440 (2007), 15-54. doi: 10.1090/conm/440/08475. [6] 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), 817-835, special issue. doi: 10.3934/dcds.2008.22.817. [7] G. Avalos and R. Triggiani, Semigroup wellposedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Cont. Dynam. Sys., 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [8] 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), 1357-1396. doi: 10.1080/00036810903278513. [9] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Eqns., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9. [10] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evolution Equations and Control Theory, 2 (2013), to appear. [11] G. Avalos and R. Triggiani, Rational decay rates for a PDE fluid-structure interaction via a resolvent operator approach,, 2012., (). [12] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions, Fluids and Waves, AMS Contemp. Math., 440 (2007), 55-82. [13] L. Bers, F. John and M. Schechter, Partial Differential Equations, John Wiley 1964, reprinted by AMS Lectures in Applied Mathematics [14] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [15] K. N. Boyadzhiev and N. Levan, Strong stability of Hilbert space contraction semigroups, Stud. Sci. Math. Hung., 30 (1995), 162-182. [16] H. Cohen and S. I. Rubinow, Some mathematical topics in biology, Proc. Symp. on System Theory Polytechnic Press, New York (1965), 321-337. [17] P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988. [18] 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), 633-650. doi: 10.3934/dcds.2003.9.633. [19] T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface, Asymptotic Analysis, 51 (2007), 17-45. [20] L. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904. doi: 10.1093/imamat/hxq038. [21] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, 1994. [22] S. G. Krein, Linear Differential Equations in Banach Space, Vol. 29, Translations of Mathematical Monographs, AMS 1971, 390 pp. [23] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Diff. Eqns., 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [24] I. Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes elastic wave interaction. Semigroup Forum, 82 (2011), 61-82. doi: 10.1007/s00233-010-9281-7. [25] I. Lasiecka and Y. Lu, Interface feedback control stabilization of a nonlinear fluid-structure interaction. Nonlinear Analysis, 75 (2012), 1449-1460. doi: 10.1016/j.na.2011.04.018. [26] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures et Appl., 65 (1986), 149-192. [27] I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1986), 243-290. (Also, preliminary version in Springer Verlag Lecture Notes, 100 (1987), 316-371.) doi: 10.1007/BF01448201. [28] 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), 189-224. doi: 10.1007/BF01182480. [29] I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equationjs of Neumann type, Part I: The $L_2$ boundary case, Annali Matem. Pura e Applicata, 157 (1990), 285-367. [30] I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler Bernoulli Equations, Applied. Math. Optimiz., 28 (1993), 277-306. doi: 10.1007/BF01200382. [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., 30 (2000), 981-1023. doi: 10.1216/rmjm/1021477256. [32] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations Vol. I, Cambridge University Press, New York, 2000. [33] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations Vol. II, Cambridge University Press, New York, 2000. [34] N. Levan, The stabilizability problem: A Hilbert space operator decomposition approach, IEEE Trans. Circuits & Sys., 25 (1978), 721-727. doi: 10.1109/TCS.1978.1084539. [35] J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Paris, 1969. [36] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, 1972. [37] W. Littman and L. Markus, Stabilization of a hybrid-type of elasticity by feedback boundary damping, Annali di Matem. Pura, 152 (1988), 281-330. [38] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach space, Stud. Math. LXXXVII (1988), 721-727. [39] V.P. Mikhailov, Partial Differential Equations, MIR Publishers Moscow, 1978. [40] L. Monauni, Exponential decay of solutions to Cauchy's Abstract problem as determined by the extended spectrum of the dynamic operator. Unpublished manuscript, 1981 (MIT Report). [41] J. Pruss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. [42] 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, 77A (1977), 97-112. [43] J. M. Rivera, private communication, May 2012. [44] J. E. M. Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, JMAA, 326 (2007), 691-707. doi: 10.1016/j.jmaa.2006.03.022. [45] J. E. M. Rivera, M. G. Naso, and F. Vagni, Asymptotic behavior of the energy for a class of weakly dissipative second-order system with memory, JMAA, 286 (2003), 692-704. doi: 10.1016/S0022-247X(03)00511-0. [46] D. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-740. doi: 10.1137/1020095. [47] H. Sohr, The Navier-Stokes Equations, An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts, 2001, 367 pp. doi: 10.1007/978-3-0348-8255-2. [48] A. E. Taylor and D. C. Lay, Introduciton to Functional Analysis, 2nd ed., John Wiley, New YHork, 1980. [49] R. Temam, Navier-Stokes Equations, North Holland, 1979. [50] R. Triggiani, A cosine operator approach to modeling boundary input problems for hyperbolic systems, Springer-Verlag Lecture Notes in Control and Information Sciences, 6 (1978), 380-390. [51] 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), 241-277. (Also, preliminary version in Springer-Verlag Lecture Notes, 102 (1987), 291-332; Proceedings of Workshop on Control for Distributed Parameter Systems, University of Graz, Austria (July 1986)). doi: 10.1007/BF01443625. [52] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl., 137 (1989), 438-461. doi: 10.1016/0022-247X(89)90255-2. [53] E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications, The Williams and Wilkins Company, Baltimore, 1976, 404 pp. [54] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system in fluid-structure interaction, Arch. Rat. Mech. Anal., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x.

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##### References:
 [1] G. Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. & Optimiz., 55 (2007), 163-184. doi: 10.1007/s00245-006-0884-z. [2] 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), 259-280. doi: 10.4064/am35-3-2. [3] G. Avalos, I. Lasiecka, and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system, invited paper, special issue of Georgian Math. J., 15 (2008), 403-437; dedicated to the memory of J. L. Lions. [4] G. Avalos, I. Lasiecka and R. Triggiani, Optimal rational decay of a parabolic-hyperbolic system with boundary interface,, 2012., (). [5] G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: Explicit semigroup generator and its spectral properties, Fluids and Waves, AMS Contemp. Math., 440 (2007), 15-54. doi: 10.1090/conm/440/08475. [6] 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), 817-835, special issue. doi: 10.3934/dcds.2008.22.817. [7] G. Avalos and R. Triggiani, Semigroup wellposedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Cont. Dynam. Sys., 2 (2009), 417-447. doi: 10.3934/dcdss.2009.2.417. [8] 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), 1357-1396. doi: 10.1080/00036810903278513. [9] G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system, J. Evol. Eqns., 9 (2009), 341-370. doi: 10.1007/s00028-009-0015-9. [10] G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evolution Equations and Control Theory, 2 (2013), to appear. [11] G. Avalos and R. Triggiani, Rational decay rates for a PDE fluid-structure interaction via a resolvent operator approach,, 2012., (). [12] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions, Fluids and Waves, AMS Contemp. Math., 440 (2007), 55-82. [13] L. Bers, F. John and M. Schechter, Partial Differential Equations, John Wiley 1964, reprinted by AMS Lectures in Applied Mathematics [14] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0. [15] K. N. Boyadzhiev and N. Levan, Strong stability of Hilbert space contraction semigroups, Stud. Sci. Math. Hung., 30 (1995), 162-182. [16] H. Cohen and S. I. Rubinow, Some mathematical topics in biology, Proc. Symp. on System Theory Polytechnic Press, New York (1965), 321-337. [17] P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988. [18] 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), 633-650. doi: 10.3934/dcds.2003.9.633. [19] T. Duyckaerts, Optimal decay rates of the energy of a hyperbolic-parabolic system coupled by an interface, Asymptotic Analysis, 51 (2007), 17-45. [20] L. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904. doi: 10.1093/imamat/hxq038. [21] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, 1994. [22] S. G. Krein, Linear Differential Equations in Banach Space, Vol. 29, Translations of Mathematical Monographs, AMS 1971, 390 pp. [23] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Diff. Eqns., 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [24] I. Lasiecka and Y. Lu, Asymptotic stability of finite energy in Navier Stokes elastic wave interaction. Semigroup Forum, 82 (2011), 61-82. doi: 10.1007/s00233-010-9281-7. [25] I. Lasiecka and Y. Lu, Interface feedback control stabilization of a nonlinear fluid-structure interaction. Nonlinear Analysis, 75 (2012), 1449-1460. doi: 10.1016/j.na.2011.04.018. [26] I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures et Appl., 65 (1986), 149-192. [27] I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1986), 243-290. (Also, preliminary version in Springer Verlag Lecture Notes, 100 (1987), 316-371.) doi: 10.1007/BF01448201. [28] 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), 189-224. doi: 10.1007/BF01182480. [29] I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equationjs of Neumann type, Part I: The $L_2$ boundary case, Annali Matem. Pura e Applicata, 157 (1990), 285-367. [30] I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler Bernoulli Equations, Applied. Math. Optimiz., 28 (1993), 277-306. doi: 10.1007/BF01200382. [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., 30 (2000), 981-1023. doi: 10.1216/rmjm/1021477256. [32] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations Vol. I, Cambridge University Press, New York, 2000. [33] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations Vol. II, Cambridge University Press, New York, 2000. [34] N. Levan, The stabilizability problem: A Hilbert space operator decomposition approach, IEEE Trans. Circuits & Sys., 25 (1978), 721-727. doi: 10.1109/TCS.1978.1084539. [35] J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Paris, 1969. [36] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, 1972. [37] W. Littman and L. Markus, Stabilization of a hybrid-type of elasticity by feedback boundary damping, Annali di Matem. Pura, 152 (1988), 281-330. [38] Y. I. Lyubich and V. Q. Phong, Asymptotic stability of linear differential equations in Banach space, Stud. Math. LXXXVII (1988), 721-727. [39] V.P. Mikhailov, Partial Differential Equations, MIR Publishers Moscow, 1978. [40] L. Monauni, Exponential decay of solutions to Cauchy's Abstract problem as determined by the extended spectrum of the dynamic operator. Unpublished manuscript, 1981 (MIT Report). [41] J. Pruss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. [42] 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, 77A (1977), 97-112. [43] J. M. Rivera, private communication, May 2012. [44] J. E. M. Rivera and M. G. Naso, Asymptotic stability of semigroups associated with linear weak dissipative systems with memory, JMAA, 326 (2007), 691-707. doi: 10.1016/j.jmaa.2006.03.022. [45] J. E. M. Rivera, M. G. Naso, and F. Vagni, Asymptotic behavior of the energy for a class of weakly dissipative second-order system with memory, JMAA, 286 (2003), 692-704. doi: 10.1016/S0022-247X(03)00511-0. [46] D. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-740. doi: 10.1137/1020095. [47] H. Sohr, The Navier-Stokes Equations, An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts, 2001, 367 pp. doi: 10.1007/978-3-0348-8255-2. [48] A. E. Taylor and D. C. Lay, Introduciton to Functional Analysis, 2nd ed., John Wiley, New YHork, 1980. [49] R. Temam, Navier-Stokes Equations, North Holland, 1979. [50] R. Triggiani, A cosine operator approach to modeling boundary input problems for hyperbolic systems, Springer-Verlag Lecture Notes in Control and Information Sciences, 6 (1978), 380-390. [51] 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), 241-277. (Also, preliminary version in Springer-Verlag Lecture Notes, 102 (1987), 291-332; Proceedings of Workshop on Control for Distributed Parameter Systems, University of Graz, Austria (July 1986)). doi: 10.1007/BF01443625. [52] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl., 137 (1989), 438-461. doi: 10.1016/0022-247X(89)90255-2. [53] E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications, The Williams and Wilkins Company, Baltimore, 1976, 404 pp. [54] X. Zhang and E. Zuazua, Long-time behavior of a coupled heat-wave system in fluid-structure interaction, Arch. Rat. Mech. Anal., 184 (2007), 49-120. doi: 10.1007/s00205-006-0020-x.
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