# American Institute of Mathematical Sciences

2013, 2(2): 233-253. doi: 10.3934/eect.2013.2.233

## 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

Received  December 2012 Revised  February 2013 Published  March 2013

In this paper, we consider a simplified version of a fluid--structure PDE model ---in fact, a heat--structure interaction PDE-model. It is intended to be a first step toward a more realistic fluid--structure PDE model which has been of longstanding interest within the mathematical and biological sciences [33, p. 121], [17], [19]. This physically more sound and mathematically more challenging model will be treated in [13]. The simplified model replaces the linear dynamic Stokes equation with a linear $n$-dimensional heat equation (heat--structure interaction). The entire dynamics manifests both hyperbolic and parabolic features. Our main result is as follows: Given smooth initial data---i.e., data in the domain of the associated semigroup generator---the corresponding solutions decay at the rate $o( t^{-\frac{1}{2}})$ (see Theorem 1.3 below). The basis of our proof is the recently derived resolvent criterion in [15]. In order to apply it, however, suitable PDE-estimates need to be established for each component by also making critical use of the interface conditions. A companion paper [6] will sharpen Lemma 5.8 of the present work by use of a lengthy and technical microlocal argument as in [26,29,30,31], to obtain the optimal value $\alpha =1$; hence, the optimal decay rate $o(t^{-1})$. See Remarks 1.2,1.3.
Citation: George Avalos, Roberto Triggiani. Rational decay rates for a PDE heat--structure interaction: A frequency domain approach. Evolution Equations & Control Theory, 2013, 2 (2) : 233-253. doi: 10.3934/eect.2013.2.233
##### 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). [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). [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). [14] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions,, in, 440 (2007), 55. [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. [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. [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. [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). [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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##### 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). [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). [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). [14] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions,, in, 440 (2007), 55. [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. [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. [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. [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). [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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