# American Institute of Mathematical Sciences

June  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 and 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), 837-852. doi: 10.2307/2000826. [3] 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. [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), 259-280. 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., dedicated to the memory of J. L. Lions, 15 (2008), 403-437. [6] G. Avalos, I. Lasiecka and R. Triggiani, Optimal rational decay of a parabolic-hyperbolic system with boundary interface, (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 "Fluids and Waves," Contemp. Math., 440, Amer. Math. Soc., Providence, RI, (2007), 15-54. 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), 817-835. 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), 737-761. 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), 417-447. 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), 1357-1396. 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), 341-370. 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). [14] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions, in "Fluids and Waves," Contemp. Math., 440, Amer. Math. Soc., Providence, RI, (2007), 55-82. [15] 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. [16] K. N. Boyadzhiev and N. Levan, Strong stability of Hilbert space contraction semigroups, Stud. Sci. Math. Hung., 30 (1995), 162-182. [17] H. Cohen and S. I. Rubinow, "Some Mathematical Topics in Biology," Proc. Symp. on System Theory, Polytechnic Press, New York, (1965), 321-337. [18] P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, The University of Chicago Press, Chicago IL, 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), 633-650. 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), 17-45. [21] 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. [22] B. Kellogg, Properties of solutions of elliptic boundary value problems, in "The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations" (ed. A. K. Aziz), Academic Press, New York, (1972), 47-81. [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, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures et Appl., 65 (1986), 149-192. [25] I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1986), 243-290; preliminary version in Springer Verlag Lecture Notes, 100 (1987), 316-371. [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), 189-224. doi: 10.1007/BF01182480. [27] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. I," Cambridge University Press, New York, 2000. [28] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. II," Cambridge University Press, New York, 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, (IV) CLVII (1990), 285-367. [30] I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler Bernoulli equations , Applied Math. Optimization , 28, (1993), 277-306. [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(3), (2000), 981-1023. [32] N. Levan, The stabilizability problem: A Hilbert space operator decomposition approach, Special issue on the mathematical foundations of system theory, IEEE Trans. Circuits & Sys., 25 (1978), 721-727. doi: 10.1109/TCS.1978.1084539. [33] J.-L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires," Dunod; Gauthier-Villars, Paris, 1969. [34] J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications, Vol. I," Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. [35] Y. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach space, Stud. Math., 88 (1988), 37-42. [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), 97-127. [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), 691-707. 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), 692-704. 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), 639-739. doi: 10.1137/1020095. [41] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 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 "Optimization Techniques" (Proc. 8th IFIP Conf., Würzburg, 1977), Part 1, Lecture Notes in Control and Information Sciences, 6, Springer, Berlin, (1978), 380-390. [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), 241-277; 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. [44] 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. [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), 49-120. 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). [2] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852. doi: 10.2307/2000826. [3] 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. [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), 259-280. 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., dedicated to the memory of J. L. Lions, 15 (2008), 403-437. [6] G. Avalos, I. Lasiecka and R. Triggiani, Optimal rational decay of a parabolic-hyperbolic system with boundary interface, (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 "Fluids and Waves," Contemp. Math., 440, Amer. Math. Soc., Providence, RI, (2007), 15-54. 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), 817-835. 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), 737-761. 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), 417-447. 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), 1357-1396. 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), 341-370. 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). [14] V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Weak and strong solutions in nonlinear fluid-structure interactions, in "Fluids and Waves," Contemp. Math., 440, Amer. Math. Soc., Providence, RI, (2007), 55-82. [15] 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. [16] K. N. Boyadzhiev and N. Levan, Strong stability of Hilbert space contraction semigroups, Stud. Sci. Math. Hung., 30 (1995), 162-182. [17] H. Cohen and S. I. Rubinow, "Some Mathematical Topics in Biology," Proc. Symp. on System Theory, Polytechnic Press, New York, (1965), 321-337. [18] P. Constantin and C. Foias, "Navier-Stokes Equations," Chicago Lectures in Mathematics, The University of Chicago Press, Chicago IL, 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), 633-650. 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), 17-45. [21] 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. [22] B. Kellogg, Properties of solutions of elliptic boundary value problems, in "The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations" (ed. A. K. Aziz), Academic Press, New York, (1972), 47-81. [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, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures et Appl., 65 (1986), 149-192. [25] I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1986), 243-290; preliminary version in Springer Verlag Lecture Notes, 100 (1987), 316-371. [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), 189-224. doi: 10.1007/BF01182480. [27] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. I," Cambridge University Press, New York, 2000. [28] I. Lasiecka and R. Triggiani, "Control Theory for Partial Differential Equations, Vol. II," Cambridge University Press, New York, 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, (IV) CLVII (1990), 285-367. [30] I. Lasiecka and R. Triggiani, Sharp trace estimates of solutions to Kirchhoff and Euler Bernoulli equations , Applied Math. Optimization , 28, (1993), 277-306. [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(3), (2000), 981-1023. [32] N. Levan, The stabilizability problem: A Hilbert space operator decomposition approach, Special issue on the mathematical foundations of system theory, IEEE Trans. Circuits & Sys., 25 (1978), 721-727. doi: 10.1109/TCS.1978.1084539. [33] J.-L. Lions, "Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires," Dunod; Gauthier-Villars, Paris, 1969. [34] J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications, Vol. I," Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. [35] Y. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach space, Stud. Math., 88 (1988), 37-42. [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), 97-127. [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), 691-707. 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), 692-704. 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), 639-739. doi: 10.1137/1020095. [41] H. Sohr, "The Navier-Stokes Equations. An Elementary Functional Analytic Approach," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 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 "Optimization Techniques" (Proc. 8th IFIP Conf., Würzburg, 1977), Part 1, Lecture Notes in Control and Information Sciences, 6, Springer, Berlin, (1978), 380-390. [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), 241-277; 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. [44] 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. [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), 49-120. doi: 10.1007/s00205-006-0020-x.
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