# American Institute of Mathematical Sciences

September  2016, 15(5): 1515-1543. doi: 10.3934/cpaa.2016001

## Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers

 1 University of Memphis, Department of Mathematical Sciences, 373 Dunn Hall, Memphis, TN 38152 2 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152

Received  November 2015 Revised  February 2016 Published  July 2016

We consider a heat--structure interaction model where the structure is subject to viscoelastic (strong) damping. This is a preliminary step toward the study of a fluid--structure interaction model where the heat equation is replaced by the linear version of the Navier--Stokes equation as it arises in applications. We prove four main results: analyticity of the corresponding contraction semigroup (which cannot follow by perturbation); sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point $\lambda=-1$ in its continuous spectrum; exponential decay of the semigroup with sharp decay rate; finally, a characterization of the domains of fractional power related to the generator.
Citation: Irena Lasiecka, Roberto Triggiani. Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1515-1543. doi: 10.3934/cpaa.2016001
##### References:
 [1] P. Ausher, S. Hofmann, M. Lacey, A. McIntosh and P. Tehamitchian, The solution of the Kato square root problemm for second order elliptic operators in $R^n$, Annals of Mathematics, 156 (2002), 633-654. doi: 10.2307/3597201. [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 Mathematical Journal, 15 (2008), 403-437; dedicated to the memory of J. L. Lions; J. Mawhin, editor. [4] G. Avalos and R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part I: Explicit semigroup generator and its spectral properties, AMS Contemporary Mathematics, Fluids and Waves, 440 (2007), 15-55. doi: 10.1090/conm/440/08475. [5] 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. Systems, 22 (2008), 817-833 (invited paper). doi: 10.3934/dcds.2008.22.817. [6] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system, Discr. & Cont. Dynam. Systems DCDS-S, 2 (2009), 417-448. doi: 10.3934/dcdss.2009.2.417. [7] G. Avalos and R. Triggiani, A 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. [8] 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. [9] 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), 233-253. doi: 10.3934/eect.2013.2.233. [10] G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast in stability, Evolution Equations and Control Theory, 2 (2013), 563-598, special issue by invitation on the occasion of W. Littman's retirement. doi: 10.3934/eect.2013.2.563. [11] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Birkhauser, 2007, 575 pages. doi: 10.1007/978-0-8176-4581-6. [12] S. Canic, A. Mikelic and J. Tambaca, A two-dimensional effective model describing fluid-structure interaction in blood flow: analysis, simulation and experimental validation, Compte Rendus Mechanique Acad. Sci. Paris, 333 (2005), 867-883. [13] S. Canic, D. Lamponi, A. Mikelic and J. Tambaca, Self-consistent effective equations modeling blood flow in medium-t-large compliant arteries, Multiscale Model. Simul., 3 (2005), 559-596. doi: 10.1137/030602605. [14] G. Chen and D.L. Russel, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., (1982), 433-454. [15] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234-256. Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987). doi: 10.1007/BFb0089601. [16] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$, Pacific J. Math., 136 (1989), 15-55. [17] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293. doi: 10.1016/0022-0396(90)90100-4. [18] S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle perturbation, Proceedings Amer. Math. Soc., 110 (1990), 401-415. doi: 10.2307/2048084. [19] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discr. Dynam. Sys., 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. [20] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82-86. [21] P. Grisvard, Characterization de qualques espaces d' interpolation, Arch. Pat. Mech. Anal., 25 (1967), 40-63. [22] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity 27 (2014), 467-499. doi: 10.1088/0951-7715/27/3/467. [23] T. Kato, Fractional powers of dissipative operators, J.Math.Soc. Japan , 13 (1961), 246-274. [24] V. Komornik, Exact controllability and stabilization. The multiplier method, Masson, Paris; John Wiley & Sons Ltd, Chichester (1994), 156 pp. [25] I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction, Indiana Univ. Math. J., 61 (2012), 1817-1859. doi: 10.1512/iumj.2012.61.4746. [26] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [27] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254. [28] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008. [29] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 31-62. doi: 10.1007/BF01442900. [30] I. Lasiecka and Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control Cybernet., 42 (2013), 155-181. [31] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems, Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. [32] J.L. Lions, Quelques Methods de Resolution des Problemes aux Limits Nonlinearies, Dunod. Paris, 1969. [33] J.L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs. J. Math Soc., 14 (1962), 233-241. [34] J.L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,, Springer-Verlag, (1972), 357 pp. [35] Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model, Nonlinear Anal. Real World Appl., 25 (2015), 51-63. doi: 10.1016/j.nonrwa.2015.02.006. [36] A. McIntosh, On the comparability of $A^{1/2}$and $A^{*1/2}$, Proceedings AMS, 32 (1972), 430-434. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [38] J. Pruss, On the spectrum of $C_0$ semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112. [39] A. Taylor, and D. Lay, Introduction to Functional Analysis, 2nd edition, 1980, Wiley. [40] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V. Balakrishnan, to appear. [41] R. Triggiani, A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: a critical subspace of $\mathcal{D}((-\mathcal{A})^{1/2})$ and $\mathcal{D}((-\mathcal{A}^*)^{1/2})$ and implications, Evolution Equations and Control Theory, vol 5, No.1, March 2016.

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##### References:
 [1] P. Ausher, S. Hofmann, M. Lacey, A. McIntosh and P. Tehamitchian, The solution of the Kato square root problemm for second order elliptic operators in $R^n$, Annals of Mathematics, 156 (2002), 633-654. doi: 10.2307/3597201. [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 Mathematical Journal, 15 (2008), 403-437; dedicated to the memory of J. L. Lions; J. Mawhin, editor. [4] G. Avalos and R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part I: Explicit semigroup generator and its spectral properties, AMS Contemporary Mathematics, Fluids and Waves, 440 (2007), 15-55. doi: 10.1090/conm/440/08475. [5] 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. Systems, 22 (2008), 817-833 (invited paper). doi: 10.3934/dcds.2008.22.817. [6] G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system, Discr. & Cont. Dynam. Systems DCDS-S, 2 (2009), 417-448. doi: 10.3934/dcdss.2009.2.417. [7] G. Avalos and R. Triggiani, A 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. [8] 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. [9] 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), 233-253. doi: 10.3934/eect.2013.2.233. [10] G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast in stability, Evolution Equations and Control Theory, 2 (2013), 563-598, special issue by invitation on the occasion of W. Littman's retirement. doi: 10.3934/eect.2013.2.563. [11] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, 2nd edition, Birkhauser, 2007, 575 pages. doi: 10.1007/978-0-8176-4581-6. [12] S. Canic, A. Mikelic and J. Tambaca, A two-dimensional effective model describing fluid-structure interaction in blood flow: analysis, simulation and experimental validation, Compte Rendus Mechanique Acad. Sci. Paris, 333 (2005), 867-883. [13] S. Canic, D. Lamponi, A. Mikelic and J. Tambaca, Self-consistent effective equations modeling blood flow in medium-t-large compliant arteries, Multiscale Model. Simul., 3 (2005), 559-596. doi: 10.1137/030602605. [14] G. Chen and D.L. Russel, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., (1982), 433-454. [15] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234-256. Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987). doi: 10.1007/BFb0089601. [16] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$, Pacific J. Math., 136 (1989), 15-55. [17] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293. doi: 10.1016/0022-0396(90)90100-4. [18] S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle perturbation, Proceedings Amer. Math. Soc., 110 (1990), 401-415. doi: 10.2307/2048084. [19] Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem, Discr. Dynam. Sys., 9 (2003), 633-650. doi: 10.3934/dcds.2003.9.633. [20] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82-86. [21] P. Grisvard, Characterization de qualques espaces d' interpolation, Arch. Pat. Mech. Anal., 25 (1967), 40-63. [22] M. Ignatova, I. Kukavica, I. Lasiecka and A. Tuffaha, On well-posedness and small data global existence for an interface damped free boundary fluid-structure model, Nonlinearity 27 (2014), 467-499. doi: 10.1088/0951-7715/27/3/467. [23] T. Kato, Fractional powers of dissipative operators, J.Math.Soc. Japan , 13 (1961), 246-274. [24] V. Komornik, Exact controllability and stabilization. The multiplier method, Masson, Paris; John Wiley & Sons Ltd, Chichester (1994), 156 pp. [25] I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction, Indiana Univ. Math. J., 61 (2012), 1817-1859. doi: 10.1512/iumj.2012.61.4746. [26] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system, J. Differential Equations, 247 (2009), 1452-1478. doi: 10.1016/j.jde.2009.06.005. [27] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system, Adv. Differential Equations, 15 (2010), 231-254. [28] I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary, Nonlinearity, 24 (2011), 159-176. doi: 10.1088/0951-7715/24/1/008. [29] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 31-62. doi: 10.1007/BF01442900. [30] I. Lasiecka and Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping, Control Cybernet., 42 (2013), 155-181. [31] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems, Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. [32] J.L. Lions, Quelques Methods de Resolution des Problemes aux Limits Nonlinearies, Dunod. Paris, 1969. [33] J.L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs. J. Math Soc., 14 (1962), 233-241. [34] J.L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,, Springer-Verlag, (1972), 357 pp. [35] Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model, Nonlinear Anal. Real World Appl., 25 (2015), 51-63. doi: 10.1016/j.nonrwa.2015.02.006. [36] A. McIntosh, On the comparability of $A^{1/2}$and $A^{*1/2}$, Proceedings AMS, 32 (1972), 430-434. [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [38] J. Pruss, On the spectrum of $C_0$ semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112. [39] A. Taylor, and D. Lay, Introduction to Functional Analysis, 2nd edition, 1980, Wiley. [40] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V. Balakrishnan, to appear. [41] R. Triggiani, A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: a critical subspace of $\mathcal{D}((-\mathcal{A})^{1/2})$ and $\mathcal{D}((-\mathcal{A}^*)^{1/2})$ and implications, Evolution Equations and Control Theory, vol 5, No.1, March 2016.
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