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.

show all references

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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