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 & 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 $R^n$,, \emph{Annals of Mathematics}, 156 (2002), 633.  doi: 10.2307/3597201.  Google Scholar

[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,, \emph{Applicationes Mathematicae}, 35 (2008), 259.  doi: 10.4064/am35-3-2.  Google Scholar

[3]

G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system,, invited paper, 15 (2008), 403.   Google Scholar

[4]

G. Avalos and R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part I: Explicit semigroup generator and its spectral properties,, \emph{AMS Contemporary Mathematics, 440 (2007), 15.  doi: 10.1090/conm/440/08475.  Google Scholar

[5]

G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface,, \emph{Discr. & Cont. Dynam. Systems}, 22 (2008), 817.  doi: 10.3934/dcds.2008.22.817.  Google Scholar

[6]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system,, \emph{Discr. & Cont. Dynam. Systems DCDS-S}, 2 (2009), 417.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[7]

G. Avalos and R. Triggiani, A coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behavior of the resolvent operator on the imaginary axis,, \emph{Applicable Analysis}, 88 (2009), 1357.  doi: 10.1080/00036810903278513.  Google Scholar

[8]

G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, \emph{J. Evol. Eqns.}, 9 (2009), 341.  doi: 10.1007/s00028-009-0015-9.  Google Scholar

[9]

G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach,, \emph{Evolution Equations and Control Theory}, 2 (2013), 233.  doi: 10.3934/eect.2013.2.233.  Google Scholar

[10]

G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast in stability,, \emph{Evolution Equations and Control Theory}, 2 (2013), 563.  doi: 10.3934/eect.2013.2.563.  Google Scholar

[11]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems,, 2$^{nd}$ edition, (2007).  doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[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,, \emph{Compte Rendus Mechanique Acad. Sci. Paris}, 333 (2005), 867.   Google Scholar

[13]

S. Canic, D. Lamponi, A. Mikelic and J. Tambaca, Self-consistent effective equations modeling blood flow in medium-t-large compliant arteries,, \emph{Multiscale Model. Simul.}, 3 (2005), 559.  doi: 10.1137/030602605.  Google Scholar

[14]

G. Chen and D.L. Russel, A mathematical model for linear elastic systems with structural damping,, \emph{Quart. Appl. Math.}, (1982), 433.   Google Scholar

[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 $\alpha = 1/2$,, \emph{Springer-Verlag Lecture Notes in Mathematics}, 1354 (1988), 234.  doi: 10.1007/BFb0089601.  Google Scholar

[16]

S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$,, \emph{Pacific J. Math.}, 136 (1989), 15.   Google Scholar

[17]

S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications,, \emph{J. Diff. Eqns.}, 88 (1990), 279.  doi: 10.1016/0022-0396(90)90100-4.  Google Scholar

[18]

S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle perturbation,, \emph{Proceedings Amer. Math. Soc.}, 110 (1990), 401.  doi: 10.2307/2048084.  Google Scholar

[19]

Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, \emph{Discr. Dynam. Sys.}, 9 (2003), 633.  doi: 10.3934/dcds.2003.9.633.  Google Scholar

[20]

D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,, \emph{Proc. Japan Acad.}, 43 (1967), 82.   Google Scholar

[21]

P. Grisvard, Characterization de qualques espaces d' interpolation,, \emph{Arch. Pat. Mech. Anal.}, 25 (1967), 40.   Google Scholar

[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,, \emph{Nonlinearity} \textbf{27} (2014), 27 (2014), 467.  doi: 10.1088/0951-7715/27/3/467.  Google Scholar

[23]

T. Kato, Fractional powers of dissipative operators,, \emph{J.Math.Soc. Japan }, 13 (1961), 246.   Google Scholar

[24]

V. Komornik, Exact controllability and stabilization. The multiplier method,, Masson, (1994).   Google Scholar

[25]

I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction,, \emph{Indiana Univ. Math. J.}, 61 (2012), 1817.  doi: 10.1512/iumj.2012.61.4746.  Google Scholar

[26]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system,, \emph{J. Differential Equations}, 247 (2009), 1452.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[27]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system,, \emph{Adv. Differential Equations}, 15 (2010), 231.   Google Scholar

[28]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary,, \emph{Nonlinearity}, 24 (2011), 159.  doi: 10.1088/0951-7715/24/1/008.  Google Scholar

[29]

I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach,, \emph{Appl. Math. & Optimiz.}, 6 (1980), 31.  doi: 10.1007/BF01442900.  Google Scholar

[30]

I. Lasiecka and Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping,, \emph{Control Cybernet.}, 42 (2013), 155.   Google Scholar

[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, (2000).   Google Scholar

[32]

J.L. Lions, Quelques Methods de Resolution des Problemes aux Limits Nonlinearies,, Dunod. Paris, (1969).   Google Scholar

[33]

J.L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs., \emph{J. Math Soc.}, 14 (1962), 233.   Google Scholar

[34]

J.L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,,, Springer-Verlag, (1972).   Google Scholar

[35]

Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model,, \emph{Nonlinear Anal. Real World Appl.}, 25 (2015), 51.  doi: 10.1016/j.nonrwa.2015.02.006.  Google Scholar

[36]

A. McIntosh, On the comparability of $A^{1/2} $and $A^{*1/2}$,, \emph{Proceedings AMS}, 32 (1972), 430.   Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[38]

J. Pruss, On the spectrum of $C_0$ semigroups,, \emph{Transactions of the American Mathematical Society}, 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

[39]

A. Taylor, and D. Lay, Introduction to Functional Analysis,, 2$^nd$ edition, (1980).   Google Scholar

[40]

R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications,, \emph{Applied Mathematics and Optimization, ().   Google Scholar

[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,, \emph{Evolution Equations and Control Theory}, (2016).   Google Scholar

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 $R^n$,, \emph{Annals of Mathematics}, 156 (2002), 633.  doi: 10.2307/3597201.  Google Scholar

[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,, \emph{Applicationes Mathematicae}, 35 (2008), 259.  doi: 10.4064/am35-3-2.  Google Scholar

[3]

G. Avalos, I. Lasiecka and R. Triggiani, Higher regularity of a coupled parabolic-hyperbolic fluid-structure interactive system,, invited paper, 15 (2008), 403.   Google Scholar

[4]

G. Avalos and R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part I: Explicit semigroup generator and its spectral properties,, \emph{AMS Contemporary Mathematics, 440 (2007), 15.  doi: 10.1090/conm/440/08475.  Google Scholar

[5]

G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface,, \emph{Discr. & Cont. Dynam. Systems}, 22 (2008), 817.  doi: 10.3934/dcds.2008.22.817.  Google Scholar

[6]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system,, \emph{Discr. & Cont. Dynam. Systems DCDS-S}, 2 (2009), 417.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[7]

G. Avalos and R. Triggiani, A coupled parabolic-hyperbolic Stokes-Lamé PDE system: Limit behavior of the resolvent operator on the imaginary axis,, \emph{Applicable Analysis}, 88 (2009), 1357.  doi: 10.1080/00036810903278513.  Google Scholar

[8]

G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic-hyperbolic Stokes-Lamé PDE system,, \emph{J. Evol. Eqns.}, 9 (2009), 341.  doi: 10.1007/s00028-009-0015-9.  Google Scholar

[9]

G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach,, \emph{Evolution Equations and Control Theory}, 2 (2013), 233.  doi: 10.3934/eect.2013.2.233.  Google Scholar

[10]

G. Avalos and R. Triggiani, Fluid-structure interaction with and without internal dissipation of the structure: A contrast in stability,, \emph{Evolution Equations and Control Theory}, 2 (2013), 563.  doi: 10.3934/eect.2013.2.563.  Google Scholar

[11]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems,, 2$^{nd}$ edition, (2007).  doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[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,, \emph{Compte Rendus Mechanique Acad. Sci. Paris}, 333 (2005), 867.   Google Scholar

[13]

S. Canic, D. Lamponi, A. Mikelic and J. Tambaca, Self-consistent effective equations modeling blood flow in medium-t-large compliant arteries,, \emph{Multiscale Model. Simul.}, 3 (2005), 559.  doi: 10.1137/030602605.  Google Scholar

[14]

G. Chen and D.L. Russel, A mathematical model for linear elastic systems with structural damping,, \emph{Quart. Appl. Math.}, (1982), 433.   Google Scholar

[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 $\alpha = 1/2$,, \emph{Springer-Verlag Lecture Notes in Mathematics}, 1354 (1988), 234.  doi: 10.1007/BFb0089601.  Google Scholar

[16]

S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$,, \emph{Pacific J. Math.}, 136 (1989), 15.   Google Scholar

[17]

S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications,, \emph{J. Diff. Eqns.}, 88 (1990), 279.  doi: 10.1016/0022-0396(90)90100-4.  Google Scholar

[18]

S. Chen and R. Triggiani, Gevrey class semigroups arising from elastic systems with gentle perturbation,, \emph{Proceedings Amer. Math. Soc.}, 110 (1990), 401.  doi: 10.2307/2048084.  Google Scholar

[19]

Q. Du, M. D. Gunzburger, L. S. Hou and J. Lee, Analysis of a linear fluid-structure interaction problem,, \emph{Discr. Dynam. Sys.}, 9 (2003), 633.  doi: 10.3934/dcds.2003.9.633.  Google Scholar

[20]

D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order,, \emph{Proc. Japan Acad.}, 43 (1967), 82.   Google Scholar

[21]

P. Grisvard, Characterization de qualques espaces d' interpolation,, \emph{Arch. Pat. Mech. Anal.}, 25 (1967), 40.   Google Scholar

[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,, \emph{Nonlinearity} \textbf{27} (2014), 27 (2014), 467.  doi: 10.1088/0951-7715/27/3/467.  Google Scholar

[23]

T. Kato, Fractional powers of dissipative operators,, \emph{J.Math.Soc. Japan }, 13 (1961), 246.   Google Scholar

[24]

V. Komornik, Exact controllability and stabilization. The multiplier method,, Masson, (1994).   Google Scholar

[25]

I. Kukavica and A. Tuffaha, Regularity of solutions to a free boundary problem of fluid-structure interaction,, \emph{Indiana Univ. Math. J.}, 61 (2012), 1817.  doi: 10.1512/iumj.2012.61.4746.  Google Scholar

[26]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a nonlinear fluid structure interaction system,, \emph{J. Differential Equations}, 247 (2009), 1452.  doi: 10.1016/j.jde.2009.06.005.  Google Scholar

[27]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions for a fluid structure interaction system,, \emph{Adv. Differential Equations}, 15 (2010), 231.   Google Scholar

[28]

I. Kukavica, A. Tuffaha and M. Ziane, Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary,, \emph{Nonlinearity}, 24 (2011), 159.  doi: 10.1088/0951-7715/24/1/008.  Google Scholar

[29]

I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach,, \emph{Appl. Math. & Optimiz.}, 6 (1980), 31.  doi: 10.1007/BF01442900.  Google Scholar

[30]

I. Lasiecka and Y. Lu, Stabilization of a fluid structure interaction with nonlinear damping,, \emph{Control Cybernet.}, 42 (2013), 155.   Google Scholar

[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, (2000).   Google Scholar

[32]

J.L. Lions, Quelques Methods de Resolution des Problemes aux Limits Nonlinearies,, Dunod. Paris, (1969).   Google Scholar

[33]

J.L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs., \emph{J. Math Soc.}, 14 (1962), 233.   Google Scholar

[34]

J.L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,,, Springer-Verlag, (1972).   Google Scholar

[35]

Y. Lu, Uniform stabilization to equilibrium of a nonlinear fluid-structure interaction model,, \emph{Nonlinear Anal. Real World Appl.}, 25 (2015), 51.  doi: 10.1016/j.nonrwa.2015.02.006.  Google Scholar

[36]

A. McIntosh, On the comparability of $A^{1/2} $and $A^{*1/2}$,, \emph{Proceedings AMS}, 32 (1972), 430.   Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[38]

J. Pruss, On the spectrum of $C_0$ semigroups,, \emph{Transactions of the American Mathematical Society}, 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar

[39]

A. Taylor, and D. Lay, Introduction to Functional Analysis,, 2$^nd$ edition, (1980).   Google Scholar

[40]

R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications,, \emph{Applied Mathematics and Optimization, ().   Google Scholar

[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,, \emph{Evolution Equations and Control Theory}, (2016).   Google Scholar

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