March  2020, 9(1): 39-60. doi: 10.3934/eect.2020016

Uniform exponential stability of a fluid-plate interaction model due to thermal effects

Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, Baguio, 2600, Philippines

Received  June 2018 Revised  August 2019 Published  March 2020 Early access  October 2019

Fund Project: This work was supported in part by the ERC advanced grant 668998 (OCLOC) under the EU-s H2020 research program and the Ernst-Mach grant of the Austrian Agency for International Cooperation in Education and Research (OeAD-GmbH).

We consider a coupled fluid-thermoelastic plate interaction model. The fluid velocity is modeled by the linearized 3D Navier-Stokes equation while the plate dynamics is described by a thermoelastic Kirchoff system. By eliminating the pressure term, the system is reformulated as an abstract evolution problem and its well-posedness is proved by semigroup methods. The dissipation in the system is due to the diffusion of the fluid and heat components. Uniform stability of the coupled system is established through multipliers and the energy method. The multipliers used for thermoelastic plate models in the literature are modified in accordance to the applicability of a certain Stokes map.

Citation: Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations and Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016
References:
[1]

G. Avalos and F. Bucci, Spectral analysis and rational decay rates of strong solutions to a fluid-structure PDE system, J. Differ. Equations, 258 (2015), 4398–4423, https://www.sciencedirect.com/science/article/pii/S002203961500056X. doi: 10.1016/j.jde.2015.01.037.

[2]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 49-78. doi: 10.1007/978-3-319-11406-4_3.

[3]

G. Avalos and T. Clark, A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction, Evol. Equ. Control Theory, 3 (2014), 557–578, http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=10481. doi: 10.3934/eect.2014.3.557.

[4]

G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction model with implications for a divergence-free finite element method, Appl. Math. (Warsaw), 35 (2008), 259–280, https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/applicationes-mathematicae/all/35/3/84340/a-new-maximality-argument-for-a-coupled-fluid-structure-interaction-with-implications-for-a-divergence-free-finite-element-method. doi: 10.4064/am35-3-2.

[5]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1–28, https://rendiconti.dmi.units.it/volumi/28s/01.pdf.

[6]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.  doi: 10.1137/S0036141096300823.

[7]

G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, I. Explicit semigroup generator and its spectral properties, Fluids and Waves, Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 440 (2007), 15-54.  doi: 10.1090/conm/440/08475.

[8]

G. Avalos and R. Triggiani, Fluid structure interaction with and without internal dissipation of the structure: A contrast study in stability, Evol. Equ. Control Theory, 2 (2013), 563-598.  doi: 10.3934/eect.2013.2.563.

[9]

I. D. Chuesov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Methods Appl. Sci., 34 (2011), 1801-1812.  doi: 10.1002/mma.1496.

[10]

I. Chuesov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656.  doi: 10.3934/cpaa.2013.12.1635.

[11]

H. Cohen and S. I. Rubinow, Some mathematical topics in Biology, Proc. Symp. on System Theory, Polytechnic Press, New York, (1965), 321–337.

[12]

A. Haraux, Decay rate of the range component of solutions to some semilinear evolution equations, Nonlinear Differ. Equ. Appl., 13 (2006), 435-445.  doi: 10.1007/s00030-006-4019-7.

[13]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[14]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[15]

I. Lasiecka and Y. J. Lu, Interface feedback control stabilisation of a nonlinear fluid-structure interaction, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 1449–1460, https://www.sciencedirect.com/science/article/pii/S0362546X11002136. doi: 10.1016/j.na.2011.04.018.

[16]

I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchoff plates with boundary control only on $\Delta \omega|_{\Sigma}$ and homogeneous boundary displacements, J. Differ. Equations, 93 (1991), 62-101.  doi: 10.1016/0022-0396(91)90022-2.

[17]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C., Abstr. Appl. Anal., 3 (1998), 153–169, https://www.hindawi.com/journals/aaa/1998/428531/abs/. doi: 10.1155/S1085337598000487.

[18]

J.-L. Lions, Quelques Méthodes de Résolution des Probelèmes aux Limites non Linéares. Tome 2. Perturbations, Dunod, Gauthier-Villars, Paris, 1969.

[19]

J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilization de Systèmes Distribués, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988.

[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[21]

J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system, ESAIM: Control, Optimisation and Calculus of Variations, 1 (1995/96), 1–15, https://www.esaim-cocv.org/articles/cocv/abs/1996/01/cocv-Vol1.1/cocv-Vol1.1.html. doi: 10.1051/cocv:1996100.

[22]

Z.-Y. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q.

[23]

G. Perla Menzala and E. Zuazua, Explicit exponential decay rates for solutions of von Kármán's system of thermoelastic plates, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 49-54.  doi: 10.1016/S0764-4442(97)80102-4.

[24]

G. Perla Menzala and E. Zuazua, Energy decay rates for the von Kármán system of thermoelastic plates, Differential and Integral Equations, 11 (1998), 755–770, https://projecteuclid.org/euclid.die/1367329669.

[25]

G. Perla Menzala and E. Zuazua, The energy decay rate for the modified von Karman system of thermoelastic plates: An improvement, App. Math. Lett., 16 (2003), 531-534.  doi: 10.1016/S0893-9659(03)00032-6.

[26]

A. Quarteroni and A. Valli, Numerical Approximations of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994.

[27]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[28]

R. Triggiani and J. Zhang, Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.  doi: 10.3934/eect.2018008.

[29]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser-Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[30]

J. Zhang, The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework, Evol. Equ. Control Theory, 6 (2017), 135-154.  doi: 10.3934/eect.2017008.

show all references

References:
[1]

G. Avalos and F. Bucci, Spectral analysis and rational decay rates of strong solutions to a fluid-structure PDE system, J. Differ. Equations, 258 (2015), 4398–4423, https://www.sciencedirect.com/science/article/pii/S002203961500056X. doi: 10.1016/j.jde.2015.01.037.

[2]

G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 49-78. doi: 10.1007/978-3-319-11406-4_3.

[3]

G. Avalos and T. Clark, A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction, Evol. Equ. Control Theory, 3 (2014), 557–578, http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=10481. doi: 10.3934/eect.2014.3.557.

[4]

G. Avalos and M. Dvorak, A new maximality argument for a coupled fluid-structure interaction model with implications for a divergence-free finite element method, Appl. Math. (Warsaw), 35 (2008), 259–280, https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/applicationes-mathematicae/all/35/3/84340/a-new-maximality-argument-for-a-coupled-fluid-structure-interaction-with-implications-for-a-divergence-free-finite-element-method. doi: 10.4064/am35-3-2.

[5]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28 (1997), 1–28, https://rendiconti.dmi.units.it/volumi/28s/01.pdf.

[6]

G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), 155-182.  doi: 10.1137/S0036141096300823.

[7]

G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, I. Explicit semigroup generator and its spectral properties, Fluids and Waves, Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 440 (2007), 15-54.  doi: 10.1090/conm/440/08475.

[8]

G. Avalos and R. Triggiani, Fluid structure interaction with and without internal dissipation of the structure: A contrast study in stability, Evol. Equ. Control Theory, 2 (2013), 563-598.  doi: 10.3934/eect.2013.2.563.

[9]

I. D. Chuesov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Methods Appl. Sci., 34 (2011), 1801-1812.  doi: 10.1002/mma.1496.

[10]

I. Chuesov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656.  doi: 10.3934/cpaa.2013.12.1635.

[11]

H. Cohen and S. I. Rubinow, Some mathematical topics in Biology, Proc. Symp. on System Theory, Polytechnic Press, New York, (1965), 321–337.

[12]

A. Haraux, Decay rate of the range component of solutions to some semilinear evolution equations, Nonlinear Differ. Equ. Appl., 13 (2006), 435-445.  doi: 10.1007/s00030-006-4019-7.

[13]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[14]

J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. doi: 10.1137/1.9781611970821.

[15]

I. Lasiecka and Y. J. Lu, Interface feedback control stabilisation of a nonlinear fluid-structure interaction, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 1449–1460, https://www.sciencedirect.com/science/article/pii/S0362546X11002136. doi: 10.1016/j.na.2011.04.018.

[16]

I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchoff plates with boundary control only on $\Delta \omega|_{\Sigma}$ and homogeneous boundary displacements, J. Differ. Equations, 93 (1991), 62-101.  doi: 10.1016/0022-0396(91)90022-2.

[17]

I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C., Abstr. Appl. Anal., 3 (1998), 153–169, https://www.hindawi.com/journals/aaa/1998/428531/abs/. doi: 10.1155/S1085337598000487.

[18]

J.-L. Lions, Quelques Méthodes de Résolution des Probelèmes aux Limites non Linéares. Tome 2. Perturbations, Dunod, Gauthier-Villars, Paris, 1969.

[19]

J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilization de Systèmes Distribués, Recherches en Mathématiques Appliquées, 9. Masson, Paris, 1988.

[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.

[21]

J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system, ESAIM: Control, Optimisation and Calculus of Variations, 1 (1995/96), 1–15, https://www.esaim-cocv.org/articles/cocv/abs/1996/01/cocv-Vol1.1/cocv-Vol1.1.html. doi: 10.1051/cocv:1996100.

[22]

Z.-Y. Liu and M. Renardy, A note on the equations of a thermoelastic plate, Appl. Math. Lett., 8 (1995), 1-6.  doi: 10.1016/0893-9659(95)00020-Q.

[23]

G. Perla Menzala and E. Zuazua, Explicit exponential decay rates for solutions of von Kármán's system of thermoelastic plates, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 49-54.  doi: 10.1016/S0764-4442(97)80102-4.

[24]

G. Perla Menzala and E. Zuazua, Energy decay rates for the von Kármán system of thermoelastic plates, Differential and Integral Equations, 11 (1998), 755–770, https://projecteuclid.org/euclid.die/1367329669.

[25]

G. Perla Menzala and E. Zuazua, The energy decay rate for the modified von Karman system of thermoelastic plates: An improvement, App. Math. Lett., 16 (2003), 531-534.  doi: 10.1016/S0893-9659(03)00032-6.

[26]

A. Quarteroni and A. Valli, Numerical Approximations of Partial Differential Equations, Springer Series in Computational Mathematics, 23. Springer-Verlag, Berlin, 1994.

[27]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, Studies in Mathematics and its Applications, Vol. 2. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[28]

R. Triggiani and J. Zhang, Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay, Evol. Equ. Control Theory, 7 (2018), 153-182.  doi: 10.3934/eect.2018008.

[29]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser-Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[30]

J. Zhang, The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework, Evol. Equ. Control Theory, 6 (2017), 135-154.  doi: 10.3934/eect.2017008.

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