Previous |  Up |  Next

Article

Keywords:
transmission problem; Kirchhoff plate; Kelvin-Voigt damping; energy decay; Carleman estimate
Summary:
We are concerned with a transmission problem for the Kirchhoff plate equation where one small part of the domain is made of a viscoelastic material with the Kelvin-Voigt constitutive relation. We obtain the logarithmic stabilization result (explicit energy decay rate), as well as the wellposedness, for the transmission system. The method is based on a new Carleman estimate to obtain information on the resolvent for high frequency. The main ingredient of the proof is some careful analysis for the Kirchhoff transmission plate equation.
References:
[1] Ammari, K., Jellouli, M., Mehrenberger, M.: Feedback stabilization of a coupled stringbeam system. Netw. Heterog. Media 4 (2009), 19-34. DOI 10.3934/nhm.2009.4.19 | MR 2480421 | Zbl 1183.93109
[2] Ammari, K., Nicaise, S.: Stabilization of a transmission wave/plate equation. J. Differ. Equations 249 (2010), 707-727. DOI 10.1016/j.jde.2010.03.007 | MR 2646047 | Zbl 1201.35128
[3] Ammari, K., Vodev, G.: Boundary stabilization of the transmission problem for the Bernoulli-Euler plate equation. Cubo 11 (2009), 39-49. MR 2568250 | Zbl 1184.35045
[4] Avalos, G., Lasiecka, I., Triggiani, R.: Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate. J. Math. Anal. Appl. 437 (2016), 782-815. DOI 10.1016/j.jmaa.2015.12.051 | MR 3456199 | Zbl 1336.35054
[5] Avalos, G., Triggiani, R.: Backward uniqueness of the s.c. semigroup arising in parabolichyperbolic fluid-structure interaction. J. Differ. Equations 245 (2008), 737-761. DOI 10.1016/j.jde.2007.10.036 | MR 2422526 | Zbl 1158.35300
[6] Avalos, G., Triggiani, R.: Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete Contin. Dyn. Syst. 22 (2008), 817-833. DOI 10.3934/dcds.2008.22.817 | MR 2434971 | Zbl 1158.35320
[7] Avalos, G., Triggiani, R.: 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 | MR 3177244 | Zbl 1277.35045
[8] Avalos, G., Triggiani, R.: Rational decay rates for a PDE heat-structure interaction: A frequency domain approach. Evol. Equ. Control Theory 2 (2013), 233-253. DOI 10.3934/eect.2013.2.233 | MR 3089718 | Zbl 1275.35035
[9] Barbu, V., Grujić, Z., Lasiecka, I., Tuffaha, A.: Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model. Fluids and Waves: Recent Trends in Applied Analysis Contemporary Mathematics 440. American Mathematical Society, Providence (2007), 55-82. DOI 10.1090/conm/440 | MR 2359449 | Zbl 1297.35234
[10] Bastos, W. D., Raposo, C. A.: Transmission problem for waves with frictional damping. Electron. J. Differ. Equ. 2007 (2007), Article ID 60, 10 pages. MR 2299614 | Zbl 1136.35315
[11] Batty, C. J. K., Duyckaerts, T.: Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008), 765-780. DOI 10.1007/s00028-008-0424-1 | MR 2460938 | Zbl 1185.47043
[12] Bellassoued, M.: Carleman estimates and distribution of resonances for the transparent obstacle and application to the stabilization. Asymptotic. Anal. 35 (2003), 257-279. MR 2011790 | Zbl 1137.35388
[13] Burq, N.: Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180 (1998), 1-29 French. DOI 10.1007/BF02392877 | MR 1618254 | Zbl 0918.35081
[14] Chai, S.: Uniform decay rate for the transmission wave equations with variable coefficients. J. Syst. Sci. Complex 24 (2011), 253-260. DOI 10.1007/s11424-011-8009-4 | MR 2802559 | Zbl 1226.93111
[15] Chai, S., Liu, K.: Boundary stabilization of the transmission of wave equations with variable coefficients. Chin. Ann. Math., Ser. A 26 (2005), 605-612 Chinese. MR 2186628 | Zbl 1090.35010
[16] Chen, S., Liu, K., Liu, Z.: Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping. SIAM J. Appl. Math. 59 (1999), 651-668. DOI 10.1137/S0036139996292015 | MR 1654395 | Zbl 0924.35018
[17] Du, Q., Gunzburger, M. D., Hou, L. S., Lee, J.: Analysis of a linear fluid-structure interaction problem. Discrete Contin. Dyn. Syst. 9 (2003), 633-650. DOI 10.3934/dcds.2003.9.633 | MR 1974530 | Zbl 1039.35076
[18] Duyckaerts, T.: Optimal decay rates of the energy of an hyperbolic-parabolic system coupled by an interface. Asymptotic. Anal. 51 (2007), 17-45. MR 2294103 | Zbl 1227.35062
[19] Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194. Springer, Berlin (2000). DOI 10.1007/b97696 | MR 1721989 | Zbl 0952.47036
[20] Hassine, F.: Stability of elastic transmission systems with a local Kelvin-Voigt damping. Eur. J. Control 23 (2015), 84-93. DOI 10.1016/j.ejcon.2015.03.001 | MR 3339649 | Zbl 1360.93600
[21] Hassine, F.: Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete Contin. Dyn. Syst., Ser. B 21 (2016), 1757-1774. DOI 10.3934/dcdsb.2016021 | MR 3543607 | Zbl 1350.35031
[22] Hassine, F.: Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping. Int. J. Control 89 (2016), 1933-1950. DOI 10.1080/00207179.2015.1135509 | MR 3568279 | Zbl 1364.35043
[23] Hassine, F.: Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping. J. Math. Anal. Appl. 455 (2017), 1765-1782. DOI 10.1016/j.jmaa.2017.06.068 | MR 3671253 | Zbl 1379.35024
[24] Lagnese, J. E.: Boundary Stabilization of Thin Plates. SIAM Studies in Applied Mathematics 10. SIAM, Philadelphia (1989). DOI 10.1137/1.9781611970821 | MR 1061153 | Zbl 0696.73034
[25] Lasiecka, I., Triggiani, R., Zhang, J.: Min-max game theory for elastic and visco-elastic fluid structure interactions. Open Appl. Math. J. 7 (2013), 1-17. DOI 10.2174/1874114220130430001 | MR 3071536 | Zbl 1322.74015
[26] Lebeau, G., Robbiano, L.: Contrôle exacte de l'équation de la chaleur. Commun. Partial Differ. Equations 20 (1995), 335-356 French. DOI 10.1080/03605309508821097 | MR 1312710 | Zbl 0819.35071
[27] Lebeau, G., Robbiano, L.: Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997), 465-491 French. DOI 10.1215/S0012-7094-97-08614-2 | MR 1432305 | Zbl 0884.58093
[28] Rousseau, J. Le, Lebeau, G.: Introduction aux inégalités de Carleman pour les opérateurs elliptiques et paraboliques: Applications au prolongement unique et au contrôle des équations paraboliques. Available at https://hal.archives-ouvertes.fr/hal-00351736v2 (2009), 27 pages French.
[29] Rousseau, J. Le, Robbiano, L.: Carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations. Arch. Ration. Mech. Anal. 195 (2010), 953-990. DOI 10.1007/s00205-009-0242-9 | MR 2591978 | Zbl 1202.35336
[30] Li, Y.-F., Han, Z.-J., Xu, G.-Q.: Explicit decay rate for coupled string-beam system with localized frictional damping. Appl. Math. Lett. 78 (2018), 51-58. DOI 10.1016/j.aml.2017.11.003 | MR 3739755 | Zbl 1383.35028
[31] Liu, K., Liu, Z.: Exponential decay of the energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping. SIAM J. Control Optim. 36 (1998), 1086-1098. DOI 10.1137/S0363012996310703 | MR 1613917 | Zbl 0909.35018
[32] Liu, K., Liu, Z.: Exponential decay of energy of vibrating strings with local viscoelasticity. Z. Angew. Math. Phys. 53 (2002), 265-280. DOI 10.1007/s00033-002-8155-6 | MR 1900674 | Zbl 0999.35012
[33] Liu, W., Williams, G.: The exponential stability of the problem of transmission of the wave equation. Bull. Aust. Math. Soc. 57 (1998), 305-327. DOI 10.1017/S0004972700031683 | MR 1617324 | Zbl 0914.35074
[34] Ramos, A. J. A., Souza, M. W. P.: Equivalence between observability at the boundary and stabilization for transmission problem of the wave equation. Z. Angew. Math. Phys. 68 (2017), Article ID 48, 11 pages. DOI 10.1007/s00033-017-0791-y | MR 3626611 | Zbl 1373.35191
[35] Rauch, J., Zhang, X., Zuazua, E.: Polynomial decay for a hyperbolic-parabolic coupled system. J. Math. Pures Appl., IX. Sér. 84 (2005), 407-470. DOI 10.1016/j.matpur.2004.09.006 | MR 2132724 | Zbl 1077.35030
[36] Wloka, J. T., Rowley, B., Lawruk, B.: Boundary Value Problems for Elliptic System. Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511662850 | MR 1343490 | Zbl 0836.35042
[37] Zhang, Q.: Exponential stability of an elastic string with local Kelvin-Voigt damping. Z. Angew. Math. Phys. 61 (2010), 1009-1015. DOI 10.1007/s00033-010-0064-5 | MR 2738301 | Zbl 1273.74120
[38] Zhang, Q.: On the lack of exponential stability for an elastic-viscoelastic waves interaction system. Nonlinear Anal., Real World Appl. 37 (2017), 387-411. DOI 10.1016/j.nonrwa.2017.02.019 | MR 3648388 | Zbl 1375.35037
[39] Zhang, X., Zuazua, E.: Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. Anal. 184 (2007), 49-120. DOI 10.1007/s00205-006-0020-x | MR 2289863 | Zbl 1178.74075
Partner of
EuDML logo