Figure 1.
Elastic-viscoelastic waves interaction models satisfying the assumption (A1)
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December
2021, 11(4): 885-904.
doi: 10.3934/mcrf.2020050
Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions
| 1. | Université de Bretagne Occidentale, LMBA, Brest, France, Lebanese University, Faculty of Sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon |
| 2. | Lebanese University, Faculty of Sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon |
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We investigate a multidimensional transmission problem between viscoelastic system with localized Kelvin-Voigt damping and purely elastic system under different types of geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. It was shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system. So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. For this aim, using frequency domain approach combined with a new multiplier technic, we will establish a polynomial energy decay estimate of type $ t^{-1} $ for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type $ t^{-2/5} $ if the strip is localized near the boundary. Else, it's of type $ t^{-1/3} $. The main novelty in this paper is that the geometric situations covered here are richer and less restrictive than those considered in [
Keywords:
Wave equation,
transmission problem,
local Kelvin-Voigt damping,
geometric conditions,
stabilization.
Mathematics Subject Classification:
Primary: 35B35, 93B52; Secondary: 93C20.
Citation:
Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control and Related Fields,
2021, 11
(4)
: 885-904.
doi: 10.3934/mcrf.2020050
![]()
References:
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Stabilization for the wave equation with singular Kelvin-Voigt damping, Arch. Ration. Mech. Anal., 236 (2020), 577-601.
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Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
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C. J. K. Batty and T. Duyckaerts,
Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.
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A. Borichev and Y. Tomilov,
Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.
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Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping, SIAM J. Appl. Math., 59 (1999), 651-668.
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F. L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces., Ann. Differential Equations, 1 (1985), 43-56.
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F. L. Huang,
On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.
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J. Le Rousseau and G. Lebeau,
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.
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Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590.
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K. Liu and Z. Liu,
Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.
doi: 10.1137/S0363012996310703. |
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K. Liu and Z. Liu,
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. |
| [18] |
K. Liu, Z. Liu and Q. Zhang,
Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.
doi: 10.1051/cocv/2015055. |
| [19] |
K. Liu and B. Rao,
Exponential stability for the wave equations with local Kelvin–Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.
doi: 10.1007/s00033-005-0029-2. |
| [20] |
Z. Liu and B. Rao,
Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.
doi: 10.1007/s00033-004-3073-4. |
| [21] |
Z. Liu and Q. Zhang,
Stability of a string with local Kelvin-Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.
doi: 10.1137/15M1049385. |
| [22] |
R. Nasser, N. Noun and A. Wehbe,
Stabilization of the wave equations with localized Kelvin-Voigt type damping under optimal geometric conditions, C. R. Math. Acad. Sci. Paris, 357 (2019), 272-277.
doi: 10.1016/j.crma.2019.01.005. |
| [23] |
S. Nicaise and C. Pignotti,
Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.
doi: 10.3934/dcdss.2016029. |
| [24] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [25] |
J. Prüss,
On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
| [26] |
J. Rauch and M. Taylor,
Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.
doi: 10.1512/iumj.1975.24.24004. |
| [27] |
R. Stahn, Optimal decay rate for the wave equation on a square with constant damping on a strip, Z. Angew. Math. Phys., 68 (2017), 10pp.
doi: 10.1007/s00033-017-0781-0. |
| [28] |
L. Tebou,
A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping, C. R. Math. Acad. Sci. Paris, 350 (2012), 603-608.
doi: 10.1016/j.crma.2012.06.005. |
| [29] |
Q. Zhang,
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. |
| [30] |
Q. Zhang,
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. |
| [31] |
Q. Zhang, Polynomial decay of an elastic/viscoelastic waves interaction system, Z. Angew. Math. Phys., 69 (2018), 10pp.
doi: 10.1007/s00033-018-0981-2. |
show all references
References:
| [1] |
K. Ammari, F. Hassine and L. Robbiano,
Stabilization for the wave equation with singular Kelvin-Voigt damping, Arch. Ration. Mech. Anal., 236 (2020), 577-601.
doi: 10.1007/s00205-019-01476-4. |
| [2] |
W. Arendt and C. J. K. Batty,
Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.
doi: 10.1090/S0002-9947-1988-0933321-3. |
| [3] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
| [4] |
C. J. K. Batty and T. Duyckaerts,
Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.
doi: 10.1007/s00028-008-0424-1. |
| [5] |
A. Borichev and Y. Tomilov,
Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.
doi: 10.1007/s00208-009-0439-0. |
| [6] |
H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. |
| [7] |
N. Burq,
Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.
doi: 10.3233/ASY-1997-14203. |
| [8] |
M. Cavalcanti, V. D. Cavalcanti and L. Tebou, Stabilization of the wave equation with localized compensating frictional and Kelvin-Voigt dissipating mechanisms, Electron. J. Differential Equations, (2017), 18pp. |
| [9] |
S. Chen, K. Liu and Z. Liu,
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. |
| [10] |
F. L. Huang,
Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces., Ann. Differential Equations, 1 (1985), 43-56.
|
| [11] |
F. L. Huang,
On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.
doi: 10.1137/0326041. |
| [12] |
J. Le Rousseau and G. Lebeau,
On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.
doi: 10.1051/cocv/2011168. |
| [13] |
G. Lebeau, Équation des ondes amorties, in Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996, 73–109. |
| [14] |
J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués: Perturbations, Recherches en Mathématiques Appliquées, 9, Masson, Paris, 1988. |
| [15] |
K. Liu,
Locally distributed control and damping for the conservative systems, SIAM J. Control Optim., 35 (1997), 1574-1590.
doi: 10.1137/S0363012995284928. |
| [16] |
K. Liu and Z. Liu,
Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.
doi: 10.1137/S0363012996310703. |
| [17] |
K. Liu and Z. Liu,
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. |
| [18] |
K. Liu, Z. Liu and Q. Zhang,
Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.
doi: 10.1051/cocv/2015055. |
| [19] |
K. Liu and B. Rao,
Exponential stability for the wave equations with local Kelvin–Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.
doi: 10.1007/s00033-005-0029-2. |
| [20] |
Z. Liu and B. Rao,
Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.
doi: 10.1007/s00033-004-3073-4. |
| [21] |
Z. Liu and Q. Zhang,
Stability of a string with local Kelvin-Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.
doi: 10.1137/15M1049385. |
| [22] |
R. Nasser, N. Noun and A. Wehbe,
Stabilization of the wave equations with localized Kelvin-Voigt type damping under optimal geometric conditions, C. R. Math. Acad. Sci. Paris, 357 (2019), 272-277.
doi: 10.1016/j.crma.2019.01.005. |
| [23] |
S. Nicaise and C. Pignotti,
Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.
doi: 10.3934/dcdss.2016029. |
| [24] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [25] |
J. Prüss,
On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
| [26] |
J. Rauch and M. Taylor,
Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.
doi: 10.1512/iumj.1975.24.24004. |
| [27] |
R. Stahn, Optimal decay rate for the wave equation on a square with constant damping on a strip, Z. Angew. Math. Phys., 68 (2017), 10pp.
doi: 10.1007/s00033-017-0781-0. |
| [28] |
L. Tebou,
A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping, C. R. Math. Acad. Sci. Paris, 350 (2012), 603-608.
doi: 10.1016/j.crma.2012.06.005. |
| [29] |
Q. Zhang,
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. |
| [30] |
Q. Zhang,
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. |
| [31] |
Q. Zhang, Polynomial decay of an elastic/viscoelastic waves interaction system, Z. Angew. Math. Phys., 69 (2018), 10pp.
doi: 10.1007/s00033-018-0981-2. |
Figure 2.
A model satisfying assumption (A2)
Figure 3.
A model satisfying both (A1) and (A2)
Figure 4.
A square model with local viscoelastic strip not satisfying any geometry cited before
Figure 5.
A model satisfying $ (m \cdot \nu_2)\vert _{\Gamma_2} \leq 0 $
Figure 6.
A model not satisfying $ (m \cdot \nu_2)\vert _{\Gamma_2} \leq 0 $
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