Stabilization of a system modeling temperature and porosity fields in a Kelvin–Voigt-type mixture
In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for the interaction between the temperature field and the porosity fields in a homogeneous and isotropic mixture from the linear theory of porous Kelvin–Voigt materials. Our main result is to est...
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| Published in | Acta mechanica Vol. 219; no. 1-2; pp. 145 - 167 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Vienna
Springer Vienna
01.06.2011
Springer Springer Nature B.V |
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| Abstract | In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for the interaction between the temperature field and the porosity fields in a homogeneous and isotropic mixture from the linear theory of porous Kelvin–Voigt materials. Our main result is to establish conditions which insure the analyticity and the exponential stability of the corresponding semigroup. We show that under certain conditions for the coefficients we obtain a lack of exponential stability. A numerical scheme is given. |
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| AbstractList | In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for the interaction between the temperature field and the porosity fields in a homogeneous and isotropic mixture from the linear theory of porous Kelvin–Voigt materials. Our main result is to establish conditions which insure the analyticity and the exponential stability of the corresponding semigroup. We show that under certain conditions for the coefficients we obtain a lack of exponential stability. A numerical scheme is given. In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for the interaction between the temperature field and the porosity fields in a homogeneous and isotropic mixture from the linear theory of porous Kelvin-Voigt materials. Our main result is to establish conditions which insure the analyticity and the exponential stability of the corresponding semigroup. We show that under certain conditions for the coefficients we obtain a lack of exponential stability. A numerical scheme is given.[PUBLICATION ABSTRACT] |
| Audience | Academic |
| Author | Rivera, Jaime E. Muñoz Vera, Octavio Sepúlveda, Mauricio Alves, Margareth S. |
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| Keywords | Exponential Stability Porosity Linear Theory Initial Boundary Discrete Fourier Transform Viscoelasticity Stabilization Initial value problem Mixture theory Semigroup Exponential stability Porous material Modeling Boundary value problem Kelvin voigt model Homogeneous mixtures Asymptotic approximation |
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| References | Ieşan, Nappa (CR8) 2008; 13 Martínez, Quintanilla (CR10) 1995; 46 Ieşan, Quintanilla (CR7) 2007; 30 Alves, Muñoz Rivera, Quintanilla (CR1) 2009; 46 Pazy (CR11) 1983 CR9 Quintanilla (CR14) 2005; 24 Alves, Muñoz Rivera, Sepúlveda, Villagrán (CR2) 2009; 46 Alves, Muñoz Rivera, Sepúlveda, Villagrán (CR3) 2009; 32 Huang (CR5) 1985; 1 Ieşan, Quintanilla (CR6) 2002; 25 Quintanilla (CR13) 2005; 18 Gearhart (CR4) 1978; 236 Prüss (CR12) 1984; 284 A. Pazy (443_CR11) 1983 R. Quintanilla (443_CR13) 2005; 18 F. Martínez (443_CR10) 1995; 46 F.L. Huang (443_CR5) 1985; 1 443_CR9 J. Prüss (443_CR12) 1984; 284 M.S. Alves (443_CR3) 2009; 32 M.S. Alves (443_CR2) 2009; 46 L.M. Gearhart (443_CR4) 1978; 236 R. Quintanilla (443_CR14) 2005; 24 M.S. Alves (443_CR1) 2009; 46 D. Ieşan (443_CR7) 2007; 30 D. Ieşan (443_CR6) 2002; 25 D. Ieşan (443_CR8) 2008; 13 |
| References_xml | – volume: 236 start-page: 385 year: 1978 end-page: 394 ident: CR4 article-title: Spectral theory for contraction semigroups on Hilbert spaces publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1978-0461206-1 contributor: fullname: Gearhart – volume: 46 start-page: 1659 year: 2009 end-page: 1666 ident: CR1 article-title: Exponential decay in a thermoelastic mixture of solids publication-title: Int. J. Solids Struct. doi: 10.1016/j.ijsolstr.2008.12.005 contributor: fullname: Quintanilla – volume: 1 start-page: 43 year: 1985 end-page: 56 ident: CR5 article-title: Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces publication-title: Ann. Differ. Equ. contributor: fullname: Huang – volume: 25 start-page: 1161 year: 2002 end-page: 1178 ident: CR6 article-title: On a theory of interacting continua with memory publication-title: J. Therm. Stress. doi: 10.1080/01495730290074586 contributor: fullname: Quintanilla – volume: 13 start-page: 55 year: 2008 end-page: 80 ident: CR8 article-title: On the theory of viscoelastic mixtures and stability publication-title: Math. Mech. Solids contributor: fullname: Nappa – volume: 18 start-page: 1381 year: 2005 end-page: 1388 ident: CR13 article-title: Exponential decay in mixtures with localized dissipative term publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2005.02.023 contributor: fullname: Quintanilla – ident: CR9 – volume: 30 start-page: 693 year: 2007 end-page: 714 ident: CR7 article-title: A theory of porous thermoviscoelastic mixtures publication-title: J. Therm. Stress. doi: 10.1080/01495730701212880 contributor: fullname: Quintanilla – volume: 32 start-page: 986 year: 2009 end-page: 1004 ident: CR3 article-title: Analyticity of semigroups associated with thermoviscoelastic mixtures of solids publication-title: J. Therm. Stress. doi: 10.1080/01495730903103028 contributor: fullname: Villagrán – volume: 284 start-page: 847 year: 1984 end-page: 857 ident: CR12 article-title: On the spectrum of -semigroups publication-title: Trans. Am. Math. Soc. doi: 10.2307/1999112 contributor: fullname: Prüss – volume: 24 start-page: 311 year: 2005 end-page: 324 ident: CR14 article-title: Existence and exponential decay in the linear theory of viscoelastic mixtures publication-title: Eur. J. Mech. A/Solids doi: 10.1016/j.euromechsol.2004.11.008 contributor: fullname: Quintanilla – volume: 46 start-page: 263 year: 1995 end-page: 277 ident: CR10 article-title: Some qualitative results for the linear theory of binary mixtures of thermoelastic solids publication-title: Collect. 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Alves – volume: 284 start-page: 847 year: 1984 ident: 443_CR12 publication-title: Trans. Am. Math. Soc. doi: 10.2307/1999112 contributor: fullname: J. Prüss – volume: 32 start-page: 986 year: 2009 ident: 443_CR3 publication-title: J. Therm. Stress. doi: 10.1080/01495730903103028 contributor: fullname: M.S. Alves – volume: 1 start-page: 43 year: 1985 ident: 443_CR5 publication-title: Ann. Differ. Equ. contributor: fullname: F.L. Huang – volume: 25 start-page: 1161 year: 2002 ident: 443_CR6 publication-title: J. Therm. Stress. doi: 10.1080/01495730290074586 contributor: fullname: D. Ieşan – volume: 30 start-page: 693 year: 2007 ident: 443_CR7 publication-title: J. Therm. Stress. doi: 10.1080/01495730701212880 contributor: fullname: D. Ieşan – volume: 13 start-page: 55 year: 2008 ident: 443_CR8 publication-title: Math. Mech. Solids doi: 10.1177/1081286506072351 contributor: fullname: D. Ieşan – volume: 46 start-page: 1659 year: 2009 ident: 443_CR1 publication-title: Int. J. Solids Struct. doi: 10.1016/j.ijsolstr.2008.12.005 contributor: fullname: M.S. Alves – volume: 236 start-page: 385 year: 1978 ident: 443_CR4 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1978-0461206-1 contributor: fullname: L.M. Gearhart – volume-title: Semigroups of Linear Operators and Applications to Partial Differential Equations year: 1983 ident: 443_CR11 doi: 10.1007/978-1-4612-5561-1 contributor: fullname: A. Pazy – volume: 18 start-page: 1381 year: 2005 ident: 443_CR13 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2005.02.023 contributor: fullname: R. Quintanilla |
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| Title | Stabilization of a system modeling temperature and porosity fields in a Kelvin–Voigt-type mixture |
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