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Infinite-energy solutions for the Navier-Stokes equations in a strip revisited
Global existence of solutions for the thermoelastic Bresse system
1. | Department of Applied Mathematics, Donghua University, Shanghai 201620 |
2. | College of Information Science and Technology, Donghua University, Shanghai 201620 |
3. | College of Science, Shanghai Second Polytechnic University, Shanghai, 201209, China |
References:
[1] |
J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, 1859.
doi: 10.3934/krm.2011.4.901. |
[2] |
C. M. Dafermos, On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 247-271.
doi: 10.1016/j.anihpc.2013.04.006. |
[3] |
C. M. Dafermos, Asymptotic stability in viscoelascity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[4] |
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity, J. Differential Equations, 7 (1990), 554-569. |
[5] |
R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations arising in linear viscoelascity, SIAM J. Math. Anal., 21 (1990), 374-393.
doi: 10.1002/cpa.3160360506. |
[6] |
L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904.
doi: 10.3934/dcds.2004.10.543. |
[7] |
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442.
doi: 10.1016/j.jfa.2005.06.009. |
[8] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
doi: 10.1007/s10255-012-0175-1. |
[9] |
K. Liu and Z. Liu, On the type of $C_0$-semigroup associated with the abstract linear viscoelastic system, Z. angew. Math. Phys., 47 (1996), 1-15.
doi: 10.3934/cpaa.2012.11.973. |
[10] |
W. J. Liu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity, ESAIM: Control Optim. Calc. Var., 3 (1998), 23-48.
doi: 10.3934/dcds.2005.12.881. |
[11] |
W. J. Liu, The exponential stabilization of higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures Appl., 77 (1998), 355-386.
doi: 10.1016/j.anihpc.2006.03.014. |
[12] |
Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. angew. Math. Phys., 60 (2009), 54-69.
doi: 10.1007/s00033-009-0023-1. |
[13] |
Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math., LIV (1996), 21-31.
doi: 10.1016/j.na.2009.12.045. |
[14] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, 389, Chapman & Hall/CRC, Boca Raton, FL, 1999.
doi: 10.1515/form.2011.079. |
[15] |
Z. Ma, Exponential stability and global attractors for a thermoelastic Bresse system, Adv. Difference Equations, 1 (2010), 1-17.
doi: 10.1016/j.nonrwa.2011.07.055. |
[16] |
S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307.
doi: 10.1016/j.jde.2009.09.020. |
[17] |
J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.
doi: 10.3934/dcds.2009.25.575. |
[18] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelbery, Tokyo, 1983. |
[19] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. |
[20] |
Y. Qin and Z. Ma, Global existence of the higher-dimensional linear system of thermoviscoelasticity, Preprint.
doi: 10.1142/S0129167X12500279. |
[21] |
Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory, Advances and Applications, Vol. 184, Birkh\"auser, Basel-Boston-Berlin. |
[22] |
Y. Qin, Universal attractor in $H^4$ for the nonlinear one-dimensional compressible Navier -Stokes equations, J. Differential Equations, 207 (2004), 21-72.
doi: 10.1007/s002090100332. |
[23] |
Y. Qin and J. E. Muñoz Rivera, Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas, Proc. Roy. Soc. Edinburgh, 132(A) (2002), 685-709.
doi: 10.1155/2011/128614. |
[24] |
Y. Qin, S. Deng and B. W. Schulze, Uniform compact attractors for a nonlinear non-autonomous equation of viscoelastisity, J. Partial Differential Equations, 22 (2009), 152-192.
doi: 10.1186/1687-2770-2011-11. |
[25] |
R. Racke, Y. Shibata and S. Zheng, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 4 (1993), 751-763.
doi: 10.1002/mma.1510. |
[26] |
R. Racke and J. E. Muñoz Rivera, Midly dissipative nonlinear Tymoshenko systems-Global existence and exponential stability, J. Math. Appl. Anal., 276 (2002), 248-278. |
[27] |
Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity, Banach Center Publication, 27 (1992), 457-480. |
[28] |
M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelascity, Arch. Rational Mech. Anal., 76 (1981), 97-133. |
[29] |
S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 6 (1921), 744-746. |
[30] |
X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math., 5 (2003), 25-83. |
[31] |
S. Zheng, Nonlinear Evolution Equations, Pitman Monogr. Survey. Pure Appl. Math., Vol. 133, CRC Press, USA, 2004. |
[32] |
S. Zheng and Y. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas, Quart. Appl. Math., 3 (2001), 579-599. |
[33] |
S. Zheng and Y. Qin, Universal attractors for the Navier-Stokes equations of compressible and heat conductive fluids in bounded annular domains in $R^n$, Arch. Rational Mech. Anal., 160 (2001), 153-179. |
show all references
References:
[1] |
J. A. C. Bresse, Cours de Méchanique Appliquée, Mallet Bachelier, 1859.
doi: 10.3934/krm.2011.4.901. |
[2] |
C. M. Dafermos, On the existence and the asymptotic stability of solution to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 247-271.
doi: 10.1016/j.anihpc.2013.04.006. |
[3] |
C. M. Dafermos, Asymptotic stability in viscoelascity, Arch. Rational Mech. Anal., 37 (1970), 297-308. |
[4] |
C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelascity, J. Differential Equations, 7 (1990), 554-569. |
[5] |
R. H. Fabiano and K. Ito, Semigroup theory and numerical approximation for equations arising in linear viscoelascity, SIAM J. Math. Anal., 21 (1990), 374-393.
doi: 10.1002/cpa.3160360506. |
[6] |
L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA J. Appl. Math., 75 (2010), 881-904.
doi: 10.3934/dcds.2004.10.543. |
[7] |
S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442.
doi: 10.1016/j.jfa.2005.06.009. |
[8] |
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
doi: 10.1007/s10255-012-0175-1. |
[9] |
K. Liu and Z. Liu, On the type of $C_0$-semigroup associated with the abstract linear viscoelastic system, Z. angew. Math. Phys., 47 (1996), 1-15.
doi: 10.3934/cpaa.2012.11.973. |
[10] |
W. J. Liu, Partial exact controllability and exponential stability in higher dimensional linear thermoelascity, ESAIM: Control Optim. Calc. Var., 3 (1998), 23-48.
doi: 10.3934/dcds.2005.12.881. |
[11] |
W. J. Liu, The exponential stabilization of higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures Appl., 77 (1998), 355-386.
doi: 10.1016/j.anihpc.2006.03.014. |
[12] |
Z. Liu and B. Rao, Energy decay rate of the thermoelastic Bresse system, Z. angew. Math. Phys., 60 (2009), 54-69.
doi: 10.1007/s00033-009-0023-1. |
[13] |
Z. Liu and S. Zheng, On the exponential stability of linear viscoelasticity and thermoviscoelasticity, Quart. Appl. Math., LIV (1996), 21-31.
doi: 10.1016/j.na.2009.12.045. |
[14] |
Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, 389, Chapman & Hall/CRC, Boca Raton, FL, 1999.
doi: 10.1515/form.2011.079. |
[15] |
Z. Ma, Exponential stability and global attractors for a thermoelastic Bresse system, Adv. Difference Equations, 1 (2010), 1-17.
doi: 10.1016/j.nonrwa.2011.07.055. |
[16] |
S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III, J. Math. Anal. Appl., 348 (2008), 298-307.
doi: 10.1016/j.jde.2009.09.020. |
[17] |
J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.
doi: 10.3934/dcds.2009.25.575. |
[18] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, Berlin, Heidelbery, Tokyo, 1983. |
[19] |
J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. |
[20] |
Y. Qin and Z. Ma, Global existence of the higher-dimensional linear system of thermoviscoelasticity, Preprint.
doi: 10.1142/S0129167X12500279. |
[21] |
Y. Qin, Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, Operator Theory, Advances and Applications, Vol. 184, Birkh\"auser, Basel-Boston-Berlin. |
[22] |
Y. Qin, Universal attractor in $H^4$ for the nonlinear one-dimensional compressible Navier -Stokes equations, J. Differential Equations, 207 (2004), 21-72.
doi: 10.1007/s002090100332. |
[23] |
Y. Qin and J. E. Muñoz Rivera, Universal attractors for a nonlinear one-dimensional heat-conductive viscous real gas, Proc. Roy. Soc. Edinburgh, 132(A) (2002), 685-709.
doi: 10.1155/2011/128614. |
[24] |
Y. Qin, S. Deng and B. W. Schulze, Uniform compact attractors for a nonlinear non-autonomous equation of viscoelastisity, J. Partial Differential Equations, 22 (2009), 152-192.
doi: 10.1186/1687-2770-2011-11. |
[25] |
R. Racke, Y. Shibata and S. Zheng, Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity, Quart. Appl. Math., 4 (1993), 751-763.
doi: 10.1002/mma.1510. |
[26] |
R. Racke and J. E. Muñoz Rivera, Midly dissipative nonlinear Tymoshenko systems-Global existence and exponential stability, J. Math. Appl. Anal., 276 (2002), 248-278. |
[27] |
Y. Shibata, Neumann problem for one-dimensional nonlinear thermoelascity, Banach Center Publication, 27 (1992), 457-480. |
[28] |
M. Slemrod, Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelascity, Arch. Rational Mech. Anal., 76 (1981), 97-133. |
[29] |
S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, 6 (1921), 744-746. |
[30] |
X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Comm. Contemp. Math., 5 (2003), 25-83. |
[31] |
S. Zheng, Nonlinear Evolution Equations, Pitman Monogr. Survey. Pure Appl. Math., Vol. 133, CRC Press, USA, 2004. |
[32] |
S. Zheng and Y. Qin, Maximal attractor for the system of one-dimensional polytropic viscous ideal gas, Quart. Appl. Math., 3 (2001), 579-599. |
[33] |
S. Zheng and Y. Qin, Universal attractors for the Navier-Stokes equations of compressible and heat conductive fluids in bounded annular domains in $R^n$, Arch. Rational Mech. Anal., 160 (2001), 153-179. |
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