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Spectral analysis for transition front solutions in Cahn-Hilliard systems
Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements
1. | Institute of Applied Physics and Computational Mathematics, PO Box 8009, Beijing 100088, China |
2. | Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088 |
References:
[1] |
V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Phys. D, 236 (2007), 13-21.
doi: 10.1016/j.physd.2007.07.009. |
[2] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295. |
[3] |
G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24. |
[4] |
G. Bonfanti, M. Frémond and F. Luterotti, Local solutions to the full model of phase transitions with dissipation, Adv. Math. Sci. Appl., 11 (2001), 791-810. |
[5] |
M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. |
[6] |
E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490.
doi: 10.1016/j.camwa.2006.02.042. |
[7] |
E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.
doi: 10.1002/mma.1089. |
[8] |
M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Internat. J. Engrg. Sci., 44 (2006), 529-539.
doi: 10.1016/j.ijengsci.2006.02.006. |
[9] |
M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model, J. Math. Phys., 49 (2008), 102902.
doi: 10.1063/1.2992478. |
[10] |
P. Germain, "Cours de Méchanique des Milieux Continus. Tome I: Théorie Générale," Masson er Cie, Éditeurs, Paris 1973. |
[11] |
B. Guo and P. Zhu, Global existence of smooth solution to nonlinear thermoviscoelastic system with clamped boundary conditions in solid-like materials, Comm. Math. Phys., 203 (1999), 365-383.
doi: 10.1007/s002200050617. |
[12] |
J. Jiang and Y. Zhang, Counting the set of equilibria for a one-dimensional full model for phase transitions with microscopic movements, Q. Appl. Math., to appear. |
[13] |
Ph. Laurençot, G. Schimperna and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442.
doi: 10.1016/S0022-247X(02)00127-0. |
[14] |
F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350. |
[15] |
F. Luterotti and U. Stefanelli, Errata and addendum to: "Existence result for the one-dimensional full model of phase transitions", [Z. Anal. Anwendungen, 21 (2002), 335-350], Z. Anal. Anwendungen, 22 (2003), 239-240. |
[16] |
F. Luterotti, G. Schimperna and U. Stefanelli, Existence results for a phase transition model based on microscopic movements, Differential Equations: Inverse and Direct Problems, (2006), 245-263. |
[17] |
F. Luterotti, G. Schimperna and U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci., 11 (2001), 809-825.
doi: 10.1142/S0218202501001112. |
[18] |
R. Racke and S. Zheng, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Diff. Equ., 134 (1997), 46-67.
doi: 10.1006/jdeq.1996.3216. |
[19] |
E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transition in thermoviscoelastic materials, Applications of Mathematics, 53 (2008), 485-520. |
[20] |
C. Shang, Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions, Asymptot. Anal., 55 (2007), 229-251. |
[21] |
C. Shang, Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions, J. Math. Anal. Appl., 343 (2008), 1-21.
doi: 10.1016/j.jmaa.2008.01.043. |
[22] |
W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727. |
[23] |
W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34.
doi: 10.1016/S0362-546X(00)00246-7. |
[24] |
J. Sprekels and S. Zheng, Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys, Phys. D, 121 (1998), 252-262.
doi: 10.1016/S0167-2789(98)00167-5. |
[25] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, 1988. |
[26] |
H. Wu and S. Zheng, Global attractor for the 1-D thin film equation, Asympt. Anal., 51, 101-111. |
[27] |
S. Zheng, "Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems," Pitman Series Monographs and Surveys in Pure and Applied Mathematics, 76, 1995. |
show all references
References:
[1] |
V. Berti, M. Fabrizio and C. Giorgi, Well-posedness for solid-liquid phase transitions with a fourth-order nonlinearity, Phys. D, 236 (2007), 13-21.
doi: 10.1016/j.physd.2007.07.009. |
[2] |
E. Bonetti, P. Colli, M. Fabrizio and G. Gilardi, Existence and boundedness of solutions for a singular phase field system, J. Differential Equations, 246 (2009), 3260-3295. |
[3] |
G. Bonfanti, M. Frémond and F. Luterotti, Global solution to a nonlinear system for irreversible phase changes, Adv. Math. Sci. Appl., 10 (2000), 1-24. |
[4] |
G. Bonfanti, M. Frémond and F. Luterotti, Local solutions to the full model of phase transitions with dissipation, Adv. Math. Sci. Appl., 11 (2001), 791-810. |
[5] |
M. Frémond, "Non-Smooth Thermomechanics," Springer-Verlag, Berlin, 2002. |
[6] |
E. Feireisl, Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl., 53 (2007), 461-490.
doi: 10.1016/j.camwa.2006.02.042. |
[7] |
E. Feireisl, H. Petzeltová and E. Rocca, Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci., 32 (2009), 1345-1369.
doi: 10.1002/mma.1089. |
[8] |
M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions, Internat. J. Engrg. Sci., 44 (2006), 529-539.
doi: 10.1016/j.ijengsci.2006.02.006. |
[9] |
M. Fabrizio, Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model, J. Math. Phys., 49 (2008), 102902.
doi: 10.1063/1.2992478. |
[10] |
P. Germain, "Cours de Méchanique des Milieux Continus. Tome I: Théorie Générale," Masson er Cie, Éditeurs, Paris 1973. |
[11] |
B. Guo and P. Zhu, Global existence of smooth solution to nonlinear thermoviscoelastic system with clamped boundary conditions in solid-like materials, Comm. Math. Phys., 203 (1999), 365-383.
doi: 10.1007/s002200050617. |
[12] |
J. Jiang and Y. Zhang, Counting the set of equilibria for a one-dimensional full model for phase transitions with microscopic movements, Q. Appl. Math., to appear. |
[13] |
Ph. Laurençot, G. Schimperna and U. Stefanelli, Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions, J. Math. Anal. Appl., 271 (2002), 426-442.
doi: 10.1016/S0022-247X(02)00127-0. |
[14] |
F. Luterotti and U. Stefanelli, Existence result for the one-dimensional full model of phase transitions, Z. Anal. Anwendungen, 21 (2002), 335-350. |
[15] |
F. Luterotti and U. Stefanelli, Errata and addendum to: "Existence result for the one-dimensional full model of phase transitions", [Z. Anal. Anwendungen, 21 (2002), 335-350], Z. Anal. Anwendungen, 22 (2003), 239-240. |
[16] |
F. Luterotti, G. Schimperna and U. Stefanelli, Existence results for a phase transition model based on microscopic movements, Differential Equations: Inverse and Direct Problems, (2006), 245-263. |
[17] |
F. Luterotti, G. Schimperna and U. Stefanelli, Existence result for a nonlinear model related to irreversible phase changes, Math. Models Methods Appl. Sci., 11 (2001), 809-825.
doi: 10.1142/S0218202501001112. |
[18] |
R. Racke and S. Zheng, Global existence and asymptotic behavior in nonlinear thermoviscoelasticity, J. Diff. Equ., 134 (1997), 46-67.
doi: 10.1006/jdeq.1996.3216. |
[19] |
E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transition in thermoviscoelastic materials, Applications of Mathematics, 53 (2008), 485-520. |
[20] |
C. Shang, Asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with constant temperature boundary conditions, Asymptot. Anal., 55 (2007), 229-251. |
[21] |
C. Shang, Global attractor for the Ginzburg-Landau thermoviscoelastic systems with hinged boundary conditions, J. Math. Anal. Appl., 343 (2008), 1-21.
doi: 10.1016/j.jmaa.2008.01.043. |
[22] |
W. Shen and S. Zheng, On the coupled Cahn-Hilliard equations, Comm. Partial Differential Equations, 18 (1993), 701-727. |
[23] |
W. Shen and S. Zheng, Maximal attractor for the coupled Cahn-Hilliard equations, Nonlinear Anal., 49 (2002), 21-34.
doi: 10.1016/S0362-546X(00)00246-7. |
[24] |
J. Sprekels and S. Zheng, Maximal attractor for the system of a Landau-Ginzburg theory for structural phase transitions in shape memory alloys, Phys. D, 121 (1998), 252-262.
doi: 10.1016/S0167-2789(98)00167-5. |
[25] |
R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, 1988. |
[26] |
H. Wu and S. Zheng, Global attractor for the 1-D thin film equation, Asympt. Anal., 51, 101-111. |
[27] |
S. Zheng, "Nonlinear Parabolic Equations and Hyperbolic-Parabolic Coupled Systems," Pitman Series Monographs and Surveys in Pure and Applied Mathematics, 76, 1995. |
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