Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements

Previous Article
On the mass-critical generalized KdV equation
DCDS Home
This Issue
Next Article
Spectral analysis for transition front solutions in Cahn-Hilliard systems

January 2012, 32(1): 167-190. doi: 10.3934/dcds.2012.32.167

Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements

Jie Jiang ^1, and Boling Guo ^2,

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

Received July 2010 Revised November 2010 Published September 2011

Abstract
Related Papers

This paper is devoted to the study of long-time behavior of the solutions to a one-dimensional full model for the first order phase transitions. Our system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\theta$, which is coupled with an evolution equation for the phase change parameter $f$ with a third-order nonlinearity $G_2'(f)$ in place of the customarily constant latent heat. The main novelty of this paper is that we perform an argument to establish Lemma 3.1 which enables us to obtain uniform estimates of the global solutions with respect to time. Asymptotic behavior of the solutions as time goes to infinity and the compactness of the orbit are obtained. Furthermore, we investigate the dynamics of the system and prove the existence of global attractors.

Keywords: global attractor., longtime behaviors, microscopic movements, First order phase transitions, existence and uniqueness.

Mathematics Subject Classification: Primary: 35B40, 80A22; Secondary: 35B41, 35K2.

Citation: Jie Jiang, Boling Guo. Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 167-190. doi: 10.3934/dcds.2012.32.167

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.

[1]	Emil Minchev. Existence and uniqueness of solutions of a system of nonlinear PDE for phase transitions with vector order parameter. Conference Publications, 2005, 2005 (Special) : 652-661. doi: 10.3934/proc.2005.2005.652
[2]	Giovanna Bonfanti, Fabio Luterotti. Global solution to a phase transition model with microscopic movements and accelerations in one space dimension. Communications on Pure and Applied Analysis, 2006, 5 (4) : 763-777. doi: 10.3934/cpaa.2006.5.763
[3]	Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2279-2290. doi: 10.3934/cpaa.2021069
[4]	Xiangjin Xu. Sub-harmonics of first order Hamiltonian systems and their asymptotic behaviors. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 643-654. doi: 10.3934/dcdsb.2003.3.643
[5]	Yirong Jiang, Nanjing Huang, Zhouchao Wei. Existence of a global attractor for fractional differential hemivariational inequalities. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1193-1212. doi: 10.3934/dcdsb.2019216
[6]	Honghu Liu. Phase transitions of a phase field model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883
[7]	Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[8]	Messoud A. Efendiev, Sergey Zelik, Hermann J. Eberl. Existence and longtime behavior of a biofilm model. Communications on Pure and Applied Analysis, 2009, 8 (2) : 509-531. doi: 10.3934/cpaa.2009.8.509
[9]	Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[10]	Ming Mei, Yau Shu Wong, Liping Liu. Phase transitions in a coupled viscoelastic system with periodic initial-boundary condition: (I) Existence and uniform boundedness. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 825-837. doi: 10.3934/dcdsb.2007.7.825
[11]	Shu-Yi Zhang. Existence of multidimensional non-isothermal phase transitions in a steady van der Waals flow. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2221-2239. doi: 10.3934/dcds.2013.33.2221
[12]	Sylvia Anicic. Existence theorem for a first-order Koiter nonlinear shell model. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1535-1545. doi: 10.3934/dcdss.2019106
[13]	A. Jiménez-Casas. Invariant regions and global existence for a phase field model. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 273-281. doi: 10.3934/dcdss.2008.1.273
[14]	Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419
[15]	Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545
[16]	Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139
[17]	Mauro Garavello, Benedetto Piccoli. Coupling of microscopic and phase transition models at boundary. Networks and Heterogeneous Media, 2013, 8 (3) : 649-661. doi: 10.3934/nhm.2013.8.649
[18]	Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151
[19]	Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179
[20]	Nobuyuki Kenmochi, Noriaki Yamazaki. Global attractor of the multivalued semigroup associated with a phase-field model of grain boundary motion with constraint. Conference Publications, 2011, 2011 (Special) : 824-833. doi: 10.3934/proc.2011.2011.824

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Jie Jiang Boling Guo
on Google Scholar
Jie Jiang Boling Guo

[Back to Top]