February  2014, 7(1): 139-159. doi: 10.3934/dcdss.2014.7.139

Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion

1. 

Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522

2. 

Department of General Education, Salesian Polytechnic, 4-6-8 Oyamagaoka, Machida-city, Tokyo, 194-0215, Japan

Received  February 2012 Revised  August 2012 Published  July 2013

In this paper, a coupled system of two parabolic initial-boundary value problems is considered. The system presented is a one-dimensional version of the Kobayashi-Warren-Carter model of grain boundary motion [15,16], that is derived as a gradient system of a governing free energy including a weighted total variation. Due to the weighted total variation, some nonstandard terms appear in the mathematical expressions of this system, and such nonstandard terms have made the mathematical treatments to be quite delicate. Recently, a certain definition of the solution have been provided in [21], together with the solvability result. The main objective in this paper is to verify that the system reproduces the foundational rules as a gradient system of parabolic PDEs, such as ``smoothing effect'' and ``energy-dissipation''. Consequently, the existence of a special solution, called ``energy-dissipative solution'', will be demonstrated in the Main Theorem of this paper.
Citation: Ken Shirakawa, Hiroshi Watanabe. Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 139-159. doi: 10.3934/dcdss.2014.7.139
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'', Oxford Science Publications, (2000). Google Scholar

[2]

F. Andreu, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'', Progress in Mathematics, 223 (2004). doi: 10.1007/978-3-0348-7928-6. Google Scholar

[3]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,'', Mathematical Surveys and Monographs, 165 (2010). Google Scholar

[4]

F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Local and nonlocal weighted $ p $-Laplacian evolution equations with Neumann boundary conditions,, Publ. Mat., 55 (2011), 27. doi: 10.5565/PUBLMAT_55111_03. Google Scholar

[5]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, Ann. Mat. Pura Appl. (4), 135 (1983), 293. doi: 10.1007/BF01781073. Google Scholar

[6]

H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces,'', Applications to PDEs and Optimization, (2001). Google Scholar

[7]

H. Brézis, "Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,'', North-Holland Mathematics Studies, 5 (1973). Google Scholar

[8]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'', Studies in Advanced Mathematics, (1992). Google Scholar

[9]

M. -H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems,, Jpn. J. Ind. Appl. Math., 27 (2010), 323. doi: 10.1007/s13160-010-0020-y. Google Scholar

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'', Monographs in Mathematics, 80 (1984). Google Scholar

[11]

A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion,, Appl. Math., 53 (2008), 433. doi: 10.1007/s10492-008-0035-8. Google Scholar

[12]

A. Ito, N. Kenmochi and N. Yamazaki, Global solvability of a model for grain boundary motion with constraint,, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 127. Google Scholar

[13]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. Google Scholar

[14]

R. Kobayashi and Y. Giga, Equations with singular diffusivity,, J. Statist. Phys., 95 (1999), 1187. doi: 10.1023/A:1004570921372. Google Scholar

[15]

R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundary,, Phys. D, 140 (2000), 141. doi: 10.1016/S0167-2789(00)00023-3. Google Scholar

[16]

R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity,, in, 14 (2000), 283. Google Scholar

[17]

J. S. Moll, The anisotropic total variation flow,, Math. Ann., 332 (2005), 177. doi: 10.1007/s00208-004-0624-0. Google Scholar

[18]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Advances in Math., 3 (1969), 510. doi: 10.1016/0001-8708(69)90009-7. Google Scholar

[19]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators: Cauchy problems,, J. Differential Equations, 46 (1982), 268. doi: 10.1016/0022-0396(82)90119-X. Google Scholar

[20]

K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients,, in, 71 (2006), 269. doi: 10.1142/9789812774293_0014. Google Scholar

[21]

K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability for one-dimensional phase field system associated with grain boundary motion,, Math. Ann., 356 (2013), 301. doi: 10.1007/s00208-012-0849-2. Google Scholar

[22]

J. Simon, Compact set in the space $ L^p(0, T; B) $,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'', Oxford Science Publications, (2000). Google Scholar

[2]

F. Andreu, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'', Progress in Mathematics, 223 (2004). doi: 10.1007/978-3-0348-7928-6. Google Scholar

[3]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,'', Mathematical Surveys and Monographs, 165 (2010). Google Scholar

[4]

F. Andreu, J. M. Mazón, J. D. Rossi and J. Toledo, Local and nonlocal weighted $ p $-Laplacian evolution equations with Neumann boundary conditions,, Publ. Mat., 55 (2011), 27. doi: 10.5565/PUBLMAT_55111_03. Google Scholar

[5]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness,, Ann. Mat. Pura Appl. (4), 135 (1983), 293. doi: 10.1007/BF01781073. Google Scholar

[6]

H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces,'', Applications to PDEs and Optimization, (2001). Google Scholar

[7]

H. Brézis, "Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,'', North-Holland Mathematics Studies, 5 (1973). Google Scholar

[8]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'', Studies in Advanced Mathematics, (1992). Google Scholar

[9]

M. -H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems,, Jpn. J. Ind. Appl. Math., 27 (2010), 323. doi: 10.1007/s13160-010-0020-y. Google Scholar

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'', Monographs in Mathematics, 80 (1984). Google Scholar

[11]

A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion,, Appl. Math., 53 (2008), 433. doi: 10.1007/s10492-008-0035-8. Google Scholar

[12]

A. Ito, N. Kenmochi and N. Yamazaki, Global solvability of a model for grain boundary motion with constraint,, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 127. Google Scholar

[13]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. Google Scholar

[14]

R. Kobayashi and Y. Giga, Equations with singular diffusivity,, J. Statist. Phys., 95 (1999), 1187. doi: 10.1023/A:1004570921372. Google Scholar

[15]

R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundary,, Phys. D, 140 (2000), 141. doi: 10.1016/S0167-2789(00)00023-3. Google Scholar

[16]

R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity,, in, 14 (2000), 283. Google Scholar

[17]

J. S. Moll, The anisotropic total variation flow,, Math. Ann., 332 (2005), 177. doi: 10.1007/s00208-004-0624-0. Google Scholar

[18]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Advances in Math., 3 (1969), 510. doi: 10.1016/0001-8708(69)90009-7. Google Scholar

[19]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators: Cauchy problems,, J. Differential Equations, 46 (1982), 268. doi: 10.1016/0022-0396(82)90119-X. Google Scholar

[20]

K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients,, in, 71 (2006), 269. doi: 10.1142/9789812774293_0014. Google Scholar

[21]

K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability for one-dimensional phase field system associated with grain boundary motion,, Math. Ann., 356 (2013), 301. doi: 10.1007/s00208-012-0849-2. Google Scholar

[22]

J. Simon, Compact set in the space $ L^p(0, T; B) $,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[1]

Emil Minchev, Mitsuharu Ôtani. $L^∞$-energy method for a parabolic system with convection and hysteresis effect. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1613-1632. doi: 10.3934/cpaa.2018077

[2]

Abbes Benaissa, Abderrahmane Kasmi. Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4361-4395. doi: 10.3934/dcdsb.2018168

[3]

Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911

[4]

Zhuangyi Liu, Ramón Quintanilla. Energy decay rate of a mixed type II and type III thermoelastic system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1433-1444. doi: 10.3934/dcdsb.2010.14.1433

[5]

Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799

[6]

Gianluca Mola. Recovering a large number of diffusion constants in a parabolic equation from energy measurements. Inverse Problems & Imaging, 2018, 12 (3) : 527-543. doi: 10.3934/ipi.2018023

[7]

Guoshan Zhang, Shiwei Wang, Yiming Wang, Wanquan Liu. LS-SVM approximate solution for affine nonlinear systems with partially unknown functions. Journal of Industrial & Management Optimization, 2014, 10 (2) : 621-636. doi: 10.3934/jimo.2014.10.621

[8]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883

[9]

Jerry L. Bona, Zoran Grujić, Henrik Kalisch. A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1121-1139. doi: 10.3934/dcds.2010.26.1121

[10]

Antti Lipponen, Aku Seppänen, Jari Hämäläinen, Jari P. Kaipio. Nonstationary inversion of convection-diffusion problems - recovery from unknown nonstationary velocity fields. Inverse Problems & Imaging, 2010, 4 (3) : 463-483. doi: 10.3934/ipi.2010.4.463

[11]

Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321

[12]

Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283

[13]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[14]

Lars Lamberg, Lauri Ylinen. Two-Dimensional tomography with unknown view angles. Inverse Problems & Imaging, 2007, 1 (4) : 623-642. doi: 10.3934/ipi.2007.1.623

[15]

Aki Pulkkinen, Ville Kolehmainen, Jari P. Kaipio, Benjamin T. Cox, Simon R. Arridge, Tanja Tarvainen. Approximate marginalization of unknown scattering in quantitative photoacoustic tomography. Inverse Problems & Imaging, 2014, 8 (3) : 811-829. doi: 10.3934/ipi.2014.8.811

[16]

Lorenzo Audibert, Alexandre Girard, Houssem Haddar. Identifying defects in an unknown background using differential measurements. Inverse Problems & Imaging, 2015, 9 (3) : 625-643. doi: 10.3934/ipi.2015.9.625

[17]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

[18]

Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691

[19]

Henri Schurz. Dissipation of mean energy of discretized linear oscillators under random perturbations. Conference Publications, 2005, 2005 (Special) : 778-783. doi: 10.3934/proc.2005.2005.778

[20]

Lisa C Flatley, Robert S MacKay, Michael Waterson. Optimal strategies for operating energy storage in an arbitrage or smoothing market. Journal of Dynamics & Games, 2016, 3 (4) : 371-398. doi: 10.3934/jdg.2016020

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]