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 and 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, The Clarendon Press, Oxford University Press, New York, 2000.

[2]

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

[3]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,'' Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, RI, 2010.

[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), 27-66. doi: 10.5565/PUBLMAT_55111_03.

[5]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4), 135 (1983), 293-318. doi: 10.1007/BF01781073.

[6]

H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces,'' Applications to PDEs and Optimization, MPS-SIAM Series on Optimization, SIAM and MPS, 2001.

[7]

H. Brézis, "Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,'' North-Holland Mathematics Studies, 5, Notas de Matemática (50), North-Holland Publishing and American Elsevier Publishing, 1973.

[8]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'' Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992.

[9]

M. -H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Jpn. J. Ind. Appl. Math., 27 (2010), 323-345. doi: 10.1007/s13160-010-0020-y.

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984.

[11]

A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433-454. doi: 10.1007/s10492-008-0035-8.

[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), 127-146.

[13]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Available from: http://ci.nii.ac.jp/naid/110004715232.

[14]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372.

[15]

R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundary, Phys. D, 140 (2000), 141-150. doi: 10.1016/S0167-2789(00)00023-3.

[16]

R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, in "Free Boundary Problems: Theory and Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 283-294.

[17]

J. S. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0.

[18]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.

[19]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators: Cauchy problems, J. Differential Equations, 46 (1982), 268-299. doi: 10.1016/0022-0396(82)90119-X.

[20]

K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions,'' Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ, (2006), 269-288. doi: 10.1142/9789812774293_0014.

[21]

K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability for one-dimensional phase field system associated with grain boundary motion, Math. Ann., 356 (2013), 301-330. doi: 10.1007/s00208-012-0849-2.

[22]

J. Simon, Compact set in the space $ L^p(0, T; B) $, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,'' Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2000.

[2]

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

[3]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,'' Mathematical Surveys and Monographs, 165, American Mathematical Society, Providence, RI, 2010.

[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), 27-66. doi: 10.5565/PUBLMAT_55111_03.

[5]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4), 135 (1983), 293-318. doi: 10.1007/BF01781073.

[6]

H. Attouch, G. Buttazzo and G. Michaille, "Variational Analysis in Sobolev and BV Spaces,'' Applications to PDEs and Optimization, MPS-SIAM Series on Optimization, SIAM and MPS, 2001.

[7]

H. Brézis, "Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,'' North-Holland Mathematics Studies, 5, Notas de Matemática (50), North-Holland Publishing and American Elsevier Publishing, 1973.

[8]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,'' Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992.

[9]

M. -H. Giga and Y. Giga, Very singular diffusion equations: second and fourth order problems, Jpn. J. Ind. Appl. Math., 27 (2010), 323-345. doi: 10.1007/s13160-010-0020-y.

[10]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984.

[11]

A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433-454. doi: 10.1007/s10492-008-0035-8.

[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), 127-146.

[13]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. Available from: http://ci.nii.ac.jp/naid/110004715232.

[14]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372.

[15]

R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundary, Phys. D, 140 (2000), 141-150. doi: 10.1016/S0167-2789(00)00023-3.

[16]

R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, in "Free Boundary Problems: Theory and Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 283-294.

[17]

J. S. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0.

[18]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585. doi: 10.1016/0001-8708(69)90009-7.

[19]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators: Cauchy problems, J. Differential Equations, 46 (1982), 268-299. doi: 10.1016/0022-0396(82)90119-X.

[20]

K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions,'' Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ, (2006), 269-288. doi: 10.1142/9789812774293_0014.

[21]

K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability for one-dimensional phase field system associated with grain boundary motion, Math. Ann., 356 (2013), 301-330. doi: 10.1007/s00208-012-0849-2.

[22]

J. Simon, Compact set in the space $ L^p(0, T; B) $, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360.

[1]

Angelo Favini, Gianluca Mola, Silvia Romanelli. Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1439-1454. doi: 10.3934/dcdss.2022017

[2]

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

[3]

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

[4]

Chong Lai, Lishan Liu, Rui Li. The optimal solution to a principal-agent problem with unknown agent ability. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2579-2605. doi: 10.3934/jimo.2020084

[5]

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

[6]

Chris Guiver, Nathan Poppelreiter, Richard Rebarber, Brigitte Tenhumberg, Stuart Townley. Dynamic observers for unknown populations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3279-3302. doi: 10.3934/dcdsb.2020232

[7]

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

[8]

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

[9]

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

[10]

Haixia Li. Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity. Evolution Equations and Control Theory, 2021, 10 (4) : 723-732. doi: 10.3934/eect.2020088

[11]

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

[12]

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

[13]

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

[14]

Chandan Pathak, Saswati Mukherjee, Santanu Kumar Ghosh, Sudhansu Khanra. A three echelon supply chain model with stochastic demand dependent on price, quality and energy reduction. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021098

[15]

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

[16]

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

[17]

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

[18]

Umakanta Mishra, Abu Hashan Md Mashud, Sankar Kumar Roy, Md Sharif Uddin. The effect of rebate value and selling price-dependent demand for a four-level production manufacturing system. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2021233

[19]

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

[20]

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

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]