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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

Ken Shirakawa 1, and  Hiroshi Watanabe 2,

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.
Keywords: unknown-dependent measure, energy dissipation., energy-dissipative solution, unknown-dependent singular diffusion, System of parabolic type, smoothing effect.
Mathematics Subject Classification: Primary: 35K87, 35R06; Secondary: 35K6.
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, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[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.  Google Scholar

[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.  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), 27-66. 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), 293-318. 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, MPS-SIAM Series on Optimization, SIAM and MPS, 2001.  Google Scholar

[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.  Google Scholar

[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.  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), 323-345. doi: 10.1007/s13160-010-0020-y.  Google Scholar

[10]

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

[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.  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), 127-146.  Google Scholar

[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. Google Scholar

[14]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. 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), 141-150. 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 "Free Boundary Problems: Theory and Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 283-294.  Google Scholar

[17]

J. S. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. 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), 510-585. 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), 268-299. doi: 10.1016/0022-0396(82)90119-X.  Google Scholar

[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.  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), 301-330. 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), 65-96. 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, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[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.  Google Scholar

[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.  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), 27-66. 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), 293-318. 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, MPS-SIAM Series on Optimization, SIAM and MPS, 2001.  Google Scholar

[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.  Google Scholar

[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.  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), 323-345. doi: 10.1007/s13160-010-0020-y.  Google Scholar

[10]

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

[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.  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), 127-146.  Google Scholar

[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. Google Scholar

[14]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. 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), 141-150. 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 "Free Boundary Problems: Theory and Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000), 283-294.  Google Scholar

[17]

J. S. Moll, The anisotropic total variation flow, Math. Ann., 332 (2005), 177-218. 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), 510-585. 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), 268-299. doi: 10.1016/0022-0396(82)90119-X.  Google Scholar

[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.  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), 301-330. 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), 65-96. doi: 10.1007/BF01762360.  Google Scholar

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