# American Institute of Mathematical Sciences

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.

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##### 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.
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