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

*(English)*Zbl 1275.35132Summary: 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, 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 [the authors and N. Yamazaki, Math. Ann. 356, No. 1, 301–330 (2013; Zbl 1270.35008)], 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.

##### MSC:

35K87 | Unilateral problems for parabolic systems and systems of variational inequalities with parabolic operators |

35R06 | PDEs with measure |

35K67 | Singular parabolic equations |

35K51 | Initial-boundary value problems for second-order parabolic systems |

35K59 | Quasilinear parabolic equations |

##### Keywords:

unknown-dependent singular diffusion; unknown-dependent measure; smoothing effect; energy dissipation; Kobayashi-Warren-Carter model; gradient system##### Citations:

Zbl 1270.35008
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\textit{K. Shirakawa} and \textit{H. Watanabe}, Discrete Contin. Dyn. Syst., Ser. S 7, No. 1, 139--159 (2014; Zbl 1275.35132)

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