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Existence of solutions to the Kobayashi–Warren–Carter system

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Abstract

In this paper, a coupled system of two parabolic type initial-boundary value problems is considered. The system is known as the Kobayashi–Warren–Carter model of grain boundary motion in a polycrystal. Kobayashi–Warren–Carter model is derived as a gradient descent flow of an energy functional, which is called “free-energy”, with respect to two unknown variables and it involves a weighted-unknown dependent total variation term. The main goal of this paper is to obtain existence of solutions to this system. We solve the problem by means of a time-discretization of a relaxed system and a highly non-trivial passage to the limit. We point out that our time-discretization method is effective not only for the original Kobayashi–Warren–Carter system but also for its relaxed versions. Therefore, we provide a uniform approach for obtaining solutions to systems associated with this model.

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Acknowledgments

S. M. has been partially supported by the Spanish MEC project MTM2012-31103. K. S. is supported by Grant-in-Aid for Encouragement of Young Scientists (B) (No. 24740099) JSPS. The authors want to thank the anonymous referees for their excellent report and helpful comments.

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Authors and Affiliations

  1. Departament d’Anàlisi Matemàtica, Universitat de València, C/Dr. Moliner, 50, Burjassot, Spain

    Salvador Moll

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

    Ken Shirakawa

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  1. Salvador Moll
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  2. Ken Shirakawa
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Correspondence to Salvador Moll.

Additional information

Communicated by Y. Giga.

Dedicated to J. M. Mazón on occasion of his 60th birthday.

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Moll, S., Shirakawa, K. Existence of solutions to the Kobayashi–Warren–Carter system. Calc. Var. 51, 621–656 (2014). https://doi.org/10.1007/s00526-013-0689-2

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