Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system

Hiroshi Watanabe; Ken Shirakawa

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 381-389
  • ISSN: 0862-7959

Abstract

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A one-dimensional version of a gradient system, known as “Kobayashi-Warren-Carter system”, is considered. In view of the difficulty of the uniqueness, we here set our goal to ensure a “stability” which comes out in the approximation approaches to the solutions. Based on this, the Main Theorem concludes that there is an admissible range of approximation differences, and in the scope of this range, any approximation method leads to a uniform type of solutions having a certain common features. Further, this is specified by using the notion of “energy-dissipative solution”, proposed in a relevant previous work.

How to cite

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Watanabe, Hiroshi, and Shirakawa, Ken. "Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system." Mathematica Bohemica 139.2 (2014): 381-389. <http://eudml.org/doc/261891>.

@article{Watanabe2014,
abstract = {A one-dimensional version of a gradient system, known as “Kobayashi-Warren-Carter system”, is considered. In view of the difficulty of the uniqueness, we here set our goal to ensure a “stability” which comes out in the approximation approaches to the solutions. Based on this, the Main Theorem concludes that there is an admissible range of approximation differences, and in the scope of this range, any approximation method leads to a uniform type of solutions having a certain common features. Further, this is specified by using the notion of “energy-dissipative solution”, proposed in a relevant previous work.},
author = {Watanabe, Hiroshi, Shirakawa, Ken},
journal = {Mathematica Bohemica},
keywords = {approximation method; stability; energy-dissipative solution; approximation method; stability; energy-dissipative solution},
language = {eng},
number = {2},
pages = {381-389},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system},
url = {http://eudml.org/doc/261891},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Watanabe, Hiroshi
AU - Shirakawa, Ken
TI - Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 381
EP - 389
AB - A one-dimensional version of a gradient system, known as “Kobayashi-Warren-Carter system”, is considered. In view of the difficulty of the uniqueness, we here set our goal to ensure a “stability” which comes out in the approximation approaches to the solutions. Based on this, the Main Theorem concludes that there is an admissible range of approximation differences, and in the scope of this range, any approximation method leads to a uniform type of solutions having a certain common features. Further, this is specified by using the notion of “energy-dissipative solution”, proposed in a relevant previous work.
LA - eng
KW - approximation method; stability; energy-dissipative solution; approximation method; stability; energy-dissipative solution
UR - http://eudml.org/doc/261891
ER -

References

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  3. Ito, A., Kenmochi, N., Yamazaki, N., Weak solutions of grain boundary motion model with singularity, Rend. Mat. Appl., VII. Ser. 29 (2009), 51-63. (2009) Zbl1183.35159MR2548486
  4. Ito, A., Kenmochi, N., Yamazaki, N., Global solvability of a model for grain boundary motion with constraint, Discrete Contin. Dyn. Syst., Ser. S 5 (2012), 127-146. (2012) Zbl1246.35100MR2836555
  5. Kobayashi, R., Warren, J. A., Carter, W. C., 10.1016/S0167-2789(00)00023-3, Physica D 140 (2000), 141-150. (2000) MR1752970DOI10.1016/S0167-2789(00)00023-3
  6. Moll, S., Shirakawa, K., 10.1007/s00526-013-0689-2, Calc. Var. Partial Differ. Equ (2013), 1-36 DOI 10.1007/s00526-013-0689-2. (2013) MR3268865DOI10.1007/s00526-013-0689-2
  7. Mosco, U., 10.1016/0001-8708(69)90009-7, Adv. Math. 3 (1969), 510-585. (1969) Zbl0192.49101MR0298508DOI10.1016/0001-8708(69)90009-7
  8. Shirakawa, K., Watanabe, H., 10.3934/dcdss.2014.7.139, Discrete Contin. Dyn. Syst., Ser. S 7 (2014), 139-159. (2014) Zbl1275.35132MR3082861DOI10.3934/dcdss.2014.7.139
  9. Shirakawa, K., Watanabe, H., Yamazaki, N., 10.1007/s00208-012-0849-2, Math. Ann. 356 (2013), 301-330. (2013) Zbl1270.35008MR3038131DOI10.1007/s00208-012-0849-2
  10. Watanabe, H., Shirakawa, K., Qualitative properties of a one-dimensional phase-field system associated with grain boundary, GAKUTO Internat. Ser. Math. Sci. Appl. 36 (2013), 301-328. (2013) MR3203495

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