Previous |  Up |  Next


approximation method; stability; energy-dissipative solution
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.
[1] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs Clarendon Press, Oxford (2000). MR 1857292 | Zbl 0957.49001
[2] Ito, A., Kenmochi, N., Yamazaki, N.: A phase-field model of grain boundary motion. Appl. Math. 53 (2008), 433-454. DOI 10.1007/s10492-008-0035-8 | MR 2469586 | Zbl 1199.35138
[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. MR 2548486 | Zbl 1183.35159
[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. MR 2836555 | Zbl 1246.35100
[5] Kobayashi, R., Warren, J. A., Carter, W. C.: A continuum model of grain boundary. Physica D 140 (2000), 141-150. DOI 10.1016/S0167-2789(00)00023-3 | MR 1752970
[6] Moll, S., Shirakawa, K.: Existence of solutions to the Kobayashi-Warren-Carter system. Calc. Var. Partial Differ. Equ (2013), 1-36 DOI 10.1007/s00526-013-0689-2. DOI 10.1007/s00526-013-0689-2 | MR 3268865
[7] Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 (1969), 510-585. DOI 10.1016/0001-8708(69)90009-7 | MR 0298508 | Zbl 0192.49101
[8] Shirakawa, K., Watanabe, H.: Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete Contin. Dyn. Syst., Ser. S 7 (2014), 139-159. DOI 10.3934/dcdss.2014.7.139 | MR 3082861 | Zbl 1275.35132
[9] Shirakawa, K., Watanabe, H., Yamazaki, N.: Solvability of one-dimensional phase field systems associated with grain boundary motion. Math. Ann. 356 (2013), 301-330. DOI 10.1007/s00208-012-0849-2 | MR 3038131 | Zbl 1270.35008
[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. MR 3203495
Partner of
EuDML logo