Open Access
| Issue |
ESAIM: COCV
Volume 31, 2025
|
|
|---|---|---|
| Article Number | 95 | |
| Number of page(s) | 43 | |
| DOI | https://doi.org/10.1051/cocv/2025081 | |
| Published online | 28 November 2025 | |
- R. Kobayashi, J.A. Warren and W.C. Carter, A continuum model of grain boundaries. Phys. D 140 (2000) 141–150. MR 1752970 [Google Scholar]
- R. Kobayashi, J.A. Warren and W.C. Carter, Grain boundary model and singular diffusivity. Free Boundary Problems: Theory and Applications, II (Chiba, 1999). GAKUTO Internat. Ser. Math. Sci. Appl., vol. 14. Gakkōtosho, Tokyo (2000) 283-294. MR 1794359 [Google Scholar]
- N.C. Admal, J. Segurado and J. Marian, A three-dimensional misorientation axis- and inclination-dependent Kobayashi-Warren-Carter grain boundary model. J. Mech. Phys. Solids 128 (2019) 32–53. MR 3937357 [Google Scholar]
- H. Antil, S. Kubota, K. Shirakawa and N. Yamazaki, Optimal control problems governed by 1-D Kobayashi-Warren- Carter type systems. Math. Control Relat. Fields 11 (2021) 253–289. MR 4218112 [Google Scholar]
- H. Antil, S. Kubota, K. Shirakawa and N. Yamazaki, Constrained optimization problems governed by PDE models of grain boundary motions. Adv. Nonlinear Anal. 11 (2022) 1249–1286. MR 4395725 [Google Scholar]
- H. Antil, K. Shirakawa and N. Yamazaki, A class of parabolic systems associated with optimal controls of grain boundary motions. Adv. Math. Sci. Appl. 27 (2018) 299–336. MR 3888633 [Google Scholar]
- Y. Giga, A. Kubo, H. Kuroda, J. Okamoto, K. Sakakibara and M. Uesaka, Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system. Proc. Roy. Soc. Edinb. A: Math. First View (2024) 1-37. [Google Scholar]
- Y. Giga, J. Okamoto, K. Sakakibara and M. Uesaka, On a singular limit of the Kobayashi-Warren-Carter energy. Indiana Univ. Math. J. 73 (2024) 1453–1491. MR 4806756 [Google Scholar]
- Y. Giga, J. Okamoto and M. Uesaka, A finer singular limit of a single-well Modica-Mortola functional and its applications to the Kobayashi-Warren-Carter energy. Adv. Calc. Var. 16 (2023) 163–182. MR 4529388 [Google Scholar]
- A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion. Appl. Math. 53 (2008) 433–454. MR 2469586 [Google Scholar]
- A. Ito, N. Kenmochi and N. Yamazaki, Weak solutions of grain boundary motion model with singularity. Rend. Mat. Appl. 29 (2009) 51–63. MR 2548486 [Google Scholar]
- 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. MR 2836555 [Google Scholar]
- A. Ito, N. Yamazaki and N. Kenmochi, Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces. Discrete Contin. Dyn. Syst. 1 (1998) 327–349. Dynamical Systems and Differential Equations, Vol. I. Springfield, MO (1996). MR 1720614 [Google Scholar]
- N. Kenmochi and N. Yamazaki, Large-time behavior of solutions to a phase-field model of grain boundary motion with constraint. Current Advances in Nonlinear Analysis and Related Topics. GAKUTO International Series Math. Sci. Appl., vol. 32. Gakkotosho, Tokyo (2010) 389-403. MR 2668289 [Google Scholar]
- S. Kubota and K. Shirakawa, Periodic solutions to Kobayashi-Warren-Carter systems. Adv. Math. Sci. Appl. 32 (2023) 511–553. MR 4683700 [Google Scholar]
- D. Mizuno, Weak solution to KWC systems of pseudo-parabolic type, arXiv preprint arXiv:2407.18561 (2024) 15. [Google Scholar]
- S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi-Warren-Carter system. Calc. Var. Partial Diff. Equ. 51 (2014) 621–656. MR 3268865 [Google Scholar]
- S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the Kobayashi-Warren-Carter system. Nonlinearity 30 (2017) 2752–2784. MR 3670006 [Google Scholar]
- S. Moll, K. Shirakawa and H. Watanabe, Kobayashi-Warren-Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation. Nonlinear Anal. 217 (2022) Paper No. 112722. MR 4352617 [Google Scholar]
- S. Moll, K. Shirakawa and H. Watanabe, Existence of solutions to a phase-field model of 3D grain boundary motion governed by a regularized 1-harmonic type flow. J. Nonlinear Sci. 33 (2023) Paper No. 68. MR 4603338 [Google Scholar]
- S. Moll, K. Shirakawa and H. Watanabe, Large-time behavior for a phase-field system of 3D-grain boundary motion. SIAM J. Math. Anal. 56 (2024) 6885–6914. MR 4809351 [Google Scholar]
- R. Nakayashiki and K. Shirakawa, Kobayashi-Warren-Carter system of singular type under dynamic boundary condition. Discrete Contin. Dyn. Syst. Ser. S 16 (2023) 3746–3783. MR 4677072 [Google Scholar]
- K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion. Math. Ann. 356 (2013) 301–330. MR 3038131 [Google Scholar]
- K. Shirakawa, H. Watanabe and N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications. Adv. Math. Sci. Appl. 24 (2014) 353–400. MR 3362773 [Google Scholar]
- N. Yamazaki, Global attractors for non-autonomous phase-field systems of grain boundary motion with constraint. Adv. Math. Sci. Appl. 23 (2013) 267–296. MR 3155454 [Google Scholar]
- R.H.W. Hoppe and J.J. Winkle, A splitting scheme for the numerical solution of the KWC system. Numer. Math. Theory Methods Appl. 12 (2019) 661–680. MR 3951294 [Google Scholar]
- T. Aiki, D. Mizuno and K. Shirakawa, A class of initial-boundary value problems governed by pseudo-parabolic weighted total variation flows. Adv. Math. Sci. Appl. 32 (2023) 311–341. MR 4683693 [Google Scholar]
- H. Antil, D. Mizuno and K. Shirakawa, Well-posedness of a pseudo-parabolic KWC system in materials science. SIAM J. Math. Anal. 56 (2024) 6422–6445. MR 4797677 [Google Scholar]
- H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North- Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York (1973). North- Holland Mathematics Studies, No. 5. Notas de Matematica (50). MR 0348562 [Google Scholar]
- U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3 (1969) 510–585. MR 0298508 [CrossRef] [Google Scholar]
- H. Attouch, Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA (1984). MR 0773850 [Google Scholar]
- N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Educ. Chiba Univ. 30 (1981) 1–87. [Google Scholar]
- P. Colli, G. Gilardi, R. Nakayashiki and K. Shirakawa, A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions. Nonlinear Anal. 158 (2017) 32–59. MR 3661429 [Google Scholar]
- Y. Giga, Y. Kashima and N. Yamazaki, Local solvability of a constrained gradient system of total variation. Abstr. Appl. Anal. 8 (2004) 651–682. MR 2096945 [Google Scholar]
- T. Aiki, A. Kadoya and N. Sato, Optimal control problem for phase-field equations with nonlinear dynamic boundary conditions. Proceedings of the Third World Congress of Nonlinear Analysts, Part 5 (Catania, 2000), vol. 47 (5) 3183-3194. MR 1979214 [Google Scholar]
- S. Kubota, K. Shirakawa and N. Yamazaki, A class of approximate optimal control problems for 1-D phase-field system with singularity and its numerical algorithm. Adv. Math. Sci. Appl. 29 (2020) 495–561. [Google Scholar]
- T. Ohtsuka, K. Shirakawa and N. Yamazaki, Convergence of numerical algorithm for optimal control problem of Allen-Cahn type equation with constraint. Nonlinear Phenomena with Energy Dissipation. GAKUTO Int. Ser. Math. Sci. Appl., vol. 29. GakkBotosho, Tokyo (2008) 441-462. MR 2509574 [Google Scholar]
- Y. Giga and S. Tsubouchi, Continuity of derivatives of a convex solution to a perturbed one-Laplace equation by p-Laplacian. Arch. Ration. Mech. Anal. 244 (2022) 253–292. MR 4408168 [Google Scholar]
- S. Tsubouchi, Local Lipschitz bounds for solutions to certain singular elliptic equations involving the one-Laplacian. Calc. Var. Partial Diff. Equ. 60 (2021) Paper No. 33. MR 4201656 [Google Scholar]
- S. Tsubouchi, Continuous differentiability of a weak solution to very singular elliptic equations involving anisotropic diffusivity. Adv. Calc. Var. 17 (2024) 881–939. MR 4767356 [Google Scholar]
- S. Tsubouchi, A weak solution to a perturbed one-Laplace system by p-Laplacian is continuously differentiable. Math. Ann. 388 (2024) 1261–1322. MR 4700370 [Google Scholar]
- S. Tsubouchi, Gradient continuity for the parabolic (1, p)-Laplace equation under the subcritical case. Ann. Mat. Pura Appl. 204 (2025) 261–287. MR 4865084 [Google Scholar]
- E. Emmrich, Discrete versions of gronwall’s lemma and their application to the numerical analysis of parabolic problems, Tech. Report 637. Institute of Mathematics, Technische Universitat Berlin (1999). [Google Scholar]
- J. Simon, Compact sets in the space Lp (0, T ; B). Ann. Mat. Pura Appl. 146 (1987) 65–96. MR 0916688 [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.
