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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2310.20507 |
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| _version_ | 1866909172204830720 |
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| author | Sato, Kotaro |
| author_facet | Sato, Kotaro |
| contents | In this paper, the global-in-time $ L^2 $-solvability of the initial-boundary value problem for differential inclusions of doubly-nonlinear type, which arises from fracture mechanics, is proved. This problem is not covered by general existence theories due to the degeneracy and singularity of a dissipation potential along with the nonlinearity of elliptic terms. The existence of solutions is proved based on a minimizing movement scheme, which also plays a crucial role for deriving qualitative and asymptotic properties of strong solutions. Moreover, the solutions to the initial-boundary value problem comply with three properties intrinsic to brittle fracture: complete irreversibility, unilateral equilibrium of an energy and an energy balance law, which cannot generally be realized in dissipative systems. Furthermore, long-time dynamics of strong solutions is also treated, i.e., stationary limits of the global-in-time solutions are characterized as a solution to the stationary problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20507 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasistatic evolution equations with irreversibility arising from fracture mechanics Sato, Kotaro Analysis of PDEs 35K86, 49J40, 74R10 In this paper, the global-in-time $ L^2 $-solvability of the initial-boundary value problem for differential inclusions of doubly-nonlinear type, which arises from fracture mechanics, is proved. This problem is not covered by general existence theories due to the degeneracy and singularity of a dissipation potential along with the nonlinearity of elliptic terms. The existence of solutions is proved based on a minimizing movement scheme, which also plays a crucial role for deriving qualitative and asymptotic properties of strong solutions. Moreover, the solutions to the initial-boundary value problem comply with three properties intrinsic to brittle fracture: complete irreversibility, unilateral equilibrium of an energy and an energy balance law, which cannot generally be realized in dissipative systems. Furthermore, long-time dynamics of strong solutions is also treated, i.e., stationary limits of the global-in-time solutions are characterized as a solution to the stationary problem. |
| title | Quasistatic evolution equations with irreversibility arising from fracture mechanics |
| topic | Analysis of PDEs 35K86, 49J40, 74R10 |
| url | https://arxiv.org/abs/2310.20507 |