2023/10/31 by Kotaro SATO, Sato, Kotaro
Engineering · Mathematics · Physics and Astronomy · #35K86 #49J40 #74R10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2310.20507
openalex publication_date 2023/10/31 · openalex created_date 2023/11/02 · openalex updated_date 2026/07/28
In this paper, the global-in-time L2 -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.