Approximation result for non-autonomous and non-local rock fracture models

  • E. F. Doungmo GoufoEmail author
  • A. Kubeka
Original Paper Area 1


The increasing number of earthquakes in the world and the unusual behavior observed in rock fracture processes across the globe have recently raised number of concerns. The way rocks undergo fractures in nature (sometime caused by human activities) is not always the idealized one as wished by geologists. This makes their mathematical modeling and analysis quite complex. In this paper a non-local and non-autonomous rock fracture model where sizes of clusters are discrete and rock fracture rate is time, position and size dependent is investigated by means of forward propagators. Such a model, where in addition, new particles are spatially randomly distributed according to some probabilistic law, has not yet been analyzed in the same work. Our analysis consists of approximating the solution of a discrete, non-local and non-autonomous rock fracture model by a sequence of solutions of cut-off problems of a similar form. Then, we make use of the classical argument of Dini (Lemma 4, Kato in Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften. Band, vol 132. Springer, New York, 1984) to show existence of strong solutions of the model in the class of Banach spaces (of functions with finite higher moments) \(\mathcal {X}_{r}:=L_{1}(\mathfrak {I},X_{r})\), with \(X_{r}=\{g:\mathbb {R}^{3}\times \mathbb {N}\ni (x,n)\rightarrow g(x,n), \Vert g\Vert _{r}:=\int _{\mathbb {R}^{3}}\sum _{n=1}^{\infty }n^{r}|g(x,n)|dx<\infty \}.\) Numerical implementations of the approximation scheme show the fracture evolution of the shear stress resulting from rock fracture as a function of slip. A comparison with the standard well-know results reveals, as expected, that the approximation scheme maintains a sharp weakening of the shear stress in the real fracture case compared to the idealized (homogenous) case. This great observation together with this approximation scheme may contribute to the proof of uniqueness of a strong solution to the discrete, non-local and non-autonomous rock fracture model which remains open and unsolved.


Evolution system Forward propagator Non-autonomous rock fracture Non-local model Discrete model Integro-differential equations 

Mathematics Subject Classification

46Txx 47Txx 


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© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaFloridaSouth Africa

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