General models of shape evolution, formulated as partial differential equations, are studied. The shape is regarded as a wave front, the propagation of which is driven either by local quantities (such as curvature) or by global quantities (such as the distance from an object fixed in space). Curvature-driven partial differential equations serve as models of collisional abrasion and have been previously investigated. Here, the first analysis of distance-driven partial differential equations (which are candidate models for frictional abrasion) is provided from the geophysical point of view. The analysis is focused on the evolution of geological shape descriptors: the evolution of axis ratios, roundness (isoperimetric ratio) and the number of static balance points is investigated under distance-driven flows. These flows were already proposed by Aristotle as models of particle shape evolution, and recent studies indicate that they may serve as models for frictional abrasion. The exact conditions under which Aristotle’s original claims are true are shown. For several geological shape descriptors, monotonic or quasiconcave time evolution is proven and compared with results from the literature on curvature-driven flows as models of collisional abrasion.
Equilibrium Convex surface Affinity Isoperimetric ratio
Mathematics Subject Classification
53A05 53Z05 52A38
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The authors gratefully acknowledge the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and support from OTKA Grant 119245. Furthermore, they express their gratitude to an anonymous referee for making Theorems 6 and 7 more general.
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