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Oceanic transform faults play an essential role in plate tectonics. Yet to date, there is no unifying explanation for the global trend in broad-scale transform fault topography, ranging from deep valleys to shallow topographic highs. Using three-dimensional numerical models, we find that spreading-rate dependent magmatism within the transform domain exerts a first-order control on the observed spectrum of transform fault depths. Low-rate magmatism results in deep transform valleys caused by transform-parallel tectonic stretching; intermediate-rate magmatism fully accommodates far-field stretching, but strike-slip motion induces across-transform tension, producing transform strength dependent shallow valleys; high-rate magmatism produces elevated transform zones due to local compression. Our models also address the observation that fracture zones are consistently shallower than their adjacent transform fault zones. These results suggest that plate motion change is not a necessary condition for reproducing oceanic transform topography and that oceanic transform faults are not simple conservative strike-slip plate boundaries.

 

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains.

 

Abstract Oceanic detachment faults represent an end-member form of seafloor creation, associated with relatively weak magmatism at slow-spreading mid-ocean ridges. We use 3-D numerical models to investigate the underlying mechanisms for why detachment faults predominantly form on the transform side (inside corner) of a ridge-transform intersection as opposed to the fracture zone side (outside corner). One hypothesis for this behavior is that the slipping, and hence weaker, transform fault allows for the detachment fault to form on the inside corner, and a stronger fracture zone prevents the detachment fault from forming on the outside corner. However, the results of our numerical models, which simulate different frictional strengths in the transform and fracture zone, do not support the first hypothesis. Instead, the model results, combined with evidence from rock physics experiments, suggest that shear-stress on transform fault generates excess lithospheric tension that promotes detachment faulting on the inside corner. 

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $ L^{p} $ vector fields defined on a domain $ \Omega $ that is either a bounded domain in $ \mathbb{R}^{d} $ or $ \mathbb{R}^{d} $ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $ L^{p} $ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals.

 

This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition.