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Abstract We study the existence and uniqueness of solutions to the vector field Peierls–Nabarro (PN) model for curved dislocations in a transversely isotropic medium. Under suitable assumptions for the misfit potential on the slip plane, we reduce the 3D PN model to a nonlocal scalar Ginzburg–Landau equation. For a particular range of elastic coefficients, the nonlocal scalar equation with explicit nonlocal positive kernel is derived. We prove that any stable steady solution has a one-dimensional profile. As a result, we obtain that solutions to the scalar equation, as well as the original 3D system, are characterized as a one-parameter family of straight dislocations. This paper generalizes results found previously for the full isotropic case to an anisotropic setting. 

We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript. 

Abstract Peridotite xenoliths entrained in magmas near the Alpine fault (New Zealand) provide the first direct evidence of deformation associated with the propagation of the Australian-Pacific plate boundary through the region at ca. 25–20 Ma. Two of 11 sampled xenolith localities contain fine-grained (40–150 μm) rocks, indicating that deformation in the upper mantle was focused in highly sheared zones. To constrain the nature and conditions of deformation, we combine a flow law with a model linking recrystallized fraction to strain. Temperatures calculated from this new approach (625–970 °C) indicate that the observed deformation occurred at depths of 25–50 km. Calculated shear strains were between 1 and 100, which, given known plate offset rates (10–20 mm/yr) and an estimated interval during which deformation likely occurred (<1.8 m.y.), translate to a total shear zone width in the range 0.2–32 km. This narrow width and the position of mylonite-bearing localities amid mylonite-free sites suggest that early plate boundary deformation was distributed across at least ∼60 km but localized in multiple fault strands. Such upper mantle deformation is best described by relatively rigid, plate-like domains separated by rapidly formed, narrow mylonite zones.