skip to main content


The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 10:00 PM ET on Friday, December 8 until 2:00 AM ET on Saturday, December 9 due to maintenance. We apologize for the inconvenience.

Title: Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions
Abstract We focus on the existence and rigidity problems of the vectorial Peierls–Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a reduced non-local scalar Ginzburg–Landau equation with an anisotropic positive (if Poisson ratio belongs to (−1/2, 1/3)) singular kernel is derived on the slip plane. We first prove that minimizers of the PN energy for this reduced scalar problem exist. Starting from H 1/2 regularity, we prove that these minimizers are smooth 1D profiles only depending on the shear direction, monotonically and uniformly converge to two stable states at far fields in the direction of the Burgers vector. Then a De Giorgi-type conjecture of single-variable symmetry for both minimizers and layer solutions is established. As a direct corollary, minimizers and layer solutions are unique up to translations. The proof of this De Giorgi-type conjecture relies on a delicate spectral analysis which is especially powerful for nonlocal pseudo-differential operators with strong maximal principle. All these results hold in any dimension since we work on the domain periodic in the transverse directions of the slip plane. The physical interpretation of this rigidity result is that the equilibrium dislocation on the slip plane only admits shear displacements and is a strictly monotonic 1D profile provided exclusive dependence of the misfit potential on the shear displacement.  more » « less
Award ID(s):
2106988 1812573
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Page Range / eLocation ID:
7778 to 7828
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ABSTRACT The circular-crack model has been widely used in seismology to infer earthquake stress drop. A common assumption is that the background medium is isotropic, although many earthquakes occur in geologically anisotropic settings. In this article, we study the effect of anisotropy on stress drop for a circular crack model and present explicit formalism in both static and kinematic cases. In the static case, we obtain the relationship between stress drop and slip for a circular crack model in an arbitrarily anisotropic medium. Special attention is given to the transversely isotropic (TI) medium. The static formalism is useful in understanding stress drop, but not all quantities are observables. Therefore, we resort to the kinematic case, from which we can infer stress drop using recorded far-field body waves. In the kinematic case, we assume that the crack ruptures circularly and reaches the final displacement determined by the static solutions. The far-field waveforms show that the corner frequency will change with different anisotropic parameters. Finally, we calculate the stress drops for cracks in isotropic and anisotropic media using the far-field waveforms. We find that in an isotropic medium, only shear stress acting on the crack surface contributes to shear slip. However, in a TI medium, if the anisotropy symmetry axis is not perpendicular or parallel to the crack surface, a normal stress (normal to the crack surface) can produce a shear slip. In calculating stress drop for an earthquake in an anisotropic medium using far-field body waves, a large error may be introduced if we ignore the possible anisotropy in the inversion. For a TI medium with about 18% anisotropy, the misfit of inferred stress drop could be up to 41%. Considering the anisotropic information, we can further improve the accuracy of stress-drop inversion. 
    more » « less
  2. null (Ed.)
    For hyperbolic systems of conservation laws, uniqueness of solutions is still largely open. We aim to expand the theory of uniqueness for systems of conservation laws. One difficulty is that many systems have only one entropy. This contrasts with scalar conservation laws, where many entropies exist. It took until 1994 to show that one entropy is enough to ensure uniqueness of solutions for the scalar conservation laws (see [E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Z. 55(5) (1994) 116–129 (in Russian), Math. Notes 55(5) (1994) 517–525]. This single entropy result was proven again by De Lellis, Otto and Westdickenberg about 10 years later [Minimal entropy conditions for Burgers equation, Quart. Appl. Math. 62(4) (2004) 687–700]. These two proofs both rely on the special connection between Hamilton–Jacobi equations and scalar conservation laws in one space dimension. However, this special connection does not extend to systems. In this paper, we prove the single entropy result for scalar conservation laws without using Hamilton–Jacobi. Our proof lays out new techniques that are promising for showing uniqueness of solutions in the systems case. 
    more » « less
  3. Abstract

    In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator$$-\Delta +V(x)$$-Δ+V(x)are raised to the power$$\kappa $$κis never given by the one-bound state case when$$\kappa >\max (0,2-d/2)$$κ>max(0,2-d/2)in space dimension$$d\ge 1$$d1. When in addition$$\kappa \ge 1$$κ1we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

    more » « less
  4. Abstract

    We present a critical analysis of physics-informed neural operators (PINOs) to solve partial differential equations (PDEs) that are ubiquitous in the study and modeling of physics phenomena using carefully curated datasets. Further, we provide a benchmarking suite which can be used to evaluate PINOs in solving such problems. We first demonstrate that our methods reproduce the accuracy and performance of other neural operators published elsewhere in the literature to learn the 1D wave equation and the 1D Burgers equation. Thereafter, we apply our PINOs to learn new types of equations, including the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we show that our approach is also applicable to learn the physics of the 2D linear and nonlinear shallow water equations, which involve three coupled PDEs. We release our artificial intelligence surrogates and scientific software to produce initial data and boundary conditions to study a broad range of physically motivated scenarios. We provide thesource code, an interactivewebsiteto visualize the predictions of our PINOs, and a tutorial for their use at theData and Learning Hub for Science.

    more » « less
  5. Abstract

    We use surface wave measurements to reveal anisotropy as a function of depth within the Juan de Fuca and Gorda plate system. Using a two‐plane wave method, we measure phase velocity and azimuthal anisotropy of fundamental mode Rayleigh waves, solving for anisotropic shear velocity. These surface wave measurements are jointly inverted with constraints fromSKSsplitting studies using a Markov chain approach. We show that the two data sets are consistent and present inversions that offer new constraints on the vertical distribution of strain beneath the plates and the processes at spreading centers. Anisotropy of the Juan de Fuca plate interior is strongest (~2.4%) in the low‐velocity zone between ~40‐ to 90‐km depth, with ENE direction driven by relative shear between plate motion and mantle return flow from the Cascadia subduction zone. In disagreement withPnmeasurements, weak (~1.1%) lithospheric anisotropy in Juan de Fuca is highly oblique to the expected ridge‐perpendicular direction, perhaps connoting complex intralithospheric fabrics associated with melt or off‐axis downwelling. In the Gorda microplate, strong shallow anisotropy (~1.9%) is consistent withPninversions and aligned with spreading and may be enhanced by edge‐driven internal strain. Weak anisotropy with ambiguous orientation in the low‐velocity zone can be explained by Gorda's youth and modest motion relative to the Pacific. Deeper (≥90 km) fabric appears controlled by regional flow fields modulated by the Farallon slab edge: anisotropy is strong (~1.8%) beneath Gorda, but absent beneath the Juan de Fuca, which is in the strain shadow of the slab.

    more » « less