Helmholtz-Hodge Decompositions in the Nonlocal Framework
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Helmholtz-Hodge Decompositions in the Nonlocal Framework Well-Posedness Analysis and Applications Marta D’Elia1 · Cynthia Flores2 · Xingjie Li3 · Petronela Radu4 · Yue Yu5 Received: 25 September 2019 / Accepted: 29 April 2020 / © National Technology & Engineering Solutions of Sandia, LLC. Under the terms of Contract DE-NA0003525, there is a non-exclusive license for use of this work by or on behalf of the U.S. Government 2020
Abstract Nonlocal operators that have appeared in a variety of physical models satisfy identities and enjoy a range of properties similar to their classical counterparts. In this paper, we obtain Helmholtz-Hodge type decompositions for two-point vector fields in three components that have zero nonlocal curls, zero nonlocal divergence, and a third component which is (nonlocally) curl-free and divergence-free. The results obtained incorporate different nonlocal boundary conditions, thus being applicable in a variety of settings. Keywords Nonlocal operators · Nonlocal calculus · Helmholtz-Hodge decompositions Mathematics Subject Classification (2010) 35R09 · 45A05 · 45P05 · 35J05 · 74B99
1 Introduction and Motivation Important applications in diffusion, elasticity, fracture propagation, image processing, subsurface transport, and molecular dynamics have benefited from the introduction of nonlocal models. Phenomena, materials, and behaviors that are discontinuous in nature have been ideal candidates for the introduction of this framework which allows solutions with no smoothness, or even continuity properties. This advantage is counterbalanced by the facts that the theory of nonlocal calculus is still being developed and that the numerical solution of these problems can be prohibitively expensive. In [10], the authors introduce a nonlocal framework with divergence, gradient, and curl versions of nonlocal operators for which they identify duality relationships via L2 inner This project was developed during the week-long workshop Women in Mathematics of Materials held at University of Michigan in May 2018 with support from the Michigan Center for Applied and Interdisciplinary Mathematics, James Madison University, and the Association for Women in Mathematics. Marta D’Elia
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Journal of Peridynamics and Nonlocal Modeling
product topology. Integration by parts and nonlocal Poincar´e inequality are some of the partial differential equation (PDE)-based techniques that play an important role in the study of nonlocal operators. In this work, we continue to explore the structure of nonlocal operators and take additional steps toward building a rigorous framework for nonlocal calculus, by obtaining Helmholtz-Hodge type decompositions for nonlocal functions. In particular, we show that functions with two independent arguments (labeled two-point functions) can be decomposed into three components: a curl-free component, a divergence-free component, and a component which is both, curl and divergence free. All operators
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