Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition

TL;DR

This paper develops a rigorous mathematical foundation for nonlocal vector calculus, establishing identities and a fractional Helmholtz decomposition that enhance understanding and modeling in various scientific fields.

Contribution

It provides a comprehensive analysis of nonlocal operators, proving identities, connecting variational frameworks, and deriving a fractional Helmholtz decomposition for smooth vector fields.

Findings

Proved nonlocal vector calculus identities.

Connected weighted and unweighted variational frameworks.

Derived a fractional Helmholtz decomposition for smooth vector fields.

Abstract

Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions.