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Nonlocal operators that have appeared in a variety of physical models satisfyidentities and enjoy a range of properties similar to their classicalcounterparts. In this paper we obtain Helmholtz-Hodge type decompositions fortwo-point vector fields in three components that have zero nonlocal curls, zerononlocal divergence, and a third component which is (nonlocally) curl-free anddivergence-free. The results obtained incorporate different nonlocal boundaryconditions, thus being applicable in a variety of settings.

Helmholtz-Hodge decompositions in the nonlocal framework. Well-posedness analysis and applications