Recent development of nonlocal vector calculus and nonlocal calculus of variations provides a systematic mathematical framework for the analysis of nonlocal continuum models in the form of partial-integral equations. In this lecture, we discuss the localization of some nonlocal models and associated nonlocal function spaces in order to study connections with traditional local models given by partial differential equations and Sobolev spaces. In particular, we present some recent results on heterogeneous localization of nonlocal space including an extension of classical trace theorems to nonlocal spaces of functions with significantly weaker regularity. We also discuss their implications in nonlocal modeling of multiscale processes.

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